MIRROR SYMMETRY FOR ABELIAN VARIETIES

V. GOLYSHEV, V. LUNTS, AND D. ORLOV

0. Introduction.

0.1. We define the relation of mirror symmetry on the class of pairs (complex abelian variety A + an element of the complexified ample cone of A ) and study its properties. More precisely, let A be a complex abelian variety, C_{A}^{a} ⊂ NS_{A}(ℝ) – the ample cone of A and put

C_{A}^{±} := NS_{A}(ℝ) ± iC_{A}^{a},
C_{A} := C_{A}^{+} ⊔ C_{A}^{−} ⊂ NS_{A}(ℂ).

If ω_{A} ∈ C_{A} we call (A, ω_{A}) an algebraic pair. In this work we define the notion of mirror symmetry for algebraic pairs . The definition is given below in 0.4.1. Our notion of mirror symmetry is defined purely in the language of algebraic geometry and algebraic groups. We explain in 0.9 below that our construction is compatible with the pictures of mirror symmetry of Kontsevich [Ko] (see also [PZ]) and Strominger, Yau, Zaslow [SYZ] (see also [Mo],[Gro1],[Gro2]).

0.2. Let us discuss some properties of this notion.

1. Not every algebraic pair has a mirror symmetric one (9.5.1), but for “general” abelian varieties A and any ω_{A} ∈ C_{A} the pair (A, ω_{A}) does have a symmetric one (9.6.3). Also for any abelian variety A there exists ω_{A} ∈ C_{A} such that the algebraic pair (A, ω_{A}) has a mirror symmetric one (9.6.1).

2. Suppose two algebraic pairs (B, ω_{B}) and (C, ω_{C}) are both mirror symmetric to the same algebraic pair (A, ω_{A}) then the derived categories of coherent sheaves on B and C are equivalent (as triangulated categories) (9.2.6). Thus in particular for any algebraic pair (A, ω_{A}) there are only finitely many isomorphism classes of abelian varieties which are mirror symmetric to (A, ω_{A}) (9.2.3).

Conversely, if some algebraic pairs (B, ω_{B}) and (A, ω_{A}) are mirror symmetric and an abelian variety C is such that the derived categories of coherent sheaves on C and B are equivalent, then there is ω_{C} such that the pairs (C, ω_{C}) and (A, ω_{A}) are mirror symmetric too.

3. Suppose that an algebraic pair (B, ω_{B}) is mirror symmetric to the algebraic pair (A, ω_{A}) . Starting with (A, ω_{A}) we cannot construct B , but we can construct the product B × B̂ , where B̂ is the dual abelian variety.

Given an abelian variety A consider its total cohomology H^{∗}(A, ℚ) . This space has a “horizontal” and “vertical” structures which are discussed below in 0.3. More precisely, H^{∗}(A, ℚ) is acted upon by two reductive algebraic ℚ –groups – the “horizontal” and the “vertical”, – and these groups commute. The “horizontal” group is the Hodge group or the special Mumford-Tate group (its action preserves each cohomology space H^{k}(A, ℚ) ). The vertical group is defined below in 0.3.

4. Suppose that a mirror symmetry between algebraic pairs (A, ω_{A}) and (B, ω_{B}) is given. Then there exists a natural isomorphism

β : H^{∗}(A, ℤ) ⟶^{∼} H^{∗}(B, ℤ)

such that β_{ℚ} “interchanges” the horizontal and vertical structures on H^{∗}(A, ℚ) and H^{∗}(B, ℚ) respectively (9.3.3). There exists a canonical choice (up to ± ) of such a β . This isomorphism will either preserve or switch the parity of the cohomology groups depending on the parity of the dimension of A and B. For example, if A and B are elliptic curves then β induces isomorphisms

β : H^{1}(A, ℤ) ⟶^{∼} H^{0}(B, ℤ) ⊕ H^{2}(B, ℤ),

the second author was supported by the NSF.

the third author was partly supported by the RFFI–96–01–00149.

β : H^{0}(A, ℤ) ⊕ H^{2}(A, ℤ) ⟶^{∼} H^{1}(B, ℤ).

The mirror symmetry works “best” for abelian varieties A which are obtained by a version of the G–construction of Gerritzen (section 10).

5. If the abelian variety A is obtained by the G– construction, then for any ω_{A} ∈ C_{A} the algebraic pair (A, ω_{A}) has a mirror symmetric pair (B, ω_{B}) , such that B is also obtained by the G–construction. Moreover, in this case we can describe the isomorphism β explicitly. Namely, β is given by an element in H^{∗}(A × B, ℤ) which is the Chern character of some natural line bundle on a certain real subtorus of A × B of dimension 3n (where n = dimA = dimB).

0.3. Let us describe the two ℚ –algebraic groups attached to any abelian variety A , which act on the total cohomology H^{∗}(A, ℚ) and commute with each other.

0.3.1. The first group is the classical Hodge group or the special Mumford-Tate group. Let us recall its definition.

Put Γ = Γ_{A} = H_{1}(A, ℤ), V = Γ_{ℝ} = H_{1}(A, ℝ). The complex structure on A induces the operator of complex structure J_{A} on V . Consider the homomorphism of algebraic ℝ -groups

h_{A} : S^{1} ⟶ GL(V), h_{A}(e^{iθ}) = cos(θ) · Id + sin(θ) · J_{A},

so that h_{A}(e^{iπ/2}) = J_{A} . Then Hdg_{A,ℚ} is defined as the smallest ℚ –algebraic subgroup G of GL(H_{1}(A, ℚ)) such that h_{A}(S^{1}) ⊂ G(ℝ) . By functoriality Hdg_{A,ℚ} acts on H^{1}(A, ℚ) and on the total cohomology H^{∗}(A, ℚ) = Λ^{·}H^{1}(A, ℚ). We call this group “horizontal” because it preserves each subspace H^{k}(A, ℚ).

0.3.2. Perhaps the main point of our work is the consideration of the second (“vertical”) algebraic group which we view as the mirror image of the Hodge group. This second group is the Zariski closure \overline{Spin(A)} in GL(H^{∗}(A, ℚ)) of a certain discrete subgroup Spin(A) ⊂ GL(H^{∗}(A, ℚ)).

Let D^{b}(A) be the bounded derived category of coherent sheaves on A . Consider its group Auteq(D^{b}(A)) of exact autoequivalences. We construct in 4.3 a natural representation

ρ_{A} : Auteq(D^{b}(A)) ⟶ GL(H^{∗}(A, ℤ))

and denote its image by Spin(A) . The elements of Spin(A) act by algebraic correspondences, hence commute with the Hodge group and preserve the Hodge verticals ⊕_{p−q=fixed}H^{p,q}(A, ℂ) . Hence the same is true for the algebraic group \overline{Spin(A)} . Let us discuss some properties of the groups Spin(A) and \overline{Spin(A)}.

0.3.3. Let Â be the dual abelian variety and ΓÂ be its first homology lattice that can be identified with Γ_{A}^{∗} := Hom(Γ_{A}, ℤ) . Consider the lattice

Λ := Γ_{A} ⊕ ΓÂ

with the canonical bilinear symmetric form

Q((a, b), (c, d)) = b(c) + d(a).

Let SO(Λ, Q) be the corresponding special orthogonal group. Let Spin(Λ, Q) be the corresponding spinorial group. We have the canonical exact sequence

0 ⟶ ℤ/2ℤ ⟶ Spin(Λ, Q) ⟶ SO(Λ, Q).

Note that the group Spin(Λ, Q) acts naturally on the total cohomology group Λ^{·}Γ_{A}^{∗} = H^{∗}(A, ℤ).

For any abelian variety A , there are two representations of Auteq(D^{b}(A)) . One, in H^{∗}(A, ℤ) , is the representation ρ_{A} considered above, in (0.3.2). The existence of the other, in Λ is implied by the explicit description of Auteq(D^{b}(A)) given in [24]. In fact, a commutative diagram exists:

Spin(A) ↪ Spin(Λ, Q)
↓  ↓
U(A) ↪ SO(Λ, Q)

where the horizontal arrows are compatible with the actions on H^{∗}(A, ℤ) and Λ respectively. The group U(A) was first introduced independently by S.Mukai [18] and by A.Polishchuk [27], so we call it the Mukai-Polishchuk group.

Let \overline{U(A)} ⊂ GL(Λ_{ℚ}) be the ℚ− algebraic subgroup which is the Zariski closure of U(A) in GL(Λ_{ℚ}). Thus the groups \overline{Spin(A)} and \overline{U(A)} are isogeneous.

0.3.4. The algebraic groups \overline{Spin(A)} and \overline{U(A)} are semisimple (7.2.1).

0.3.5. Consider the set C_{A} = C_{A}^{+} ⊔ C_{A}^{−} ⊂ NS_{A}(ℂ) as in 0.1. Both C_{A}^{+} and C_{A}^{−} can be considered as Siegel domains of the first kind: the Lie group \overline{U(A)}(ℝ) acts naturally on C_{A} preserving C_{A}^{+} and C_{A}^{−} . Moreover, C_{A}^{+} and C_{A}^{−} are single \overline{U(A)}(ℝ) –orbits and the stabilizer K_{ω} of a point ω ∈ C_{A} is a maximal compact subgroup of the semisimple Lie group \overline{U(A)}(ℝ). Further, for each ω ∈ C_{A} there is a natural choice of an element I_{ω} ∈ K_{ω} (14) such that I_{ω} defines a complex structure on the space Λ_{ℝ} (0.3.3).

0.4. Now we are ready to give the main definition of mirror symmetry for algebraic pairs.

Let (A, ω_{A}) be an algebraic pair, Λ_{A} = Γ_{A} ⊕ ΓÂ with the symmetric form Q_{A} as in 0.3.3. Note that the Hodge group Hdg_{A,ℚ} = Hdg_{A×Â,ℚ} is naturally a subgroup of SO(Λ_{A,ℚ}, Q_{A,ℚ}). Moreover, it commutes with \overline{U(A)}. Thus we obtain two commuting complex structures on Λ_{A,ℝ} : J_{A×Â} ∈ Hdg_{A,ℚ}(ℝ) = Hdg_{A×Â,ℚ}(ℝ) and I_{ω_{A}} ∈ \overline{U(A)}(ℝ).

0.4.1 Definition. We call algebraic pairs (A, ω_{A}) and (B, ω_{B}) mirror symmetric if there is given an isomorphism of lattices

α : Λ_{A} ⟶^{∼} Λ_{B},

which identifies the forms Q_{A} and Q_{B} and satisfies the following conditions:

α_{ℝ} · J_{A×Â} = I_{ω_{B}} · α_{ℝ},
α_{ℝ} · I_{ω_{A}} = J_{B×B̂} · α_{ℝ}.

0.4.2. Note that if we identify Λ_{A} and Λ_{B} by means of α then

Hdg_{A,ℚ} ⊆ \overline{U(B)}, Hdg_{B,ℚ} ⊆ \overline{U(A)}.

0.5. Actually we work in a more general context. Namely, we consider weak pairs (A, ω_{A}) , where A is a complex torus and ω_{A} = η_{1} +iη_{2} ∈ NS_{A}(ℂ) is such that η_{2} is nondegenerate. (Thus an algebraic pair is in particular a weak pair.) And we define the notion of mirror symmetry for weak pairs. However we prove that if two weak pairs are mirror symmetric and one of them is algebraic then so is the other.

0.6. For an abelian variety A the algebraic group \overline{Spin(A)} or rather its Lie algebra has a different description. It turns out to be isomorphic (as a Lie subalgebra of 𝔤𝔩(H^{∗}(A, ℚ))) to the Neron-Severi Lie algebra 𝔤_{NS}(A) defined in [LL].

0.6.1. Let us recall the definition of 𝔤_{NS}(X) for a smooth complex projective variety X . If κ ∈ H^{1,1}(X) ∩ H^{2}(X, ℚ) is an ample class, then cupping with it defines an operator e_{κ} in the total cohomology H^{∗}(X) = H^{∗}(X, ℂ) of degree 2 and the hard Lefschetz theorem asserts that for s = 0, 1, ..., n , e_{κ}^{s} maps H^{n−s}(X) isomorphically onto H^{n+s}(X) . As is well known, this is equivalent to the existence of a (unique) operator f_{κ} on H^{∗}(X) of degree −2 such that the commutator [e_{κ}, f_{κ}] is the operator h which on H^{k}(X) is multiplication by k − n . The elements e_{κ}, f_{κ}, h make up a Lie subalgebra 𝔤_{κ} of 𝔤𝔩(H^{∗}(X)) isomorphic to sl(2) . Define the Neron-Severi Lie algebra 𝔤_{NS}(X) as the Lie subalgebra of 𝔤𝔩(H^{∗}(X)) generated by 𝔤_{κ} ’s with κ an ample class. This Lie subalgebra is defined over ℚ and is evenly graded by the adjoint action by the semisimple element h. The Lie algebra 𝔤_{NS}(X) is semisimple.

0.6.2. Note that the equality

Lie\overline{Spin(A)} = 𝔤_{NS}(A) (∗)

implies one of the standard conjectures of Grothendieck for A . Namely, the algebraicity of the operator f_{κ} (0.5.1) that has been proved for abelian varieties by Kleiman [12].

0.6.3. Question. Let X be a smooth complex projective variety. Assume that the canonical sheaf K_{X} is trivial. Does the analogue of (*) hold for X ?

We believe that the answer is “yes”. Note that if K_{X} or K_{X}^{−1} is ample then the analogue of (*) cannot hold. Indeed, by a theorem in [BO] the group Auteq(D^{b}(X)) is then generated by Aut(X), the shift operator [1], and by operators of tensoring with a line bundle on X . Thus (if Aut(X) is trivial) Auteq(D^{b}(X)) is abelian, whereas the Lie algebra 𝔤_{NS}(X) is semisimple.

0.7. Our picture of the mirror symmetry provides an explanation of the phenomenon in Hodge theory, which was noticed long ago (see for example p.68 in [Gr]). Namely, consider a variation of Hodge structures which degenerates along a divisor. Then the logarithm of the monodromy operator has properties which are similar to the properties of a Lefschetz operator on the cohomology. But in our construction this logarithm of the monodromy is transformed by the mirror symmetry to a Lefschetz operator indeed.

0.8. A similar picture exists for hyperkahler manifolds and will be described in our next paper.

0.9. Our notion of mirror symmetry for abelian varieties is compatible with other (conjectural) pictures of mirror symmetry. We will explain the compatibility with the ideas of Kontsevich [Ko] on one hand and with the T –duality picture of Strominger, Yau and Zaslow [SYZ] on the other hand.

0.9.1. In [Ko] Kontsevich proposes the following thesis. If a symplectic manifold (V, ω) is mirror symmetric to an algebraic variety W , then the Fukaya category F (V, ω) of (V, ω) is equivalent to the derived category D^{b}(W) of coherent sheaves on W . This has been proved for elliptic curves in [PZ].

Assume that the algebraic pairs (A, ω_{A}) and (B, ω_{B}) are mirror symmetric (0.4.1). Let ω_{A} = η_{1} + iη_{2} ∈ C_{A} . Then η_{2} is a Kahler form on A and in particular (A, η_{2}) is a symplectic manifold. (The class η_{1} plays the role of a B-field). The Hodge group Hgd_{A,ℚ} acts on H_{1}(A) preserving the form η_{2} . Hence its arithmetic subgroup

Hdg_{A}(ℤ) := {g ∈ Hdg_{A,ℚ}| g(Λ_{A}) ⊂ Λ_{A}}

acts by symplectomorphisms of the symplectic manifold (A, η_{2}) . Thus

Hdg_{A}(ℤ) ⊂ AuteqF (A, η_{2}).

But our definition of mirror symmetry implies the natural inclusion of groups

Hdg_{A}(ℤ) ⊂ U(B)

(0.3.3, 0.4.2), where the group U(B) captures the essential part of AuteqD^{b}(B) . Thus we do not establish the equivalence of categories F (A, η_{2}) and D^{b}(B) as suggested by Kontsevich, but rather compare their groups of autoequivalences.

0.9.2. Let us show the compatibility with the mirror symmetry proposed in [SYZ] and more specifically in [Mo] (see also [Gro1],[Gro2]). Let algebraic pairs (A, ω_{A}) and (B, ω_{B}) be mirror symmetric. Assume that the pair (A, ω_{A}) is obtained by the G–construction (10.2.1) (this is so for “most” pairs). Then there exists a real subtorus T ⊂ A × B of dimension 3n ( n = dim_{ℂ}A = dim_{ℂ}B ) with the following property: The fibres of the projection of T onto B (resp. A ) are special Lagrangian subtori of A (resp. B ) of dimension n (10.3.2, 10.7). The existence of such a correspondence is part of the “geometric mirror symmetry” of Morrison (Definition 4 in [Mo]). Moreover, there is a natural complex line bundle L on T such that its Chern class c_{1}(L) considered as an element in H^{∗}(A × B) ≃ Hom(H^{∗}(A), H^{∗}(B)) is the isomorphism β mentioned in 0.2.

0.10. Let us briefly discuss the contents of each section.

Section 1 contains some background material on abelian varieties and complex tori. In section 2 we recall the Hodge group of a complex torus and its basic properties. Section 3 contains a review of the Clifford algebra (of a certain quadratic form which we define there) and its spinorial representation. This Clifford algebra plays the key role in the definition of the isomorphism β discussed in 0.2.

Section 4 is devoted to the bounded derived category D^{b}(X) of coherent sheaves on a smooth projective variety X . We define the group Auteq(D^{b}(X)) of autoequivalences of D^{b}(X) and construct a canonical representation

ρ_{X} : Auteq(D^{b}(X)) ⟶ GL(H^{∗}(X, ℚ)).

We then show that if X = A is an abelian variety then Imρ_{A} ⊂ GL(H^{∗}(A, ℤ)). The Mukai-Polishchuk group is introduced and then the main effort is applied to prove the commutativity of the diagram in 0.3.3 (see Proposition 4.3.7).

In section 5 we discuss various ℚ –algebraic subgroups of GL(H_{1}(A × Â, ℚ)) , which enter the picture of mirror symmetry. In particular we recall the algebraic group U_{A,ℚ} which was introduced by A. Polishchuk and present his result on the group of ℝ –points of U_{A,ℚ}. This group U_{A,ℚ} is closely related to the algebraic groups \overline{Spin(A)} and \overline{U(A)} . Namely, the semisimple group \overline{U(A)} is a subgroup of the reductive group U_{A,ℚ} which consists (up to isogeny) of all noncompact factors of U_{A,ℚ} (7.2.1).

In section 6 we recall the Neron-Severi Lie algebra 𝔤_{NS}(X) of a smooth projective variety X and describe 𝔤_{NS}(A) for an abelian variety A following [LL]. Then in section 7 we establish the relationship between this Lie algebra and the group Auteq(D^{b}(A)).

In section 8 we define a natural action of the Lie group U_{A,ℚ}(ℝ) on the Siegel domain C_{A}^{+} (and C_{A}^{−} ). We then show that this action is transitive and the stabilizer of a point is a maximal compact subgroup of U_{A,ℚ}(ℝ). To each ω ∈ C_{A} we associate a natural element I_{ω} ∈ U_{A,ℚ}(ℝ) , which is the main ingredient in the definition of mirror symmetry.

Finally, in section 9 we define the notion of mirror symmetry for complex tori and abelian varieties and study its properties.

Section 10 contains a variant of the G–construction and its relations with the mirror symmetry.

0.10. It is our pleasure to thank Joseph Bernstein, Alexei Bondal, Yuri Manin, Andrei Tyurin and Yuri Zarhin for useful discussions. We also were inspired by the pleasant atmosphere at the Moscow Steklov Institute, the Moscow Independent University, and cafe houses on the Old Arbat street.

1. Some preliminaries on complex tori and abelian varieties.

1.1. Let Γ ≅ ℤ^{2n} be a lattice, V = Γ ⊗ ℝ ≅ ℝ^{2n} and J ∈ End(V) , s.t. J^{2} = −1 . That is J is a complex structure on V . This way we obtain an n -dimensional complex torus

A = (V/Γ, J).

Note the canonical isomorphisms

Γ = H_{1}(A, ℤ), V = H_{1}(A, ℝ).

Sometimes we will add the subscript “A” to the symbols Γ, V, J.

Given another complex torus B = (V_{B}/Γ_{B}, J_{B}) , the group Hom(A, B) consists of homomorphisms f : Γ_{A} → Γ_{B} such that

J_{B} · f_{ℝ} = f_{ℝ} · J_{A} : V_{A} ⟶ V_{B}.

Thus the abelian group Hom(A, B) can be considered as a subgroup of Hom(Γ_{A}, Γ_{B}).

1.2. One has the dual torus Â defined as follows. Put Γ^{∗} = Hom_{ℤ}(Γ, ℤ) , V^{∗} = Γ^{∗}⊗ℝ = Hom(V, ℝ) and Ĵ: V^{∗} →^{∼} V^{∗} , s.t. (Ĵw)(v) = w(−Jv) for v ∈ V , w ∈ V^{∗} . Then by definition

Â = (V^{∗}/Γ^{∗}, Ĵ).

1.3. Denote by Pic_{A} the Picard group of A . Let Pic_{A}^{0} ⊂ Pic_{A} be the subgroup of line bundles with the trivial Chern class. It has a natural structure of a complex torus. Moreover, there exists a natural isomorphism of complex tori

Â ≅ Pic_{A}^{0}.

Every line bundle L on A defines a morphism φ_{L} : A → Â by the formula

φ_{L}(a) = T_{a}^{∗}L ⊗ L^{−1}.

(Here T_{a} : A → A is the translation by a .) We have φ_{L} = 0 iff L ∈ Pic_{A}^{0} and φ_{L} is an isogeny if L is ample. Thus the correspondence L ↦ φ_{L} identifies the Neron-Severi group NS_{A} := Pic_{A}/Pic_{A}^{0} as a subgroup in Hom(A, Â). Also NS_{A} is naturally a subgroup of H^{2}(A, ℤ) : to a line bundle L there corresponds its first Chern class that can be considered as a skew-symmetric bilinear form on Γ . Put c_{1}(L) = c . Then the morphism ϕ_{L} is given by the map

V_{A} → VÂ, v ↦ c(v, ·).

We will identify NS_{A} either as a subgroup of Hom(A, Â) or Hom(Γ_{A}, ΓÂ) or as a set of (integral) skew-symmetric forms c on Γ_{A} such that the extension c_{ℝ} on V_{A} is J -invariant. Put NS_{A}(ℝ) = NS_{A} ⊗ ℝ . We will denote by NS_{A}(ℝ)^{0} ⊂ NS_{A}(ℝ) the subset consisting of nondegenerate forms.

1.4. There is a unique line bundle P_{A} on the product A × Â such that

i) for any point α ∈ Â its restriction P_{α} on A × {α} represents an element of Pic_{A}^{0}, corresponding to α with respect to the fixed isomorphism Pic_{A}^{0} = Â;

ii) the restriction P |_{{0}×Â} is trivial.

Such P_{A} is called Poincare line bundle.

Given a morphism of complex tori f : A → B, the dual morphism f̂ : B̂ → Â, is defined. Pointwise, for α ∈ Â and β ∈ B̂ one has f̂(β) = α if and only if the line bundle f^{∗}P_{β} on A coincides with P_{α} .

The double dual torus ˆ Â is naturally identified with A by means of the Poincare line bundle on

A × Â and Â × ˆ Â . In other words,there exists a unique isomorphism κ_{A} : A →^{∼} ˆ Â such that under

the isomorphism 1 × κ_{A} : Â × A ⟶^{∼} Â × ˆ Â the Poincare line bundle pulls back to the Poincare line bundle: (1 × κ_{A})^{∗}PÂ ≅ P_{A}.

Thus is an antiinvolution on the category of complex tori, i.e. a contravariant functor whose square is isomorphic to the identity: κ : Id →^{∼}ˆ . It is known that φ̂_{L} = φ_{L} for any L ∈ Pic_{A} (see next remarks).

1.5 Remark. Let A be a complex torus. Consider the Poincare line bundles P_{A} and PÂ on A×Â

and Â × ˆ Â respectively. Choose a basis l_{1}, ..., l_{2n} of Γ_{A} and the dual basis l_{1}^{∗}, ..., l_{2n}^{∗} of ΓÂ = Γ_{A}^{∗}

(1.2). Let l_{1}^{∗∗}, ..., l_{2n}^{∗∗} be the basis of ΓˆÂ = Γ ∗ ; Â dual to l_{1}^{∗}, ..., l_{2n}^{∗} . The isomorphism κ_{A} : A ⟶^{∼} ˆ Â induces an identification of Γ_{A} and ΓˆÂ that takes l_{i} to −l_{i}^{∗∗} (!), due to the fact that the forms c_{1}(P_{A}) and c_{1}(PÂ) are skew-symmetric.

1.6 Remark. Let f : A → B be a morphism of complex tori and f̂: Â → B̂ be the dual morphism Choose basises l_{1}, ..., l_{2n} , m_{1}, ..., m_{2n} for Γ_{A} , Γ_{B} and the dual basises l_{1}^{∗}, ..., l_{2n}^{∗} , m_{1}^{∗}, ..., m_{2n}^{∗} for ΓÂ , ΓB̂ . Denote by F and F̂ the matrices of linear maps f : Γ_{A} → Γ_{B} and f̂: ΓB̂ → ΓÂ in these basises. The matrices F and F̂ are transposes of each other, i.e. F̂ = F^{t} .

Now, let f : A → Â . Using the isomorphism κ , we will consider f̂ as morphism form A to Â too. If, as above, F and F̂ are the matrices of linear maps f, f̂ : Γ_{A} → ΓÂ , then they are skew-transposes of each other, i.e. F̂ = −F^{t} . This immediately follows from the previous remark.

Thus, in particular, φ̂_{L} = φ_{L} for any L ∈ Pic_{A} .

1.7. A complex torus A = (V/Γ, J) is algebraic, i.e. an abelian variety, iff there exists c ∈ NS_{A} such that the symmetric bilinear form c_{ℝ}(J·, ·) on V is positive definite.

For the rest of this section 1 we assume that A is an abelian variety.

The endomorphism ring End(A) is a finitely generated ℤ –module, and we denote , as usual, End^{0}(A) := End(A)⊗ℚ . By the Poincare reducibility theorem the variety A is isogeneous to a product of simple abelian varieties. So End^{0}(A) is a semisimple finite ℚ –algebra. Put Γ_{A,ℚ} = Γ_{A} ⊗ ℚ .

Let L ∈ Pic_{A} be ample. Then the induced map

φ_{L} : Γ_{A,ℚ} → Γ_{Â,ℚ}

is an isomorphism. The map

′ : End^{0}(A) → End^{0}(A), a ↦ φ_{L}^{−1} · â · φ_{L}

is a positive (anti-)involution of End^{0}(A) called the Rosati involution.

1.8. Assume that A is simple, so that End^{0}(A) is a division algebra. Finite division ℚ –algebras with positive involutions were classified by Albert. Below is his classification which consists of four cases.

Let us introduce some notation. Put F = End^{0}(A) , K - the center of F . Fix a Rosati involution ′ of F . The involution that it induces on K is independent of ′ : for any embedding of K in ℂ it is given by complex conjugation. The fixed subfield K_{0} = K′ is totally real. Put [F : K] = d^{2}, [K_{0} : ℚ] = e_{0}.

I. The case of totally real multiplication. Here F = K = K_{0} .

II. The case of totally indefinite quaternion multiplication. Here K = K_{0} and F is a K_{0} -form of End(2) : there is an ℝ –algebra isomorphism F ⊗_{ℚ}ℝ ≅ End(2, ℝ)^{e_{0}} such that the involution corresponds to the transpose in every summand.

III. The case of totally definite quaternion multiplication. Here also K = K_{0} and F is a K_{0} –form of the quaternion algebra ℍ over K_{0} : there is an ℝ –algebra isomorphism F ⊗_{ℚ} ℝ ≅ ℍ^{e_{0}} such that the involution corresponds to the quaternion conjugation in every summand.

IV. The case of totally complex multiplication. The field K has no real embedding (so is a totally imaginary quadratic extension of K_{0} ) and F is a K –form of End(d) : there is an ℝ –algebra isomorphism F ⊗_{ℚ} ℝ ≅ End(d, ℂ)^{e_{0}} such that the involution corresponds to the conjugate transpose in every summand.

2. The Hodge group Hdg_{A,ℚ}.

2.1. Let A = (V/Γ, J) be a complex torus. Consider the homomorphism of algebraic ℝ -groups

h_{A} : S^{1} ⟶ GL(V), h_{A}(e^{iθ}) = cos(θ) · Id + sin(θ) · J,

so that h_{A}(e^{iπ/2}) = J . This defines a representation of S^{1} on the exterior algebra Λ^{·}H_{1}(A, ℝ) , hence on the total (co-)homology of A . Decomposing the representation of S^{1} on the k-th cohomology space with respect to the characters of S^{1}

S^{1} ⟶ ℂ^{∗}, e^{iθ} ↦ e^{i(q−p)θ}, p + q = k,

we get the usual Hodge decomposition

H^{k}(A, ℂ) = ⨁_{p+q=k} H^{p,q}(A, ℂ)

2.2 Definition. The Hodge group (or special Mumford-Tate group) Hdg_{A,ℚ} of A is defined as a minimal algebraic ℚ -subgroup G of GL(H_{1}(A, ℚ)) such that h_{A}(S^{1}) ⊂ G(ℝ) .

The group Hdg_{A,ℚ} acts naturally on the homology H_{k}(A, ℚ) and the cohomology H^{k}(A, ℚ). It is clear that Hdg_{A,ℚ} acts trivially on the Hodge spaces H^{p,p}(A, ℚ) := H^{p,p}(A, ℂ) ∩ H^{2p}(A, ℚ). Consider the dual torus Â . We have canonical identifications

Hdg_{A,ℚ} = Hdg_{Â,ℚ} = Hdg_{A×Â,ℚ}.

The representation of Hdg_{A,ℚ} on Γ_{Â,ℚ} is the contragradient of its representation on Γ_{A,ℚ} . Depending upon a context, we will view Hdg_{A,ℚ} as a subgroup of GL(Γ_{A,ℚ}) or GL(Γ_{Â,ℚ}) or GL(Γ_{A,ℚ} ⊕ Γ_{Â,ℚ}) .

2.3. The following facts about Hdg_{A,ℚ} are known (see [20], [4]).

2.3.1 Theorem. Assume that A is an abelian variety.

a) Hdg_{A,ℚ} is a connected reductive algebraic ℚ -group.

b) h_{A}(−1) is in the center of Hdg_{A,ℚ}.

c) The involution Ad(h_{A}(i)) on Hdg_{A,ℚ}(ℝ)^{0} is a Cartan involution, i.e. its fixed subgroup is a maximal compact subgroup K of Hdg_{A,ℚ}(ℝ)^{0}.

d) The symmetric space Hdg_{A,ℚ}(ℝ)^{0}/K is of hermitian type.

e) The reductive group Hdg_{A,ℚ} has no simple factors of exceptional type.

3. The Clifford algebra and the spinor representation.

3.1. Fix a lattice Γ ≅ ℤ^{2n} for some n ≥ 1 . (In the future applications Γ will be Γ_{A} for a complex torus A ). Put Λ = Γ ⊕ Γ^{∗} with the canonical symmetric bilinear form Q : Λ × Λ ⟶ ℤ defined by

Q((a_{1}, b_{1}), (a_{2}, b_{2})) = b_{1}(a_{2}) + b_{2}(a_{1}).

This form is even and unimodular. Moreover, Q ≃ U^{2n} , where U = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Consider the Clifford algebra Cl(Λ, Q) defined as follows

Cl(Λ, Q) := T (Λ)/ < x ⊗ x = \frac{1}{2}Q(x, x) >,

where T (Λ) is the tensor algebra of the lattice Λ .

3.2. Choose a basis l_{1}, ..., l_{2n} of Γ and the dual basis x_{1}, ..., x_{2n} of Γ^{∗} . Let I ⊂ Cl(Λ, Q) be the left ideal generated by the product l := l_{1} ⋯ l_{2n} . The left multiplication in Cl(Λ, Q) defines a homomorphism of algebras

σ : Cl(Λ, Q) ⟶ End_{ℤ}(I).

3.2.1 Proposition.

a) Cl(Λ, Q) is a free abelian group of rank 2^{4n} with the basis consisting of monomials

x_{j_{1}} ⋯ x_{j_{l}}l_{i_{1}} ⋯ l_{i_{k}}, 1 ≤ k, s ≤ 2n, i_{1} < ⋯ < i_{k}, j_{1} < ⋯ < j_{s}.

In particular, Λ is naturally a subgroup of Cl(Λ, Q) .

b) Cl(Λ, Q) is a ℤ_{2} -graded algebra, Cl(Λ, Q) = Cl^{+}(Λ, Q)⊕Cl^{−}(Λ, Q), where Cl^{+}(Λ, Q) (resp. Cl^{−}(Λ, Q) ) is the span of monomials with even (resp. odd) number of elements. Cl^{+}(Λ, Q) is a subalgebra of Cl(Λ, Q) .

c) The ideal I is independent of the choice of the basis l_{1}, ..., l_{2n} of H_{1}(A, ℤ).

d) The ideal I has a ℤ -basis consisting of the monomials

x_{j_{1}} ⋯ x_{j_{s}}l_{1} ⋯ l_{2n}, 1 ≤ s ≤ 2n, j_{1} < ⋯ < j_{s}.

Hence I ≅ Λ^{·}Γ^{∗} as abelian groups.

e) The homomorphism σ : Cl(Λ, Q) ⟶ End_{ℤ}(I) is an isomorphism, hence Cl(Λ, Q) ≅ M_{2^{2n}}(ℤ).

Proof. Assertions a), b) are standard and easy

c) is obvious since l_{i} , l_{j} anticommute for i ≠ j .

d) follows from a). Let us prove e). Since Cl(Λ, ℚ) and End_{Z}(I) are both free Z -modules of rank 2^{4n} it suffices to prove that σ is a surjection. For a subset p = {i_{1}, ..., i_{k}} ⊂ [1, 2n] , i_{1} ≤ ... ≤ i_{k} , denote by l_{p} (resp. x_{p} ) the ordered monomial l_{i_{1}} ⋯ l_{i_{k}} (resp. x_{i_{1}} ⋯ x_{i_{k}} ). Let p′ = [1, 2n] − p be the complementary set. One immediately checks that for subsets p, h, m ⊂ [1, 2n]

x_{h}lx_{p′}(x_{m}l) = \{ \begin{matrix}±x_{h}l & if p = m \\ 0 & otherwise \end{matrix} \} ,

where l = l_{1} ⋯ l_{2n} . This implies the surjectivity of σ in view of d) above (see also [1]). □

3.3 Corollary. Let Λ = M_{1} ⊕M_{2} be a decomposition such that M_{1}, M_{2} are (maximal) Q -isotropic sublattices. Let m_{1}, ..., m_{2n} be a basis of M_{1} , m = m_{1} ⋯ m_{2n} and I′ = Cl(Λ, Q)m be the corresponding left ideal. There exists a unique (up to ±1 ) isomorphism of left Cl(Λ, Q) - modules I and I′ .

Proof. The form Q identifies M_{2} with the dual M_{1}^{∗} of M_{1} . Moreover, the form Q′ on Λ = M_{1} ⊕ M_{1}^{∗} defined by

Q′((m_{1}, n_{1}), (m_{2}, n_{2})) = n_{1}(m_{2}) + n_{2}(m_{1})

coincides with Q , hence gives rise to the same Clifford algebra. Therefore we may apply the previous proposition (parts d),e)) to the ideal I′ instead of I . Since End_{Z}(I) = Cl(Λ, Q) = End_{Z}(I′) there exists a unique (up to a sign) isomorphism I ≅ I′ of left Cl(Λ, Q) -modules. This proves the corollary.

3.4. Let us recall the spinorial group of the Clifford algebra.

There exists a unique involution ′ on Cl(Λ, Q) that is the identity on Λ . In terms of the monomial basis as in Proposition 3.2.1a) it is defined by rule

(x_{j_{1}} ⋯ x_{j_{s}}l_{i_{1}} ⋯ l_{i_{k}})′ = l_{i_{k}} ⋯ l_{i_{l}}x_{j_{s}} ⋯ x_{j_{1}}

3.4.1 Definition. The spinorial group Spin(Λ, Q) is the multiplicative subgroup of such elements z in Cl^{+}(Λ, Q) that

1. zΛz^{−1} = Λ ,

2. N(z) := zz′ = 1 .

3.4.2 Proposition. For z ∈ Spin(Λ, Q) denote by r_{z} the conjugation by z restricted to Λ . Then r_{z} ∈ SO(Λ, Q) . The kernel of the map r = { 1,-1 }. In other words, there is an exact sequence

0 → Z/2ℤ → Spin(Λ, Q) → SO(Λ, Q).

Proof. See [PR].

3.5. Let us extend the scalars from ℤ to ℚ . That is we consider Γ_{ℚ} , Λ_{ℚ} , Q_{ℚ} , SO(Λ_{ℚ}, Q_{ℚ}) , etc. The conditions 1, 2 in the Definition 3.4.1 above define a ℚ -algebraic group Spin_{ℚ}(Λ_{ℚ}, Q_{ℚ}) . The map r as in Proposition 3.4.2 induces an exact sequence of ℚ -algebraic groups ([PR])

0 ⟶ ℤ/2ℤ ⟶ Spin_{ℚ}(Λ_{Q}, Q_{ℚ}) ⟶ SO(Λ_{ℚ}, Q_{ℚ}) ⟶ 1.

3.5.1. Since Spin_{ℚ}(Λ_{ℚ}, Q_{ℚ}) is a subgroup of the multiplicative group Cl^{+}(Λ_{ℚ}, Q_{ℚ})^{∗} , it acts on the space I_{ℚ} (3.2). This is called the spinorial representation of the group Spin_{ℚ}(Λ_{ℚ}, Q_{ℚ}) . It is a direct sum of two (nonequivalent) irreducible semispinorial representations I_{ℚ} = I_{ℚ}^{ev} ⊕ I_{ℚ}^{odd} , where I_{ℚ}^{ev} (resp. I_{ℚ}^{odd} ) is the direct sum of monomials x_{j_{1}} ⋯ x_{j_{s}}l_{1} ⋯ l_{2n} with s even (resp. odd) (3.2.1 d)). We have a similar Spin_{ℚ}(Λ_{ℚ}, Q_{ℚ}) – decomposition I′_{ℚ} = I_{ℚ}^{′ev} ⊕ I_{ℚ}^{′odd} (3.3). The isomorphism of Cl(Λ, Q) -modules β : I →^{∼} I′ induces an isomorphism of Spin_{ℚ}(Λ_{ℚ}, Q_{ℚ}) –modules β_{ℚ} : I_{ℚ} →^{∼} I′_{ℚ} . Thus

β_{ℚ} : I_{ℚ}^{ev} →^{∼} I_{ℚ}^{′ev}, I_{ℚ}^{ev} →^{∼} I_{ℚ}^{′odd}

or

β_{ℚ} : I_{ℚ}^{ev} →^{∼} I_{ℚ}^{′odd}, I_{ℚ}^{ev} →^{∼} I_{ℚ}^{′ev}.

3.5.2 Corollary. The isomorphism of Cl(Λ, Q) - modules β : I ⟶^{∼} I′ (3.3) satisfies

β(I^{ev}) = I^{′ev}, β(I^{odd}) = I^{′odd}

or

β(I^{ev}) = I^{′odd}, β(I^{odd}) = I^{′ev}

3.5.3 Definition. We call β even (resp. odd) if β(I^{ev}) = I^{′ev} (resp. β(I^{ev} = I^{′odd}) ).

3.5.4 Proposition. The isomorphism β is even (resp. odd) if dim_{ℚ}(Γ_{ℚ} ∩ M_{1ℚ}) is even (resp. odd).

Proof. Let d = dim_{ℚ}(Γ_{ℚ} ∩ M_{1ℚ}). Consider the obvious ℚ -versions of 3.2.1 and 3.3. We may assume that l_{1} = m_{1}, ..., l_{d} = m_{d} . Put m′ := m_{d+1} ⋯ m_{2n} . Then the right multiplication R_{m′} by m′ induces an isomorphism R_{m′} : I_{ℚ} ⟶^{∼} I′_{ℚ} of left Cl(Λ_{ℚ}, Q_{ℚ}) –modules. Hence R_{m′} is a scalar multiple of β_{ℚ} . But R_{m′}(l) = l_{d+1} ⋯ l_{2n}m′ . Hence the parity of β is equal to the parity of d . This proves the proposition. □

3.6 Remark. In the notations of 3.2 the choice of a basis x_{1}, ...x_{2n} of Γ induces an isomorphism of abelian groups

I ≅ Λ^{·}Γ^{∗}

(see Prop. 3.2.1 d)). A different choice of a basis of Γ may change this isomorphism by −1 . Thus we get a canonical left Cl(Λ, Q) -module structure on Λ^{·}Γ^{∗} . Similarly in the notation of Corollary 3.3 the group Λ^{·}M_{2} is canonically a left Cl(Λ, Q) -module. Moreover, this corollary asserts that there exists a unique (up to ±1 ) isomorphism of Cl(Λ, Q) -modules

Λ^{·}Γ^{∗} ≅ Λ^{·}M_{2}.

3.6.1. More generally, we can introduce the following equivalence relation ∼ on the set of maximal Q_{ℚ} -isotropic subspaces of Λ_{ℚ} . Namely, let L ⊂ Λ_{ℚ} be a maximal isotropic subspace. Then the ℚ -versions of 3.2.1 and 3.6 provide a canonical Cl(Λ_{ℚ}, Q_{ℚ}) -module structure on the exterior algebra Λ^{·}L^{∗} of the dual space L^{∗} . Given two maximal isotropic subspaces L_{1} , L_{2} we get a unique (up to a scalar) isomorphism β_{ℚ} : Λ^{·}L_{1}^{∗} ⟶^{∼} Λ^{·}L_{2}^{∗} of Cl(Λ_{ℚ}, Q_{ℚ}) -modules (by the ℚ -version of 3.3). We say that L_{1} ∼ L_{2} if dim(L_{1} ∩ L_{2}) is even. Then by 3.5.4 L_{1} ∼ L_{2} iff β_{ℚ}(Λ^{ev}L_{1}^{∗}) = Λ^{ev}L_{2}^{∗} . Thus ∼ is indeed an equivalence relation, which divides the set of maximal isotropic subspaces into two equivalence classes.

3.7. We will be interested in the following situation. Let A be a complex torus. Put Γ = Γ_{A} , Γ^{∗} = ΓÂ , Λ_{A} = Γ_{A} ⊕ ΓÂ with the symmetric bilinear form Q_{A} on Λ_{A} as in 3.1. Then by Remark 3.4 above Λ^{·}ΓÂ is naturally a Cl(Λ_{A}, Q_{A}) -module. Note the canonical isomorphisms ΓÂ = H_{1}(Â, ℤ) = H^{1}(A, ℤ) . Thus the total cohomology of A Λ^{·}H^{1}(A, ℤ) = H^{∗}(A, ℤ) is naturally a Cl(Λ_{A}, Q_{A}) -module.

Assume that B is another complex torus and there exists an isomorphism

α : Λ_{A} ⟶^{∼} Λ_{B}

which identifies the forms Q_{A} and Q_{B} . Let us identify

Λ_{A} = Λ = Λ_{B}, Q_{A} = Q = Q_{B}
Cl(Λ_{A}, Q_{A}) = Cl(Λ, Q) = Cl(Λ_{B}, Q_{B})

by means of α . Then by Remark 3.6 there exists a unique (up to ±1 ) isomorphism of Cl(Λ, Q) – modules

β : H^{∗}(A, ℤ) ⟶^{∼} H^{∗}(B, ℤ).

By Corollary 3.5.2 this isomorphism either preserves even and odd cohomology groups or interchanges them. By Proposition 3.5.4 the parity of β (3.5.3) is equal to the parity of the dimension dim_{ℚ}(Γ_{A,ℚ} ∩ Γ_{B,ℚ}) .

4. Derived categories and their groups of autoequivalences.

4.1 Categories of coherent sheaves and functors between them. 4.1.1. Let X be an algebraic variety over an arbitrary field k . By O_{X} denote the structure sheaf on X . Let coh(X) (resp. Qcoh(X) ) be the category of (quasi)–coherent sheaves on X . Recall that a quasicoherent (coherent) sheaf is a sheaf of O_{X} -modules, which locally on X has a (finite) presentation by free O_{X} -modules.

It is more convenient to work with derived categories instead of abelian categories. Let us denote by D^{b}(X) the bounded derived category of coh(X) .

Any derived category D has a structure of a triangulated category. This means that there are fixed a) a translation functor [1] : D ⟶ D that is an additive autoequivalence,

b) a class of distinguished (or exact) triangles:

X ⟶^{u} Y ⟶^{v} Z ⟶^{w} X[1]

that satisfies a certain set of axioms (see [29]). An additive functor F : D ⟶ D′ between two triangulated categories is called exact if it commutes

with the translation functors, and it takes every distinguished triangle in D to a distinguished triangle in D′ .

A natural example of an exact functor is related to a map of algebraic varieties. Every such map f : X → Y induces the functor of direct image f_{∗} : Qcoh(X) → Qcoh(Y) . Since a category Qcoh(X) has enough injectives there exists the right derived functor Rf_{∗} : D^{b}(Qcoh(X)) ⟶ D^{b}(Qcoh(Y)) . This gives us an example of exact functor between derived categories. If the map f is proper then for any i the functor R^{i}f_{∗} takes a coherent sheaf to a coherent one. It is known that D^{b}(coh(X)) is equivalent to the full subcategory D^{b}(Qcoh(X))_{coh} ⊂ D^{b}(Qcoh(X)) objects of which have coherent cohomologies. Hence any proper map f induces the exact functor Rf_{∗} : D^{b}(coh(X)) ⟶ D^{b}(coh(Y)) . Further, the functor f_{∗} has a left adjoint functor f^{∗} : coh(Y) ⟶ coh(X) . If the map f has finite Tor–dimension, then there exists a left derived functor Lf^{∗} : D^{b}(coh(Y)) ⟶ D^{b}(coh(X)) , which is left adjoint for Rf_{∗} . This is another example of an exact functor. (see [9], [2] Ex.I,II,III).

4.1.2. From now on we will assume that all varieties are smooth and projective and will denote D^{b}(coh(X)) by D^{b}(X) . In this case every map f : X → Y induces the functors Rf_{∗} : D^{b}(X) ⟶ D^{b}(Y) and Lf^{∗} : D^{b}(Y) ⟶ D^{b}(X) . Also each complex F ∈ D^{b}(X) induces an exact functor

⊗^{L} F : D^{b}(X) ⟶ D^{b}(X) . Using these functors one can introduce a larger class of exact functors.

Let X and Y be smooth projective varieties of dimension n and m respectively. Consider the projections

X ⟵^{p} X × Y ⟶^{q} Y

Every object E ∈ D^{b}(X × Y) defines an exact functor Φ_{E} : D^{b}(X) ⟶ D^{b}(Y) by the following formula:

Φ_{E}(·) := Rq_{∗}(E ⊗^{L} p^{∗}(·)).  (1)

Obviously, the same object defines another functor Ψ_{E} : D^{b}(Y) ⟶ D^{b}(X) by the similar formula

Ψ_{E}(·) := Rp_{∗}(E ⊗^{L} q^{∗}(·)).

The functor Φ_{E} has left and right adjoint functors Φ_{E}^{∗} and Φ_{E}^{!} respectively, defined by the rule:

\begin{aligned}Φ_{E}^{∗} & ≅ & Ψ_{E^{∨}⊗q^{∗}K_{Y} [m]} \\ Φ_{E}^{!} & ≅ & Ψ_{E^{∨}⊗p^{∗}K_{X}[n]} \end{aligned}  (2)

Here K_{X} and K_{Y} are the canonical sheaves on X and Y respectively, and E^{∨} is notation for R\underline{Hom}(E, O_{X×Y}) .

4.1.2.1 Example. Let f : X → Y be a morphism of varieties. Denote by Γ (f) a subvariety of X × Y that is the graph of f . It is clear that the functor Rf_{∗} : D^{b}(X) ⟶ D^{b}(Y) is isomorphic to Φ_{O_{Γ (f)}} , where O_{Γ (f)} is the structure sheaf of the graph Γ (f) . In the same way the functor Lf^{∗} : D^{b}(Y) ⟶ D^{b}(X) is isomorphic to Ψ_{O_{Γ (f)}} .

4.1.2.2 Example. Let δ : Δ ↪ X × X be the embedding of the diagonal. First, since the Δ is the graph of the identity morphism the identity functor from the derived category D^{b}(X) to itself is

represented by the structure sheaf O_{Δ} . Moreover, for any object F ∈ D^{b}(X) the functor ⊗^{L} F : D^{b}(X) ⟶ D^{b}(X) is isomorphic to Φ_{δ_{∗}F}

4.1.2.3. Thus there is a reasonably large class of exact functors between derived categories of smooth projective varieties that consists of functors having the form Φ_{E} for some complex E on the product. This class is closed under composition of functors. Actually, let X, Y, Z be three (smooth projective) varieties and let

Φ_{I} : D^{b}(X) ⟶ D^{b}(Y), Φ_{J} : D^{b}(Y) ⟶ D^{b}(Z)

be two functors, where I and J are objects of D^{b}(X × Y) and D^{b}(Y × Z) respectively. Denote by p_{XY} , p_{Y Z}, p_{XZ} the projections of X × Y × Z on the corresponding pair of factors.

4.1.2.4 Lemma.[17] In above notations, the composition Φ_{J} ∘ Φ_{I} is isomorphic to Φ_{K} , where K ∈ D^{b}(X × Z) is given by the formula:

K ≅ Rp_{XZ∗}(p_{Y Z}^{∗}(J) ⊗ p_{XY}^{∗} (I))

Presumably, the class of exact functors described above embraces all exact functors between bounded derived categories of coherent sheaves on smooth projective varieties. We do not know if it is true or not. However it is definitely true for exact equivalences.

4.1.2.5 Theorem.([23]) Let X and Y be smooth projective varieties. Suppose F : D^{b}(X) ⟶^{∼} D^{b}(Y) is an exact equivalence. Then there exists a unique (up to an isomorphism) object E ∈ D^{b}(X × Y) such that the functors F and Φ_{E} are isomorphic.

4.1.2.6 Definition. The group of isomorphism classes of exact equivalences F : D^{b}(X) ⟶^{∼} D^{b}(X) is called the group of autoequivalences of D^{b}(X) and is denoted by Auteq(D^{b}(X)) .

4.1.3. Assume for simplicity that k = ℂ. Analogously to derived categories and functors between them one can consider cohomologies and maps between them. The latter is much simpler.

For every element ξ ∈ H^{∗}(X × Y, ℚ) let us define linear maps

v_{ξ} : H^{∗}(X, ℚ) ⟶ H^{∗}(Y, ℚ), w_{ξ} : H^{∗}(Y, ℚ) ⟶ H^{∗}(X, ℚ)

by the formula :

v_{ξ}(·) = q_{∗}(ξ ∪ p^{∗}(·)),  w_{ξ}(·) = p_{∗}(ξ ∪ q^{∗}(·)).  (3)

It follows from the Künneth formula that any linear map between cohomologies have this form for some ξ .

The composition formula for cohomological correspondence is well-known and is similar to the composition of functors. Let as above X, Y, Z be three varieties and let ξ and η be elements of H^{∗}(X ×Y) and H^{∗}(Y × Z) respectively.

4.1.3.1 Lemma. The composition v_{η} ∘ v_{ξ} is isomorphic to v_{ζ} , where ζ ∈ H^{∗}(X × Z) is given by the formula:

ζ = p_{XZ∗}(p_{Y Z}^{∗}(η) ∪ p_{XY}^{∗} (ξ)).

4.1.4. There is a natural correspondence that attaches to a functor Φ_{E} : D^{b}(X) ⟶ D^{b}(Y) a linear map of vector spaces ϕ_{E} : H^{∗}(X, ℚ) ⟶ H^{∗}(Y, ℚ) . To describe it note that each exact functor F : D^{b}(X) ⟶ D^{b}(Y) induces a homomorphism \overline{F} : K(X) ⟶ K(Y) , where K(X) and K(Y) are the Grothendieck groups of the categories D^{b}(X) and D^{b}(Y) . On the other side, there is a map

ch : K(X) ⟶ H^{∗}(X, ℚ)

that is called the Chern character. The map ch is a ring homomorphism, if to recall that K(X) and H^{∗}(X, ℚ) are endowed with multiplications induced by ⊗ –product and ∪ –product respectively.

4.1.4.1 Definition. For every object E ∈ D^{b}(X × Y) put

\begin{matrix}ϕ_{E}(·) := v_{ch(E)∪p^{∗}(td_{X})}(·) = q_{∗}(ch(E) ∪ p^{∗}(td_{X}) ∪ p^{∗}(·)), \\ ψ_{E}(·) := w_{ch(E)∪q^{∗}(td_{Y} )}(·) = p_{∗}(ch(E) ∪ q^{∗}(td_{Y}) ∪ q^{∗}(·)), \end{matrix}  (4)

where td_{X} and td_{Y} are the Todd classes of X and Y .

4.1.4.2 Lemma. For given E ∈ D^{b}(X × Y) the following diagram

K(X) ⟶^{\overline{Φ}_{E}} K(Y)
ch↓  ↓ch
H^{∗}(X, ℚ) ⟶^{ϕ_{E}} H^{∗}(Y, ℚ)

is commutative.

4.1.4.3 Lemma. Assume that a functor Φ_{K} : D^{b}(X) ⟶ D^{b}(Z) is isomorphic to Φ_{J} ∘ Φ_{I} for some

Φ_{I} : D^{b}(X) ⟶ D^{b}(Y), Φ_{J} : D^{b}(Y) ⟶ D^{b}(Z).

Then ϕ_{K} = ϕ_{J} ∘ ϕ_{I} . Consequently, the correspondence Φ_{E} ↦ ϕ_{E} induces a representation

ρ_{X} : Auteq(D^{b}(X)) ⟶ GL(H^{∗}(X, ℚ)).

Both lemmas immediately follows from the Riemann-Roch-Grothendieck theorem.

4.1.4.4 Example. Let f : X → Y be a morphism. The functors Lf^{∗} ≅ Ψ_{O_{Γ (f)}} and Rf_{∗} ≅ Φ_{O_{Γ (f)}} give the maps ψ_{O_{Γ (f)}} and ϕ_{O_{Γ (f)}} between cohomologies. Also the morphism f induces the map: f^{∗} : H^{i}(Y, ℚ) ⟶ H^{i}(X, ℚ). It can be checked that

ψ_{O_{Γ (f)}} = f^{∗}.

On the other side, there is a map of homologies: f_{∗} : H_{i}(X, ℚ) ⟶ H_{i}(Y, ℚ) . Using the Poincare isomorphism H^{2n−i}(X, ℚ) ≅ H_{i}(X, ℚ) , the map f_{∗} can be considered as a map between cohomologies: f_{∗} : H^{2n−i}(X, ℚ) ⟶ H^{2m−i}(Y, ℚ). In this case ϕ_{O_{Γ (f)}} does not coincide with f_{∗} , but they are connected by the following relation:

ϕ_{O_{Γ (f)}}(x) ∪ td_{Y} = f_{∗}(x ∪ td_{X})

In particular, if td_{X} = td_{Y} = 1 , then these maps coincide.

4.1.4.5 Example. Let as in Example 4.1.2.2 δ : Δ ↪ X × X be an embedding of the diagonal. For

any F ∈ D^{b}(X) a functor ⊗^{L} F is isomorphic to Φ_{δ_{∗}F} . Since ch is a ring homomorphism Lemma 4.1.4.2 implies that

ϕ_{δ_{∗}F}(·) = (·) ∪ ch(F).

4.1.5. For any variety X the group Auteq(D^{b}(X)) contains a subgroup G(X) = (ℤ × Pic(X)) ⋊ Aut(X) that is the semi-direct product of the normal subroup (ℤ × Pic(X)) and Aut(X) naturally acting on the former. Under the inclusion G(X) ⊂ Auteq(D^{b}(X)) the generator of ℤ goes to the translation functor [1] , a line bundle L ∈ Pic(X) maps to the functor ⊗L = Φ_{δ_{∗}L} and an automorphism f : X → X induces the autoequivalence Rf_{∗} ≅ Φ_{O_{Γ (f)}} .

4.1.5.1 Lemma. In the above notation ρ_{X}([1]) = −Id_{H^{∗}(X,ℚ)} and ρ_{X}(⊗L) = ∪ch(L) .

The proof follows from the Example 4.1.4.5 above and the equality ch(E[1]) = −ch(E) . 4.1.6. Now let us show that any equivalence Φ_{E} : D^{b}(X) ⟶^{∼} D^{b}(Y) defines a functor between derived categories of coherent sheaves on X × X and Y × Y , which will be called adjoined. Consider two functors

Φ_{E} : D^{b}(X)⟶D^{b}(Y), Ψ_{F} : D^{b}(Y)⟶D^{b}(X),

where E and F are objects of D^{b}(X × Y) . By F ⊠ E denote the object p_{13}^{∗}(F) ⊗ p_{24}^{∗}(E) from D^{b}(X × X × Y × Y) , which is called the external tensor product of E and F . It defines the functor

Φ_{F⊠E} : D^{b}(X × X) ⟶ D^{b}(Y × Y).

Let us take an object G ∈ D^{b}(X × X) and denote Φ_{F⊠E}(G) ∈ D^{b}(Y × Y) by H for short. These two objects define the functors

Φ_{G} : D^{b}(X) ⟶ D^{b}(X), Φ_{H} : D^{b}(Y) ⟶ D^{b}(Y).

4.1.6.1 Lemma. In above notation, there is an isomorphism of functors Φ_{H} ≅ Φ_{E}Φ_{G}Ψ_{F} . This follows from Lemma 4.1.2.4. □

4.1.6.2 Lemma. If Φ_{E} and Ψ_{F} are equivalences, then the functor Φ_{F⊠E} is an equivalence too. The proof is straightforward (see for example [24]).

4.1.6.3 Definition. Assume that Φ_{E} : D^{b}(A) ⟶ D^{b}(B) is an equivalence and Ψ_{F} ≅ Φ_{E}^{−1} . In this case the functor Φ_{F⊠E} and the map ϕ_{F⊠E} will be denoted by Ad_{E} and by ad_{E} respectively.

Thus any exact equivalence Φ_{E} : D^{b}(X) ⟶^{∼} D^{b}(Y) defines an exact equivalence Ad_{E} : D^{b}(X × X) ⟶^{∼} D^{b}(Y × Y) such that , by Lemma 4.1.6.1, there is an isomorphism:

Φ_{Ad_{E}(G)} ≅ Φ_{E}Φ_{G}Φ_{E}^{−1}.  (5)

4.2 Category of coherent sheaves on an abelian variety.

4.2.1. Let A be an abelian variety of dimension n over ℂ. Denote by m : A × A → A the multiplication morphism. For any point a ∈ A by T_{a} denote the translation automorphism m(·, a) : A → A .

Let Â be dual abelian variety (1.2). It is canonically isomorphic to Pic^{0}(A) (1.3). It is well-known that there is a unique line bundle P on the product A × Â such that for any point α ∈ Â the restriction P_{α} on A × {α} represents an element of Pic^{0}A , corresponding to α and , in addition, the restriction P |_{{0}×Â} should be trivial. Such P is called Poincare line bundle (1.4). As in 1.4, 1.5

we identify A with ˆ Â using the Poincare line bundles on A × Â and Â × ˆ Â .

4.2.2. The Poincare line bundle gives an example of an exact equivalence between derived categories of coherent sheaves of two non-isomorphic varieties. Let us consider the projections

A ⟵^{p} A × Â ⟶^{q} Â

and the functor Φ_{P} : D^{b}(A) ⟶ D^{b}(Â) , defined as in 4.1.2, i.e. Φ_{P} (·) = Rq_{∗}(P ⊗ p^{∗}(·)) . The following proposition was proved by Mukai.

4.2.2.1 Proposition.([17]) Let P be an Poincare line bundle on A × Â . Then the functor Φ_{P} : D^{b}(A) ⟶ D^{b}(Â) is an exact equivalence, and there is an isomorphism of functors:

Ψ_{P} ∘ Φ_{P} ≅ (−1_{A})^{∗}[n],

where (−1_{A}) is the inverse map on the group A .

4.2.3. Let x_{1}, ..., x_{2n} be a basis of H^{1}(A, ℤ). Let l_{1}, ..., l_{2n} be the dual basis of H^{1}(Â, ℤ) . It is clear that p^{∗}(x_{1}), ..., p^{∗}(x_{2n}), q^{∗}(l_{1}), ..., q^{∗}(l_{2n}) is a basis of H^{1}(A × Â, ℤ) .

4.2.3.1 Lemma.([19]) The first Chern class of the Poincare line bundle P on A × Â satisfies the equality:

c_{1}(P) = ∑_{i=1}^{2n} p^{∗}(x_{i}) ∪ q^{∗}(l_{i}).  (6)

Now consider the map ϕ_{P} : H^{∗}(A, ℚ) ⟶ H^{∗}(Â, ℚ) , given by formula (4).

4.2.3.2 Lemma. For any k the map ϕ_{P} sends k -th cohomology to (2n − k) -th cohomology and induces isomorphisms

H^{k}(A, ℤ) ⟶^{∼} H^{2n−k}(Â, ℤ),

under which a monomial x_{i_{1}} ∪ ⋯ ∪ x_{i_{k}} goes to (−1)^{ε}l_{j_{1}} ∪ ⋯ ∪ l_{j_{2n−k}} , where (j_{1} < ⋯ < j_{2n−k}) is the complement of (i_{1} < ⋯ < i_{k}) in the set (1, ..., 2n) , and ε = ∑_{t} j_{t} . Proof. Take α ∈ H^{k}(A, ℤ) . Since ch(P) = exp(c_{1}(P)) and c_{1}(P) has the form (6), we obtain

ϕ_{P} (α) = q_{∗}(ch(P) ∪ p^{∗}(α)) = \frac{1}{(2n − k)!}q_{∗}(c_{1}(P)^{2n−k} ∪ p^{∗}(α))

This implies that ϕ_{P} sends k -th cohomology to (2n − k) -th. Moreover, it is easy to see that

\frac{1}{s!}c_{1}(P)^{s} = ∑_{j_{1}<···<j_{s}} p^{∗}(x_{j_{s}} ∪ ⋯ ∪ x_{j_{1}}) ∪ q^{∗}(l_{j_{1}} ∪ ⋯ ∪ l_{j_{s}})

Substituting this expression in previous formula, we get

\begin{aligned}ϕ_{P} (x_{i_{1}} ∪ ⋯ ∪ x_{i_{k}}) & = q_{∗}p^{∗}(x_{i_{1}} ∪ ⋯ ∪ x_{i_{k}} ∪ x_{j_{2n−k}} ∪ ⋯ ∪ x_{j_{1}}) ∪ l_{j_{1}} ∪ ⋯ ∪ l_{j_{2n−k}} = \\ & (−1)^{ε}q_{∗}p^{∗}(x_{1} ∪ ⋯ ∪ x_{2n}) ∪ l_{j_{1}} ∪ ⋯ ∪ l_{j_{2n−k}} = (−1)^{ε}l_{j_{1}} ∪ ⋯ ∪ l_{j_{2n−k}} \end{aligned}

where (j_{1} < ⋯ < j_{2n−k}) is the complement of (i_{1} < ⋯ < i_{k}) , and ε = ∑_{t} j_{t} . The monomials x_{i_{1}} ∪ ⋯ ∪ x_{i_{k}} and l_{j_{1}} ∪ ⋯ ∪ l_{j_{2n−k}} form basises of H^{k}(A, ℤ) and H^{2n−k}(Â, ℤ) respectively. Hence the map ϕ_{P} gives an isomorphism between these lattices. □

4.2.4. For any point (a, α) ∈ A × Â one can introduces a functor from D^{b}(A) to itself, defined by the rule:

Φ_{(a,α)}(·) := T_{a}^{∗}(·) ⊗ P_{α}.  (7)

A functor Φ_{(a,α)} is represented by a sheaf S_{(a,α)} = O_{Γ_{a}} ⊗ p_{2}^{∗}(P_{α}) on A × A , where Γ_{a} is the graph of the automorphism T_{a} : A ⟶ A . It is clear that the functor Φ_{(a,α)} is an equivalence. Since an automorphism T_{a} acts identically on the cohomology group and ch(P_{α}) = 1 the functors Φ_{(a,α)} belongs to kernel of the homomorphism ρ (4.1, 4.3).

The set of the functors Φ_{(a,α)} , parametrized by A × Â , can be unified in one functor from D^{b}(A × Â) to D^{b}(A × A) that takes a skyscraper O_{(a,α)} to S_{(a,α)} . (Note that this condition does not define the functor by uniquely, because it can be changed by tenzoring with any line bundle pulling back from A × Â .) Let us define such a functor Φ_{S_{A}} : D^{b}(A × Â) ⟶ D^{b}(A × A) by the representing object

S_{A} = (m · p_{13}, p_{4})^{∗}O_{Δ} ⊗ p_{23}^{∗}P_{A}

on the product (A × Â) × (A × A) . Here (m · p_{13}, p_{4}) is a morphism onto A × A that takes (a_{1}, α, a_{3}, a_{4}) to (m(a_{1}, a_{3}), a_{4}) . A direct check shows that Φ_{S_{A}} takes the skyscraper sheaf O_{(a,α)} to S_{(a,α)} .

The functor Φ_{S_{A}} can be consider as a composition of two functors. Denote by R_{A} = p_{14}^{∗}O_{Δ} ⊗ p_{23}^{∗}P ∈ D^{b}((A × Â) × (A × A)). By μ_{A} : A × A ⟶ A × A denote a morphism that takes (a_{1}, a_{2}) to (a_{1}, m(a_{1}, a_{2})) . Consider two functors

Φ_{R_{A}} : D^{b}(A × Â) ⟶ D^{b}(A × A), Rμ_{A∗} : D^{b}(A × A) ⟶ D^{b}(A × A).

4.2.4.1 Lemma. The functor Φ_{S_{A}} is isomorphic to the composition Rμ_{A∗} ∘ Φ_{R_{A}} .

The proof is omitted. Also this statement can be used as another definition of the functor Φ_{S_{A}} .

4.2.4.2 Lemma. The functor Φ_{S_{A}} is an equivalence.

Proof. By Lemma 4.1.6.2 the functor Φ_{R_{A}} is an equivalence. Since μ_{A} is an automorphism of A × A the functor Rμ_{A∗} is an equivalence too. This implies that Φ_{S_{A}} is also an equivalence.

4.2.5. Let A and B be two abelian varieties. Fix an equivalence Φ_{E} : D^{b}(A) ⟶^{∼} D^{b}(B) . Consider the functor Φ_{S_{B}}^{−1}Ad_{E}Φ_{S_{A}} from D^{b}(A × Â) to D^{b}(B × B̂) (4.1, 6.3). By J (E) denote an object that represents this functor. Thus there is the commutative diagram:

\begin{matrix}D^{b}(A × Â) & ⟶^{Φ_{S_{A}}} & D^{b}(A × A) \\ Φ_{J (E)}↓ & & ↓Ad_{E} \\ D^{b}(B × B̂) & ⟶^{Φ_{S_{B}}} & D^{b}(B × B) \end{matrix}  (8)

Let us describe the object J (E).

4.2.5.1 Theorem.([24]) There exists a group isomorphism f_{E} : A × Â ⟶ B × B̂ and a line bundle L_{E} on A × Â such that the object J (E) is isomorphic to i_{∗}(L_{E}) , where i is the embedding of A × Â in (A × Â) × (B × B̂) as the graph Γ (f_{E}) of the isomorphism f_{E} .

In particular, it immediately follows from the theorem that if two abelian varieties A and B have equivalent derived categories of coherent sheaves, then the varieties A×Â and B ×B̂ are isomorphic.

4.2.5.2 Corollary. An isomorphism f_{E} takes a point (a, α) ∈ A × Â to a point (b, β) ∈ B × B̂ iff the equivalences

Φ_{(a,α)} : D^{b}(A) ⟶^{∼} D^{b}(A), Φ_{(b,β)} : D^{b}(B) ⟶^{∼} D^{b}(B),

defined by formula (7), are connected by the relation

Φ_{(b,β)} ≅ Φ_{E}Φ_{(a,α)}Φ_{E}^{−1}.

Proof. By the theorem, Φ_{J (E)} takes a skyscraper O_{(a,α)} to O_{(b,β)} , where (b, β) = f_{E}(a, α) . By construction of the functor Φ_{S_{A}} it takes a skyscraper O_{(a,α)} to S_{(a,α)} , where the object S_{(a,α)} represents the functor Φ_{(a,α)} . Thus diagram (8) implies that the morphism f_{E} takes (a, α) to (b, β) iff S_{(b,β)} ≅ Ad_{E}(S_{(a,α)}). Now the formula (5) implies that Φ_{(b,β)} ≅ Φ_{E}Φ_{(a,α)}Φ_{E}^{−1} . □

4.2.5.3 Proposition.([24]) Let B ≅ A , the correspondence Φ_{E} ↦ f_{E} induces a group homomorphism

γ_{A} : Autoeq(D^{b}(A)) ⟶ Aut(A × Â).

Proof. Let E_{1}, E_{2}, E_{3} be objects of D^{b}(A × A) such that Φ_{E_{i}} are equivalences and Φ_{E_{3}} ≅ Φ_{E_{2}} ∘ Φ_{E_{1}} . This implies that Ad_{E_{3}} ≅ Ad_{E_{2}} ∘ Ad_{E_{1}} . Further we have the sequence of isomorphisms:

Φ_{J (E_{2})} ∘ Φ_{J (E_{1})} ≅ Φ_{S_{A}}^{−1}Ad_{E_{2}}Φ_{S_{A}}Φ_{S_{A}}^{−1}Ad_{E_{1}}Φ_{S_{A}} ≅ Φ_{S_{A}}^{−1}Ad_{E_{3}}Φ_{S_{A}} ≅ Φ_{J (E_{3})}

By theorem 4.2.5.1 the object J (E) is a line bundle over the graph of some automorphism f_{E} . Thus we obtain that f_{E_{3}} = f_{E_{2}} · f_{E_{1}} . □

4.2.5.4 Proposition.([24]) The kernel of the homomorphism γ_{A} coincides with ℤ×A×Â , consisting of all equivalences Φ_{(a,α)}[i] = T_{a}^{∗}(·) ⊗ P_{α}[i] .

4.3 The Mukai-Polishchuk group U(A) and the spinorial group Spin(A) .

In his recent preprint S.Mukai [18] was interested in the group of autoequivalences Auteq(D^{b}(A)) of the bounded derived category D^{b}(A) of coherent sheaves on an abelian variety A and he defined a certain related discrete group (which he called U(A × Â) ). Independently A.Polishchuk considered the same group, which he called SL_{2}(A) , and proved that SL_{2}(A) acts naturally on D^{b}(A) ”up to the shift functor”(see [Po2]). Let us recall the definition of this group for complex tori. We call this group U(A).

4.3.1 Definition. Let A be a complex torus. Put

U(A) := \{g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ( \begin{matrix}End(A) & Hom(Â, A) \\ Hom(A, Â) & End(Â) \end{matrix} ) | g^{−1} = ( \begin{matrix}d̂ & −b̂ \\ −ĉ & â \end{matrix} )\}

and call it the Mukai-Polishchuk group.

We may consider U(A) as a subgroup of GL(Γ_{A} ⊕ΓÂ) . Consider the canonical symmetric bilinear form Q on Λ := Γ_{A} ⊕ ΓÂ as in 3.1 above. Here is another description of the group U(A) .

4.3.2 Proposition. There are equalities

U(A) = O(Λ, Q) ∩ Aut(A × Â) = SO(Λ, Q) ∩ Aut(A × Â)

of subgroups of GL(Λ) .

Proof. The second equality follows from the fact that elements in Aut(A × Â) preserve the complex structure on V_{A} ⊕ VÂ , hence have a positive determinant.

To prove the first equality it suffices to show that the group

T := \{g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL(Λ) | g^{−1} = ( \begin{matrix}d̂ & −b̂ \\ −ĉ & â \end{matrix} )\}

coincides with the orthogonal group O(Λ, Q) . Choose a basis l_{1}, ..., l_{2n} of Γ_{A} and the dual basis x_{1}, ..., x_{2n} of ΓÂ . Consider the basis l_{1}, ..., l_{2n}, x_{1}, ..., x_{2n} of the lattice Λ . Then the group O(Λ, Q) consists of matrices

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ,

such that

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}α^{t} & γ^{t} \\ β^{t} & δ^{t} \end{matrix} ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

The equality T = O(Λ, Q) now follows from Remark 1.6. □

4.3.3 Proposition. ([Or2]) The image of the homomorphism γ_{A} : Autoeq(D^{b}(A)) ⟶ Aut(A × Â) coincides with the Mukai-Polishchuk group U(A) . Moreover there is an exact sequence of groups

0 ⟶ ℤ × A × Â ⟶ Auteq(D^{b}(A)) ⟶ U(A) ⟶ 1.

This was essentially conjectured by Polishchuk in [Po2].

By Lemma 4.1.4.3 the correspondence Φ_{E} ↦ ϕ_{E} induces a homomorphism

ρ_{A} : Auteq(D^{b}(A)) ⟶ GL(H^{∗}(A, ℚ)).

As a matter of fact, this representation preserves the integral cohomology lattice.

4.3.4 Proposition. For any equivalence Φ_{E} : D^{b}(A) ⟶^{∼} D^{b}(B) the linear map ϕ_{E} preserves the integral cohomology, i.e.

ϕ_{E}(H^{∗}(A, ℤ)) = H^{∗}(B, ℤ).

Proof. By definition of ϕ_{E} , one needs to show that ch(E) belongs to integral cohomology H^{∗}(A × B, ℤ). This is equivalent to checking that ch(E^{∨} ⊠ E) belongs to the integral cohomology H^{∗}((A × A)×(B ×B), ℤ) . This object represents the functor Ad_{E} . Since Ad_{E} ≅ Φ_{S_{B}}Φ_{J (E)}Φ_{S_{A}}^{−1} it is sufficient to show that ch(S_{A}) , ch(S_{B}^{∨}) and ch(J (E)) are integral. But , by construction, all these objects are line bundles on abelian subvarieties. Thus, the following lemma implies the proposition. □

4.3.4.1 Lemma. For any line bundle L on an abelian variety A the Chern character ch(L) is integral, i.e belongs to H^{∗}(A, ℤ) .

Proof. Denote by c_{1}(L) the first Chern class of L . The Chern character ch(L) is equal to exp(c_{1}(L)) . Thus we must show that \frac{1}{k!}c_{1}(L)^{k} belong to integral cohomology for any k . If x_{1}, ..., x_{2n} be a basis of H^{1}(A, ℤ) , then the first Chern class as an element of H^{2}(A, ℤ) can be written as:

c_{1}(L) = ∑_{i<j} d_{ij} · (x_{i} ∪ x_{j}), d_{ij} ∈ ℤ

Taking k -power, we obtain

c_{1}(L)^{k} = k! ∑_{{(i_{1}<j_{1});...;(i_{k}<j_{k}) \atop (i_{1},j_{1},...,i_{k},j_{k})⊂(1,...,2n)}} d_{i_{1}j_{1}} ⋯ d_{i_{k}j_{k}}(x_{i_{1}} ∪ x_{j_{1}} ∪ ⋯ ∪ x_{i_{k}} ∪ x_{j_{k}})

That proves this lemma and the proposition. □

This way, the correspondence Φ_{E} ↦ ϕ_{E} gives us the homomorphism

ρ_{A} : Auteq(D^{b}(A)) ⟶ GL(H^{∗}(A, ℤ)),

denoted by the same letter as in Lemma 4.1.4.3.

4.3.5 Definition. The subgroup of GL(H^{∗}(A, ℤ)) that is the image of ρ_{A} will be called the spinor group of A and will be denoted by Spin(A) .

4.3.6. To justify this definition consider the Clifford algebra Cl(Λ, Q) (3.1) and the homomorphism of algebras

cor_{A} : Cl(Λ, Q) ⟶ End(H^{∗}(A, ℤ)) ≅ H^{∗}(A × A, ℤ)

defined on the Λ ⊂ Cl(Λ, Q) by rule

cor_{A}((l, x))(·) = l(·) + x ∪ (·),

where (l, x) ∈ Γ_{A} ⊕ Γ_{A}^{∗} = Λ and l(·) is the convolution with l , i.e. for a monomial α = x_{1} ∪ ... ∪ x_{k} it is defined as

l(α) := ∑(−1)^{i−1}l(x_{i}) · x_{1} ∪ ... ∪ x_{i−1} ∪ x_{i+1} ∪ ... ∪ x_{k}.

By Proposition 3.2.1e) the homomorphism cor_{A} is an isomorphism. Clearly it sends Cl^{+}(Λ, Q) onto H^{ev}(A × A, ℤ) . The homomorphism cor_{A} induces the action of the group Spin(Λ, Q) (3.4.1) on cohomology lattice H^{∗}(A, ℤ) . Henceforth, we will consider Spin(Λ, Q) as a subgroup of GL(H^{∗}(A, ℤ)) with respect to cor_{A} .

There is the standard involution ′ on Cl(Λ) . It is defined as a unique ring involution on Cl(Λ) that is the identity on Λ . The isomorphism cor_{A} induces an involution on End(H^{∗}(A, ℤ)) , which will be denoted the same symbol ′ . It is not hard to check that for any ξ ∈ H^{d}(A × A, ℤ) there is the equality

v_{ξ}′ = (−1)^{n+ \frac{d(d−1)}{2}} w_{ξ},

where v_{ξ} and w_{ξ} defined in 4.1.3 and n = dim A . In particular, this implies that

ϕ_{E}′ = ψ_{E^{∨}[n]}.  (9)

for any object E ∈ D^{b}(A × A) .

Thus there are two correspondences Φ_{E} ↦ f_{E} and Φ_{E} ↦ ϕ_{E} , which gives two group homomorphisms

γ_{A} : Auteq(D^{b}(A)) ⟶ U(A) ⊂ SO(Λ, Q), ρ_{A} : Auteq(D^{b}(A)) ⟶ Spin(A) ⊂ GL(H^{∗}(A, ℤ))

such that γ_{A}(Φ_{E}) = f_{E} and ρ_{A}(Φ_{E}) = ϕ_{E} . The following proposition relates these two homomorphisms.

4.3.7 Proposition.

a) The group Spin(A) is contained in Spin(Λ, Q) ⊂ GL(H^{∗}(A, ℤ)) .

b) Let π : Spin(Λ, Q) ⟶ SO(Λ, Q) be the canonical map. Then π · ρ_{A} = γ_{A} and π^{−1}(U(A)) = Spin(A) .

Proof. Let Φ_{E} : D^{b}(A) ⟶^{∼} D^{b}(A) be an autoequivalence. The following diagram is a cohomological analog of diagram (8) with B = A :

\begin{matrix}H^{∗}(A × Â, ℤ) & ⟶^{ϕ_{S_{A}}} & H^{∗}(A × A, ℤ) \\ ϕ_{J (E)}↓ & & ↓ad_{E} \\ H^{∗}(A × Â, ℤ) & ⟶^{ϕ_{S_{A}}} & H^{∗}(A × A, Z) \end{matrix}  (10)

By Theorem 2.4.5.1 the sheaf J (E) is a line bundle over the subvariety Γ (f_{E}) that is the graph of the isomorphism f_{E} . This implies that ϕ_{J (E)} sends H^{4n−1}(A × Â, ℤ) to itself. Moreover, the restriction of ϕ_{J (E)} on the (4n − 1) − th cohomology coincides with

f_{E} : H_{1}(A × Â, ℤ) ⟶ H_{1}(A × Â, ℤ)

under the Poincare isomorphism D : H_{1}(A × Â, ℤ) = Λ ⟶^{∼} H^{4n−1}(A × Â, ℤ) . Fix this identification of H^{4n−1}(A × Â, ℤ) with Λ.

Let us assume that the restriction of ϕ_{S_{A}} on H^{4n−1}(A × Â, ℤ) coincides with the restriction of cor_{A} on Λ . Under this assumption, it follows from the commutativity of the diagram (10) that ad_{E} takes cor_{A}(Λ) to itself. Hence ϕ_{E}cor(Λ)ϕ_{E}^{−1} = cor(Λ) . Further, since the inverse of an equivalence Φ_{E} is isomorphic to Ψ_{E^{∨}[n]} we get from (9) that ϕ_{E}′ = ϕ_{E}^{−1} . Therefore N(ϕ_{E}) = ϕ_{E} · ϕ_{E}′ = id . By Definition 3.4.1 ϕ_{E} belongs to Spin(Λ, Q) and a) is proved.

Moreover, we have that action of f_{E} on Λ coincides with the action of ad_{E} on cor_{A}(Λ) . This implies that γ_{A} = πρ_{A} . Finally, since Kerπ = ℤ/2ℤ and ρ([1]) = −Id the full inverse image π^{−1}(U(A)) not only contains but also coincides with Spin(A) .

Thus to complete the proof of the proposition it is sufficient to prove the following lemma.

4.3.7.1 Lemma. The restriction of the map ϕ_{S_{A}}D on H_{1}(A × Â, ℤ) = Λ coincides with the restriction of the map cor_{A} on Λ .

Proof. As above let x_{1}, ..., x_{2n} and l_{1}, ..., l_{2n} be the dual bases of Γ_{A}^{∗} and Γ_{A} respectively. For any α ∈ H_{i}(A × Â, ℤ) and β ∈ H^{i}(A × Â, ℤ) the following equality holds

⟨β, α⟩ = ⟨β ∪ D(α), [A × Â]⟩,

where ⟨, ⟩ is the canonical pairing of cohomology with homology and [A × Â] ∈ H_{4n}(A × Â, ℤ) is the fundamental class. It is easy to check that

\begin{matrix}D(x_{i}) = (−1)^{i−1}p^{∗}(x_{1} ∪ ⋯ ∪ x_{2n}) ∪ q^{∗}(l_{1} ∪ ⋯ ∪ l_{i−1} ∪ l_{i+1} ∪ ⋯ ∪ l_{2n}), \\ D(l_{i}) = (−1)^{i−1}p^{∗}(x_{1} ∪ ⋯ ∪ x_{i−1} ∪ x_{i+1} ∪ ⋯ ∪ x_{2n}) ∪ q^{∗}(l_{1} ∪ ⋯ ∪ l_{2n}). \end{matrix}

Denote μ_{A} = μ (4.2.4). By Lemma 4.2.4.1 the functor Φ_{S_{A}} is the composition Rμ_{∗}Φ_{R_{A}} . Hence ϕ_{S_{A}} = μ_{∗}ϕ_{R_{A}} . By construction of Φ_{R_{A}} , the map ϕ_{R_{A}} takes q^{∗}(l) ∪ p^{∗}(x) to p_{1}^{∗}(ψ_{P} (l)) ∪ p_{2}^{∗}(x) . Combining Lemma 4.2.3.2 and Proposition 4.2.2.1 we get

ϕ_{R_{A}}D(x_{i}) = p_{1}^{∗}(x_{i}) ∪ p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n})  (11)
ϕ_{R_{A}}D(l_{i}) = (−1)^{i−1}p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{i−1} ∪ x_{i+1} ∪ ⋯ ∪ x_{2n})  (12)

Further, to compute μ_{∗}(α) it is sufficient to note that μ_{∗} is a ring homomorphism because μ is an isomorphism and, consequently, μ_{∗} is the inverse of μ^{∗} . Besides, since μ(a_{1}, a_{2}) = (a_{1}, m(a_{1}, a_{2})) we have

μ^{∗}(p_{1}^{∗}(α)) = p_{1}^{∗}(α), μ^{∗}(p_{2}^{∗}(α)) = m^{∗}(α)

for any α ∈ H^{∗}(A, ℤ) . Moreover, if α = x ∈ H^{1}(A, ℤ) then

μ^{∗}(p_{2}^{∗}(x)) = m^{∗}(x) = p_{1}^{∗}(x) + p_{2}^{∗}(x)

Therefore, for any x ∈ H^{1}(A, ℤ) there are equalities

μ_{∗}(p_{1}^{∗}(x)) = p_{1}^{∗}(x) and μ_{∗}(p_{2}^{∗}(x)) = p_{2}^{∗}(x) − p_{1}^{∗}(x).

To find an element ϕ_{S_{A}}(λ) for some λ ∈ H^{4n−1}(A × Â, ℤ) it is sufficient to compute the map that is defined by this element, i.e.

ϕ_{S_{A}}(λ)(·) := p_{2∗}(ϕ_{S_{A}}(λ) ∪ p_{1}^{∗}(·)).

Substituting (11) in the last expression, we get

\begin{matrix}ϕ_{S_{A}}D(x_{i})(α) = p_{2∗}(μ_{∗}(p_{1}^{∗}(x_{i}) ∪ p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n})) ∪ p_{1}^{∗}(α)) = m_{∗}(p_{1}^{∗}(x_{i} ∪ α) ∪ p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n})) \\ = m_{∗}(m^{∗}(x_{i} ∪ α) ∪ p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n})) = x_{i} ∪ α ∪ m_{∗}(p_{2}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n})) = x_{i} ∪ α \end{matrix}

Hence ϕ_{S_{A}}D(x) coincides with cor_{A}(x) .

Let us compute the action of ϕ_{S_{A}}D(l_{i}) on the cohomology. We have

\begin{matrix}ϕ_{S_{A}}D(l_{i})(x_{s_{1}} ∪ ⋯ ∪ x_{s_{k}}) = p_{2∗}(μ_{∗}((−1)^{i−1}p_{2}^{∗}( ⋃_{{j=1, \atop j≠i}}^{2n} x_{j})) ∪ p_{1}^{∗}(x_{s_{1}} ∪ ⋯ ∪ x_{s_{k}})) = \\ (−1)^{i}p_{2∗}( ⋃_{{j=1, \atop j≠i}}^{2n}(p_{1}^{∗}(x_{j}) − p_{2}^{∗}(x_{j})) ∪ p_{1}^{∗}(x_{s_{1}} ∪ ⋯ ∪ x_{s_{k}})) \end{matrix}

If the set (s_{1}, ..., s_{k}) does not contain i , then this expression equals 0 . Suppose that s_{1} = i . In this case the last expression can be simplified

ϕ_{S_{A}}D(l_{i})(x_{i} ∪ x_{s_{2}} ∪ ⋯ ∪ x_{s_{k}}) = p_{2∗}(p_{1}^{∗}(x_{1} ∪ ⋯ ∪ x_{2n}) ∪ p_{2}^{∗}(x_{s_{2}} ∪ ⋯ ∪ x_{s_{k}})) = x_{s_{2}} ∪ ⋯ ∪ x_{s_{k}}

Thus, ϕ_{S_{A}}D(l_{i}) coincides with cor_{A}(l_{i}) , because

\begin{matrix}cor_{A}(l_{i})(x_{s_{1}} ∪ ⋯ ∪ x_{s_{k}}) = 0 if i ∉ (s_{1}, ..., s_{k}), \\ cor_{A}(l_{i})(x_{i} ∪ x_{s_{2}} ∪ ⋯ ∪ x_{s_{k}}) = x_{s_{2}} ∪ ⋯ ∪ x_{s_{k}}. \end{matrix}

This concludes the proof of the lemma and, consequently, completes the proof of the proposition. □

4.3.8 Corollary. Kerρ_{A} coincides with 2ℤ × A × Â , consisting of all equivalences Φ_{(a,α)}[2i] = T_{a}^{∗}(·) ⊗ P_{α}[2i] .

Proof. It is clear that Φ_{(a,α)}[2i] are contained in Kerρ_{A} . On the other side, it follows from Proposition 4.3.7 that Kerρ_{A} is a subgroup of Kerγ_{A} ≅ ℤ × A × Â . Note that by Lemma 4.1.5.1 ρ_{A}([1]) = −Id . Corollary is proved. □

4.3.9 Corollary. There is the exact sequence of groups

0 ⟶ ℤ/2ℤ ⟶ Spin(A) ⟶ U(A) ⟶ 1.

The last theorem of this chapter gives description of abelian varieties that have equivalent derived categories of coherent sheaves.

4.3.10 Theorem.([Or2]) Let B and C be abelian varieties. Then the derived categories D^{b}(B) and D^{b}(C) are equivalent if and only if there exists an isomorphism

γ : B × B̂ ⟶^{∼} C × Ĉ

which identifies the forms Q_{B} and Q_{C} on Γ_{B} ⊕ ΓB̂ and Γ_{C} ⊕ ΓĈ. .

The “if” direction in the above theorem was first proved by A. Polishchuk in [Po1].

5. The algebraic group U_{A,ℚ} .

5.1. Fix a complex torus A . Put Λ := Γ_{A} ⊕ ΓÂ and consider the group GL(Λ_{ℚ}) . We have the following ℚ -algebraic subgroups of GL(Λ_{ℚ}) .

1) Hdg_{A,ℚ} = Hdg_{A×Â,ℚ}.

2) Aut_{ℚ}(A × Â) , which is defined as the group of invertible elements in End^{0}(A × Â) .

3) O(Λ_{ℚ}, Q_{ℚ}) , SO(Λ_{ℚ}, Q_{ℚ}) , where Q_{ℚ} is the extension to Λ_{ℚ} of the canonical symmetric bilinear form on Λ (3.1).

4) U_{A,ℚ} , which is defined as follows

U_{A,ℚ} = \{g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ Aut_{ℚ}(A × Â) | g^{−1} = ( \begin{matrix}d̂ & −b̂ \\ −ĉ & â \end{matrix} )\}

Thus the discrete group U(A) defined in 4.3.1 is the arithmetic subgroup of U_{A,ℚ} consisting of elements which preserve the lattice Λ .

The following proposition summarizes the interrelations of these subgroups.

5.2 Proposition.

1) Hdg_{A,ℚ} ⊂ SO(Λ_{ℚ}, Q_{ℚ}).

2) Aut_{ℚ}(A × Â) is the centralizer of Hdg_{A,ℚ} in GL(Λ_{ℚ}) .

3) U_{A,ℚ} = Aut_{ℚ}(A × Â) ∩ SO(Λ_{ℚ}, Q_{ℚ}).

4) U_{A,ℚ} is the centralizer of Hdg_{A,ℚ} in SO(Λ_{ℚ}, Q_{ℚ}).

Proof. 1) The operator J_{A} of complex structure on Λ_{ℝ} has determinant 1 and preserves the form Q . Thus h_{A}(S^{1}) lies in the group of ℝ -points of SO(Λ_{ℚ}, Q_{ℚ}). Since Hdg_{A,ℚ} is the ℚ -closure of h_{A}(S^{1}), we have Hdg_{A,ℚ} ⊂ SO(Λ_{ℚ}, Q_{ℚ}).

2) By definition End^{0}(A × Â) is the centralizer of Hdg_{A,ℚ} in End(Λ_{ℚ}) . Thus Aut_{ℚ}(A × Â) is the centralizer of Hdg_{A,ℚ} in GL(Λ_{ℚ}) .

3) The proof is the same as that of proposition 4.3.2 above.

4) This follows from 1), 2), 3). □

5.3. Let us study the ℚ -algebraic group U_{A,ℚ} in case A is an abelian variety.

Clearly, the group U_{A,ℚ} depends only on the isogeny class of A . If A = A_{1}^{m_{1}} × ... × A_{k}^{m_{k}} , where A_{i} are pairwise nonisogeneous simple abelian varieties, then

U_{A,ℚ} = ∏_{i} U_{A_{i}^{m_{i}} ,ℚ}.

Below we describe the ℚ -algebraic group U_{A^{m},ℚ} and its group of ℝ -points U_{A^{m},ℚ}(ℝ) for a simple abelian variety A .

5.3.1. Let us use the notations of section 1.8. In particular denote F = End^{0}(A). Let φ ∈ Hom(A, Â) be a polarisation and ′ : F → F be the corresponding Rosati involution. We identify naturally M_{m}(F) = End^{0}(A^{m}) , which makes Γ_{A^{m},ℚ} a left M_{m}(F) module. Consider the diagonal polarisation σ = (φ, ...φ) of A^{m} . Then the corresponding Rosati involution on M_{m}(F) is

M ↦ ^{t}M′.

Consider the following homomorphism of algebras

τ : End^{0}(A × Â) ⟶ End(Γ_{A^{m},ℚ} ⊕ Γ_{A^{m},ℚ})

τ : ( \begin{matrix}B & C \\ D & E \end{matrix} ) ↦ ( \begin{matrix}B & Cσ \\ σ^{−1}D & σ^{−1}Eσ \end{matrix} ) .

Obviously, the image of τ is contained in M_{2m}(F) . We have

τ(U_{A^{m},ℚ}) = \{g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL(2m, F) | g^{−1} = ( \begin{matrix}^{t}d̂ & −^{t}b̂ \\ −^{t}ĉ & ^{t}â \end{matrix} )\}

Thus τ(U_{A^{m},ℚ}) is the group of isometries of the skew-hermitian form on F^{m} ⊕ F^{m}

ϑ((z_{1}, ..., z_{m}, z_{−1}, ..., z_{−m}), (w_{1}, ..., w_{m}, w_{−1}, ..., w_{−m})) = ∑_{k=1}^{m} z_{k}w′_{−k} − z_{k}w′_{k}.

5.3.2. Denote by U_{2m,F}^{∗} ⊂ GL(2m, F) the K_{0} -algebraic group of isometries of the skew-hermitian form ϑ . Then the ℚ -algebraic group U_{A^{m},ℚ} is obtained from U_{2m,F}^{∗} by restriction of scalars from K_{0} to ℚ . Fix an embedding K_{0} ↪ ℝ . The corresponding group of real points U_{2m,F}^{∗} (ℝ) was computed by A. Polishchuk ([Po2]). His result according to the four cases of Albert’s classification is the following

I. Sp_{2m}(ℝ) , II. Sp_{4m}(ℝ) , III. U_{2m}^{∗}(ℍ) – the group of automorphisms of ℍ^{2m} preserving the standard skew-hermitian form, IV. U(md, md) .

5.3.3 Corollary. The reductive Lie group U_{2m,F}^{∗} (ℝ) is semisimple unless we are in case IV. This group has no compact factors unless we are in case IV (then it is isogeneous to the product S^{1}×SU(md, md) ) or in case III and m = 1 (then it is isogeneous to the product SU(2) × SL(2, ℝ) ).

5.3.4. We have

U_{A^{m},ℚ}(ℝ) ≅ ∏_{e_{0}} U_{2m,F}^{∗} (ℝ),

hence the above provides the corresponding description of the group U_{A^{m},ℚ}(ℝ) .

Corollary. The reductive group U_{A^{m},ℚ}(ℝ) is semisimple unless we are in case IV. This group has no compact factors unless we are in case IV (then the compact part is isogeneous to the product of e_{0} copies of S^{1} ) or in case III and m = 1 (then the compact part is isogeneous to the product of e_{0} copies of SU(2) ).

5.3.5 Corollary. For any abelian variety A the Lie group U_{A,ℚ}(ℝ) is connected.

6. The Neron-Severi Lie algebra 𝔤_{NS}(X) .

6.1. Let X be a smooth projective variety of dimension n . Let us recall the definition of the Neron-Severi Lie algebra 𝔤_{NS}(X) from [LL]. If κ ∈ H^{1,1}(X) ∩ H^{2}(X, ℚ) is an ample class, then cupping with it defines an operator e_{κ} in the total cohomology H^{∗}(X) = H^{∗}(X, ℂ) of degree 2 and the hard Lefschetz theorem asserts that for s = 0, 1, ..., n , e_{κ}^{s} maps H^{n−s}(X) isomorphically onto H^{n+s}(X) . As is well known, this is equivalent to the existence of a (unique) operator f_{κ} on H^{∗}(X) of degree −2 such that the commutator [e_{κ}, f_{κ}] is the operator h which on H^{k}(X) is multiplication by k −n . The elements e_{κ}, f_{κ}, h make up a Lie subalgebra 𝔤_{κ} of 𝔤𝔩(H^{∗}(X)) isomorphic to sl(2) . Define the Neron-Severi Lie algebra 𝔤_{NS}(X) as the Lie subalgebra of 𝔤𝔩(H^{∗}(X)) generated by 𝔤_{κ} ’s with κ an ample class. This Lie subalgebra is defined over ℚ and is evenly graded by the adjoint action by the semisimple element h .

In the above definition we could replace an ample class κ by a Kahler class or by any element κ ∈ H^{2}(X) , such that e_{κ} satisfies the conclusion of the hard Lefschetz theorem. The resulting Lie subalgebras of 𝔤𝔩(H^{∗}(X)) are called the Kahler Lie algebra and the total Lie algebra respectively and denoted by 𝔤_{K}(X) and 𝔤_{tot}(X) respectively. Obviously we have the inclusions 𝔤_{NS}(X) ⊂ 𝔤_{K}(X) ⊂ 𝔤_{tot}(X) .

The above Lie algebras preserve (infinitesimally) the following form on the cohomology H^{∗}(X) .

χ(α, β) := (−1)^{q} ∫_{X} α ∪ β,

where α is homogeneous of degree n + 2q or n + 2q + 1 . The main fact about these Lie algebras is that they are semisimple ([LL],p.369).

6.2. In case X is a complex torus (resp. an abelian variety) the Lie algebras 𝔤_{tot}(X) , 𝔤_{K}(X) (resp. 𝔤_{NS}(X) ) were computed explicitly in [LL], p.381. Let us recall the result.

6.2.1. Let X = (V_{X}/Γ_{X}, J_{X}) be a complex torus. Put Λ = Γ_{X} ⊕ ΓX̂ with the canonical symmetric bilinear form Q (3.1). There exists a natural isomorphism of Lie algebras

𝔤_{tot}(X) ≅ 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}),

such that the semisimple element h ∈ 𝔤_{tot}(X) corresponds to the element −1_{Γ_{X}} ⊕1_{ΓX̂} ∈ 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) . The natural representation of 𝔤_{tot} on H^{∗}(X, ℚ) under the above isomorphism is the spinorial representation of 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) (which is the direct sum of two semi-spinorial ones corresponding to H^{ev} and H^{odd} respectively).

Let J = (J_{X}, JX̂) be the complex structure on V_{X} ⊕ VX̂ . Then the form Q_{ℝ} is J -invariant, hence defines a hermitian form on Λ_{ℝ} . Let 𝔰𝔲(Λ_{ℝ}) denote the Lie algebra of the corresponding special unitary Lie group. Then the isomorphism mentioned above 𝔤_{tot}(X) ≅ 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) identifies the Lie subalgebras 𝔤_{K}(X; ℝ) ≅ 𝔰𝔲(Λ_{ℝ}) .

6.2.2. Finally, let X be an abelian variety. The Lie algebra 𝔤_{NS}(X) depends only on the isogeny type of X . If

X = X_{1}^{m_{1}} × ... × X_{k}^{m_{k}},

then

𝔤_{NS}(X) = 𝔤_{NS}(X_{1}^{m_{1}}) × ... × 𝔤_{NS}(X_{k}^{m_{k}}).

We therefore assume that X is a power A^{m} of a simple abelian variety A . Let us stick to notations of section 1.8. In particular, F = End^{0}(A) . Choose a polarization φ ∈ Hom(A, Â) of A . Denote by ′ the corresponding Rosati involution on F . Consider the diagonal polarisation σ = (φ, ..., φ) of A^{m} . If we naturally identify M_{m}(F) = End^{0}(A^{m}) then the Rosati involution defined by σ is M ↦ ^{t}M′ .

Define a K_{0} –Lie subalgebra of End(2m, F) by

𝔰𝔩𝔲(2m, F,′) = \{( \begin{matrix}B & C \\ D & −^{t}B′ \end{matrix} ) |B, C, D ∈ End(m, F); C = ^{t}C′, D = ^{t}D′\}.

This is the Lie algebra of infinitesimal isometries of the skew-hermitian form ϑ on F^{m} ⊕ F^{m} , which was defined in 5.3.1 above. It is a reductive K_{0} –Lie algebra whose center is the space of scalars λ ∈ K with λ′ = −λ . So 𝔰𝔩𝔲(2m, F,′) is semisimple unless we are in the case of totally complex multiplication (case IV). We grade this Lie algebra by means of the semisimple element

u_{m} := ( \begin{matrix}−1_{m} & 0 \\ 0 & 1_{m} \end{matrix} ) ∈ 𝔰𝔩𝔲(2m, F,′),

so that B , C and D parametrize the summands of degree 0 , −2 and 2 respectively. Let 𝔤(2m, F,′) denote the K_{0} –Lie subalgebra of 𝔰𝔩𝔲(2m, F,′) generated by summands of degree 2 and -2; let 𝔲(m, F,′) denote the union of GL(m, F) –conjugacy classes in End(m, F) made up of antiinvariants with respect to the involution M ↦ ^{t}M′ and identify 𝔲(m, F,′) with the subspace of 𝔰𝔩𝔲(2m, F,′) in an obvious way. We have the following result.

6.2.2.1 Lemma. ([LL], Lemma 3.9) We have

𝔰𝔩𝔲(2m, F,′) = 𝔤(2m, F,′) × 𝔲(m, F,′).

The summand 𝔲(m, F,′) is trivial except in the following cases

1) m = 1 and F is totally definite quaternion (case III): then 𝔤(2, F,′) ≅ sl(2, K_{0}) and 𝔲(1, F,′) can be identified with pure quaternians in F (i.e. the ′ -antiinvariants in F ) or

2) K is totally complex (case IV): then 𝔤(2m, F,′) consists of the matrices for which B has its K –trace in K_{0} , whereas 𝔲(m, F.′) can be identified with the purely imaginary scalars in K (i.e. the λ ∈ K with λ̄ = −λ ).

6.2.2.2 Remark. It was noted in [LL] that in the exceptional cases the connected Lie subgroup of GL(m, F ⊗_{ℚ} ℝ) with the Lie algebra 𝔲(m, F,′)⊗_{ℚ} ℝ is a product of e_{0} copies of U(1) , resp SU(2) , and hence is compact.

In view of 6.2.1 above we may consider 𝔤_{NS}(X) as a Lie subalgebra of End(Λ_{ℚ}) = End(Γ_{X,ℚ} ⊕ Γ_{X̂,ℚ}) . Recall the homomorphism of algebras τ as in 5.3.1 above.

τ : End^{0}(Γ_{X,ℚ} ⊕ Γ_{X̂,ℚ}) ⟶ End(Γ_{X,ℚ} ⊕ Γ_{X,ℚ})

τ : ( \begin{matrix}B & C \\ D & E \end{matrix} ) ↦ ( \begin{matrix}B & Cσ \\ σ^{−1}D & σ^{−1}Eσ \end{matrix} ) .

Note that Γ_{X,ℚ} ⊕ Γ_{X,ℚ} is naturally a End(2m, F) - module.

6.2.2.3 Proposition. ([LL], Prop.3.10) The above homomorphism τ identifies the Lie algebras 𝔤_{NS}(X) and 𝔤(2m, F.′) . In particular, we get an isomorphism

𝔰𝔩𝔲(2m, F,′) ≅ 𝔤_{NS}(X) × 𝔲(m, F,′).

7. Relation between the group Auteq(D^{b}(A)) and the Neron-Severi Lie algebra 𝔤_{NS}(A) for an abelian variety A .

7.1. Let A be an abelian variety. We know that the group Auteq(D^{b}(A)) acts on the total cohomology H^{∗}(A, ℤ) of A and we denoted its image in GL(H^{∗}(A, ℤ)) by Spin(A) (4.3.5). Put Λ = Γ_{A} ⊕ ΓÂ and let Q be the canonical symmetric bilinear form on Λ (3.1). Recall the exact sequence of group homomorphisms

0 ⟶ ℤ/2ℤ ⟶ Spin(A) ⟶ U(A) ⟶ 1

(4.3.9) and the commutative diagram (4.3.7)

\begin{matrix}Spin(A) & ↪ & Spin(Λ, Q) \\ ↓ & & ↓ \\ U(A) & ↪ & SO(Λ, Q) \end{matrix} ,

where horizontal arrows are compatible with the actions on H^{∗}(A, ℤ) and Λ respectively.

7.1.1 Definition. Let \overline{Spin(A)} (resp. \overline{U(A)} ) be the Zariski closure of Spin(A) (resp. U(A) ) in GL(H^{∗}(A, ℚ)) (resp GL(Λ_{ℚ}) ).

These are ℚ -algebraic groups. These groups are isogeneous, hence have isomorphic Lie algebras.

7.1.2 Remark. Since the group Spin(A) acts on H^{∗}(A, ℚ) by algebraic correspondences its action commutes with Hdg_{A,ℚ} . Hence the same is true for \overline{Spin(A)}.

On the other hand we know that 𝔤_{NS}(A) can be identified as a Lie subalgebra of 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) so that the representation of 𝔤_{NS}(A) in H^{∗}(A, ℚ) is the restriction of the spinorial representation of 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) (6.2.1).

7.2 Theorem.

a) The Lie algebra of \overline{Spin(A)} is 𝔤_{NS}(A) considered as a Lie subalgebra of 𝔤𝔩(H^{∗}(A, ℚ)) .

b) The Lie algebra of \overline{U(A)} is 𝔤_{NS}(A) considered as a Lie subalgebra of 𝔤𝔩(Λ_{ℚ}) .

Proof. The discussion in 7.1 above implies that a) and b) are equivalent. Let us prove b).

Both \overline{U(A)} and 𝔤_{NS}(A) depend only on the isogeny type of A . If A is a product of pairwise nonisogeneous abelian varieties then both \overline{U(A)} and 𝔤_{NS}(A) are products of the corresponding factors. Thus it suffices to prove that 𝔤_{NS}(A^{m}) is the Lie algebra of the algebraic group \overline{U(A^{m})} for a simple abelian variety A .

By definition U(A^{m}) is an arithmetic subgroup of the reductive ℚ -algebraic group U_{A^{m},ℚ} (5.1). By Corollary 5.3.4 all abelian factors of U_{A^{m},ℚ} are compact. Thus by the density theorem ([PR]) the ℚ -algebraic group \overline{U(A^{m})} consists (up to isogeny) of all noncompact factors of U_{A^{m},ℚ} . Let 𝔤(A^{m}) be the Lie algebra of U_{A^{m},ℚ} . As follows from the results in 5.3.1, 6.2.2, 6.2.2.3 that it is a product of Lie algebras

𝔤(A^{m}) = 𝔤_{NS}(A^{m}) × 𝔲_{A^{m}},

where 𝔲_{A^{m}} = τ^{−1}(𝔲(m, F,′)) in the notation of 5.3.1, 6.2.2 above. Moreover, it follows from the results in 5.3.4, 6.2.2.1, 6.2.2.2 above that the ℚ -algebraic subgroup of U_{A^{m},ℚ} corresponding to 𝔲_{A^{m}} consists (up to isogeny) of all compact factors of U_{A^{m},ℚ} . Thus 𝔤_{NS}(A^{m}) is the Lie algebra of the group \overline{U(A^{m})} . This proves the theorem. □

7.2.1 Corollary.

a) The algebraic groups \overline{Spin(A)} and \overline{U(A)} are semisimple.

b) The group \overline{U(A)} consists (up to isogeny) of all noncompact factors of U_{A,ℚ}.

Proof. a) Indeed, this follows from the above theorem, since the Lie algebra 𝔤_{NS}(A) is semisimple.

b) This was obtained in the proof of the above theorem. □

7.3. Let A be an abelian variety. Consider 𝔤_{NS}(A; ℝ) as a Lie subalgebra of 𝔤𝔩(Λ_{ℝ}) . Let φ ∈ NS_{A}(ℝ)^{0} ⊂ Hom(V_{A}, VÂ) . Then φ, φ^{−1} ∈ 𝔤_{NS}(A; ℝ) and

[φ, φ^{−1}] = h = ( \begin{matrix}−1_{A} & 0 \\ 0 & 1Â \end{matrix} ) .

The elements φ, φ^{−1}, h make up a Lie subalgebra 𝔤_{φ} of 𝔤_{NS}(A; ℝ) isomorphic to sl(2) . By definition the Lie algebra 𝔤_{NS}(A; ℝ) is generated by these subalgebras.

Consider the group of ℝ -points \overline{U(A)}(ℝ) of the ℚ -algebraic group \overline{U(A)} . It is a semisimple Lie subgroup of the reductive Lie group U_{A,ℚ}(ℝ) (5.3.4) which consists (up to isogeny) of all noncompact factors of U_{A,ℚ}(ℝ) . By the above theorem the Lie algebra of \overline{U(A)}(ℝ) is 𝔤_{NS}(A; ℝ) . For φ ∈ NS_{A}(ℝ)^{0} denote by G_{φ} ⊂ \overline{U(A)}(ℝ) the connected Lie subgroup corresponding to the Lie subalgebra 𝔤_{φ} . We have G_{φ} ≅ SL(2; ℝ) .

8. Action of U_{A,ℚ}(ℝ) on a Siegel domain.

8.1. Let A be a complex torus. Let us define a rational (i.e. not defined everywhere) action of the group U_{A,ℚ}(ℝ) (5.1) on the complex space NS_{A}(ℂ) = NS_{A} ⊗ ℂ ⊂ Hom(A, Â) ⊗ ℂ. It is given by the formula

( \begin{matrix}a & b \\ c & d \end{matrix} ) ω := (c + dω)(a + bω)^{−1}, ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{A,ℚ}(ℝ), ω ∈ NS_{A}(ℂ) ⊂ Hom(A, Â) ⊗ ℂ.(13)

Here multiplication is understood as composition of maps.

In case A is an abelian variety NS_{A}(ℂ) contains Siegel domains of the first kind on which this action is well-defined. Namely, let C_{A}^{a} ⊂ NS_{A}(ℝ) be the ample cone of A, which is defined as ℝ^{+} -linear combinations of ample classes in NS_{A}. It is an open subset in NS_{A}(ℝ). Put

C_{A}^{±} := NS_{A}(ℝ) ± iC_{A}^{a} ⊂ NS_{A}(ℂ),
C_{A} := C_{A}^{+} ⨿ C_{A}^{−}.

Thus C_{A} ≠ ∅ if and only if the complex torus A is an abelian variety.

8.2 Theorem. Let A be an abelian variety.

a) The action of U_{A,ℚ}(ℝ) on C_{A} is well defined.

b) This action on C_{A}^{+} (resp. C_{A}^{−} ) is transitive.

c) The stabilizer of a point in C_{A} is a maximal compact subgroup of the Lie group U_{A,ℚ}(ℝ) .

Proof. This theorem is proved in Appendix.

8.3 Remark. Let A be an abelian variety. The action of U_{A,ℚ}(ℝ) on C_{A} restricts to an action of the Lie subgroup \overline{U(A)}(ℝ) . Since this subgroup consists (up to isogeny) of all noncompact factors of U_{A,ℚ}(ℝ) its action of C_{A}^{+} (resp. C_{A}^{−} ) is also transitive. Thus C_{A}^{+} (resp. C_{A}^{−} ) is a hermitian symmetric space for the semisimple Lie group \overline{U(A)}(ℝ).

8.4. Let A be a complex torus. As above, denote by NS_{A}(ℝ)^{0} ⊂ NS_{A}(ℝ) the subset consisting of nondegenerate forms. Assume that NS_{A}(ℝ)^{0} ≠ ∅ . Put

NS_{A}(ℂ)^{0} := {φ_{1} + iφ_{2} ∈ NS_{A}(ℂ) | φ_{2} ∈ NS_{A}(ℝ)^{0}}

Thus, NS_{A}(ℂ)^{0} is a nonempty Zariski open subset in NS_{A}(ℂ) . (Note that if A is an abelian variety then C_{A}^{+} (resp. C_{A}^{−} ) is a connected component of NS_{A}(ℂ)^{0}) .

Given ω = φ_{1} + iφ_{2} ∈ NS_{A}(ℂ)^{0} consider the following element

I_{ω} := (\begin{matrix}φ_{2}^{−1}φ_{1} & −φ_{2}^{−1} \\ φ_{2} + φ_{1}φ_{2}^{−1}φ_{1} & −φ_{1}φ_{2}^{−1} \end{matrix}) = (\begin{matrix}1 & 0 \\ φ_{1} & 1 \end{matrix}) (\begin{matrix}0 & −φ_{2}^{−1} \\ φ_{2} & 0 \end{matrix}) (\begin{matrix}1 & 0 \\ −φ_{1} & 1 \end{matrix}) ∈ U_{A,ℚ}(ℝ)  (14)

and the morphism of algebraic ℝ -groups

μ_{ω} : S^{1} ⟶ U_{A,ℚ}(ℝ)

given by the formula μ_{ω}(e^{iθ}) = cos θ · Id + sin θ · I_{ω}, so that μ_{ω}(e^{i \frac{π}{2}} ) = I_{ω} .

The correspondence ω ↦ I_{ω} is injective.

8.4.1 Properties of I_{ω} and μ_{ω} .

1. I\overline{ω} = −I_{ω} , hence μω̄ = μ_{ω} · (−Id_{S}^{1}) .

This is obvious.

2. I_{ω}^{2} = −Id .

This is a direct computation.

By K_{ω} ⊂ U_{A,ℚ}(ℝ) denote the stabilizer of ω with respect to the action (13), and Z(K_{ω}) denote its center. One checks directly that

K_{ω} = \{( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{A,ℚ}(ℝ) | \begin{matrix}φ_{2}a + φ_{1}bφ_{2} + φ_{2}bφ_{1} = dφ_{2}; \\ φ_{1}a + φ_{1}bφ_{1} − φ_{2}bφ_{2} = c + dφ_{1} \end{matrix} \}

3. μ_{ω}(S^{1}) ⊆ Z(K_{ω}) .

This is also a direct computation.

4. Z_{U_{A,ℚ}(ℝ)}(I_{ω}) = Z_{U_{A,ℚ}(ℝ)}(μ_{ω}(S^{1})) = K_{ω} .

The first equality is obvious and the second is a direct computation.

Let T_{ω}NS_{A}(ℂ) be the tangent space to NS_{A}(ℂ) at the point ω . Each element g ∈ K_{ω} defines a linear operator on T_{ω}NS_{A}(ℂ) .

5. μ_{ω}(e^{i \frac{π}{4}} ) defines the complex structure (i.e. multiplication by i ) on T_{ω}NS_{A}(ℂ) . Indeed, given γ + iδ ∈ NS_{A}(ℂ) it suffices to check that

lim_{ϵ∈ℝ,ϵ→0} \frac{1}{ϵ} [( \begin{matrix}1 + φ_{2}^{−1}φ_{1} & −φ_{2} \\ φ_{2} + φ_{1}φ_{2}^{−1}φ_{1} & 1 − φ_{1}φ_{2}^{−1} \end{matrix} ) (φ_{1} + iφ_{2} + ϵ(γ + iδ)) − (φ_{1} + iφ_{2})] = i(γ + iδ)

This is a straightforward computation, which we omit.

6. If A is an abelian variety and ω ∈ C_{A} , then Ad_{I_{ω}} is a Cartan involution in U_{A,ℚ}(ℝ) , i.e.

U_{A,ℚ}(ℝ)^{Ad_{Iω}} is a maximal compact subgroup K_{ω} of U_{A,ℚ}(ℝ) . Indeed, this follows from properties 2, 3, 4 above and from Theorem 8.2 c).

7. If A is an abelian variety, then μ_{ω}(S^{1}) ⊂ \overline{U(A)}(ℝ) (7.3).

Let us prove this. If ω′ = iφ_{2} , then μ_{ω} = g_{φ_{1}}μ_{ω′}g_{φ_{1}}^{−1} where g_{φ_{1}} = ( \begin{matrix}1 & 0 \\ φ_{1} & 1 \end{matrix} ) ∈ G_{φ_{1}} (7.3).

So we may assume that ω = iφ and hence

μ_{ω}(e^{iθ}) = cos θ ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) + sin θ ( \begin{matrix}0 & −φ^{−1} \\ φ & 0 \end{matrix} ) .

But then μ_{φ}(S^{1}) ⊂ G_{φ}. (7.3). □

9. Mirror symmetry.

Here we introduce the notion of mirror symmetry for abelian varieties and complex tori.

9.1 Definition. An algebraic pair (resp. a weak pair) (A, ω_{A}) consists of an abelian variety (resp. a complex torus) A and an element ω_{A} ∈ C_{A} (8.1) (resp. ω_{A} ∈ NS_{A}(ℂ)^{0} (8.4)).

Thus a algebraic pair is a special case of a weak pair.

Let (A, ω_{A}) be a weak pair. Recall the canonical symmetric bilinear form Q_{A} (3.1) on the lattice Γ_{A} ⊕ ΓÂ . The group Hdg_{A,ℚ}(ℝ) × U_{A,ℚ}(ℝ) acts on the space V_{A} ⊕ VÂ , preserving the form Q_{A,ℝ} . In fact U_{A,ℚ}(ℝ) is the centralizer of Hdg_{A,ℚ}(ℝ) in SO(V_{A} ⊕ VÂ, Q_{A,ℝ}) (Prop. 5.2.4). We have the elements

J_{A×Â} ∈ Hdg_{A,ℚ}(ℝ) = Hdg_{A×Â,ℚ}(ℝ), I_{ω_{A}} ∈ U_{A,ℚ}(ℝ)

(see 2.1, 8.4) which commute and define two complex structures on V_{A} ⊕ VÂ .

9.2 Definition. We say that algebraic pairs (resp. weak pairs) (A, ω_{A}), (B, ω_{B}) are mirror symmetric if there is an isomorphism

α : Γ_{A} ⊕ ΓÂ ⟶^{∼} Γ_{B} ⊕ ΓB̂

which identifies bilinear forms Q_{A} and Q_{B} and satisfies the following conditions

\begin{aligned}α_{ℝ} · J_{A×Â} & = & I_{ω_{B}} · α_{ℝ}, \\ α_{ℝ} · I_{ω_{A}} & = & J_{B×B̂} · α_{ℝ}. \end{aligned}

9.2.1 Remark. Let the weak pairs (A, ω_{A}) and (B, ω_{B}) be mirror symmetric. We will identify the lattices Γ_{A} ⊕ ΓÂ and Γ_{B} ⊕ ΓB̂ by means of α . That is we have

Γ_{A} ⊕ ΓÂ = Λ = Γ_{B} ⊕ ΓB̂

with the bilinear form Q_{A} = Q = Q_{B} .

Then the ℚ -algebraic groups Hdg_{A,ℚ}×U_{A,ℚ} and Hdg_{B,ℚ}×U_{B,ℚ} are subgroups of SO(Λ_{ℚ}, Q_{ℚ}) . By the definition of the Hodge group we have the following group inclusions

Hdg_{A,ℚ} ⊆ U_{B,ℚ}, Hdg_{B,ℚ} ⊆ U_{A,ℚ}.

If A and B are abelian varieties, then stronger inclusions hold (see 8.4.1 (7))

Hdg_{A,ℚ} ⊆ \overline{U(B)}, Hdg_{B,ℚ} ⊆ \overline{U(A)}.

9.2.2 Remark. Let (A, ω_{A}) be a weak pair. Assume that weak pairs (B, ω_{B}) and (C, ω_{C}) are mirror symmetric to (A, ω_{A}) . If B is an abelian variety, then so is C . Indeed, composing the α ’s for B and C we obtain an isomorphism of complex tori

B × B̂ ≅ C × Ĉ.

Thus C is an abelian variety (as a subtorus of an abelian variety).

9.2.3 Remark. Let (A, ω_{A}) be a weak pair. The collection of isomorphism classes of abelian varieties B for which there exists ω_{B} ∈ NS_{B}(ℂ)^{0} such that the pairs (A, ω_{A}) and (B, ω_{B}) are mirror symmetric, is finite. Indeed, fix one such B . Then for any other such C there exists an isomorphism of abelian varieties

B × B̂ ≅ C × Ĉ,

hence C is isomorphic to an abelian subvariety of B × B̂ . The finiteness follows from the following theorem of Lenstra, Oort and Zarhin.

Theorem.([LOZ]) Let A be an abelian variety. There are finitely many isomorphism classes of abelian varieties which admit an embedding as an abelian subvariety in A .

9.2.4 Proposition. Let (A, ω_{A}) and (B, ω_{B}) be two mirror symmetric weak pairs. Then the set of α ’s that establish a mirror symmetry between the pairs is a torsor over the stabilizer of ω_{A} in U(A) . In particular, if the pair (A, ω_{A}) is algebraic then this set is finite.

Proof. Let α_{1}, α_{2} be two isomorphisms that establish a mirror symmetry of (A, ω_{A}) and (B, ω_{B}). Then γ := α_{2}^{−1}α_{1} is an automorphism of the torus A × Â which preserves the form Q_{A} . Thus by Proposition 4.3.2 γ ∈ U(A). Also γI_{ω_{A}}γ^{−1} = I_{ω_{A}}. Therefore by property 4 in 8.4.1 γω_{A} = ω_{A} . This proves the first assertion. If ω_{A} ∈ C_{A} then its stabilizer in U_{A,ℚ}(ℝ) is compact (Theorem 8.2 c). This implies the second assertion since the group U(A) is discrete. □

9.2.5 Proposition. Two weak pairs (B, ω_{1}) and (B, ω_{2}) are both mirror symmetric to the same weak pair (A, ω_{A}) if and only if there exists g ∈ U(B) such that gω_{1} = ω_{2}.

Proof. ⇒ Similarly to the previous proof we find g ∈ U(B) such that gI_{ω_{1}}g^{−1} = I_{ω_{2}}. This means that gω_{1} = ω_{2}.

⇐ If α establishes a mirror symmetry of (A, ω_{A}) and (B, ω_{1}) , then gα induces a mirror symmetry of (A, ω_{A}) and (B, ω_{2}) . □

9.2.6 Proposition. Let A, B, C be abelian varieties.

a) If some algebraic pairs (B, ω_{B}), (C, ω_{C}) are both mirror symmetric to an algebraic pair (A, ω_{A}), then the derived categories D^{b}(B) and D^{b}(C) are equivalent.

b) If some algebraic pairs (B, ω_{B}) and (A, ω_{A}) are mirror symmetric and the derived categories D^{b}(B) and D^{b}(C) are equivalent, then there exists ω_{C} ∈ C_{C} such that the pairs (C, ω_{C}) and (A, ω_{A}) are mirror symmetric.

Proof. This follows immediately from the Theorem 4.3.10 and Proposition 9.4.3 below. □

9.3. Assume that weak pairs (A, ω_{A}), (B, ω_{B}) are mirror symmetric. Use the identification of 9.2.1. Consider the Clifford algebra Cl(Λ, Q) (3.1). The total cohomology groups H^{∗}(A, ℤ) and H^{∗}(B, ℤ) are naturally Cl(Λ, Q) –modules (3.7). There exists a unique (up to ±1 ) isomorphism of these Cl(Λ, Q) –modules

β : H^{∗}(A, ℤ) ⟶^{∼} H^{∗}(B, ℤ).

This isomorphism either preserves the even and odd cohomology groups or interchanges them (3.7). In the first case we called β even and in the second – odd.

9.3.1. Assume that the weak pair (A, ω_{A}) is mirror symmetric to yet another weak pair (C, ω_{C}) by an isomorphism

α′ : Γ_{A} ⊕ ΓÂ ⟶^{∼} Γ_{C} ⊕ ΓĈ.

Identify Γ_{C} ⊕ ΓĈ with Γ_{B} ⊕ ΓB̂ by means of α and α′ . We claim that dim(Γ_{B,ℚ} ∩ Γ_{C,ℚ}) is even and hence Γ_{B,ℚ} ∼ Γ_{C,ℚ} in the sense of 3.6.1. Indeed, both V_{B} and V_{C} are complex subspaces of Λ_{ℝ} with respect to the complex structure I_{ω_{A}} (8.4). Hence dim_{ℝ}(V_{B} ∩ V_{C}) is even.

9.3.2 Proposition. Let weak pairs (A, ω_{A}) and (B, ω_{B}) be mirror symmetric. Let n = dimA = dimB. Then the parity of β is equal to the parity of n .

Proof. Recall that Λ^{·}V_{A}^{∗} = H^{∗}(A, ℝ) and Λ^{·}V_{B}^{∗} = H^{∗}(B, ℝ) are both spinorial representations of the Lie algebra 𝔰𝔬(Λ_{ℝ}, Q_{ℝ}) , and β_{ℝ} : Λ^{·}V_{A}^{∗} ⟶^{∼} Λ^{·}V_{B}^{∗} is an 𝔰𝔬(Λ_{ℝ}, Q_{ℝ}) -morphism. Thus the parity of β_{ℝ} will not change if we replace V_{B} by a subspace gV_{B} for some g ∈ SO(Λ_{ℝ}, Q_{ℝ}) . Let ω_{A} = φ_{1} + iφ_{2} , where φ_{1} ∈ NS_{A}(ℝ) , φ_{2} ∈ NS_{A}(ℝ)^{0} . Choose g = g_{φ_{1}}^{−1} ∈ G_{φ_{1}} ⊂ SO(Λ_{ℝ}, Q_{ℝ}) as in 8.4.1 (7). Then gV_{B} is a subspace of Λ_{ℝ} which is preserved by the complex structure

I_{iφ_{2}} = ( \begin{matrix}o & −φ_{2}^{−1} \\ φ_{2} & 0 \end{matrix} ) .

It suffices to find a maximal isotropic subspace V ⊂ Λ_{ℝ} which is preserved by I_{iφ_{2}} (hence V ∼ gV_{B} ) and such that dim_{ℝ}(V ∩ V_{A}) = n . Let us construct such a subspace V .

Consider φ_{2} as a skew-symmetric form on V_{A} . There exists a basis e_{1}, ..., e_{n}, e_{−1}, ..., e_{−n} of V_{A} in which

φ_{2} = ( \begin{matrix}0 & −Δ \\ Δ & 0 \end{matrix} ) , Δ = ( \begin{matrix}δ_{1} \\ & δ_{2} \\ & & ... \\ & & & δ_{n} \end{matrix} ) .

Let e_{1}^{∗}, ..., e_{n}^{∗}, e_{−1}^{∗}, ..., e_{−n}^{∗} be the dual basis of VÂ . Then V :=< e_{1}, ..., e_{n}, e_{−1}^{∗}, ..., e_{−n}^{∗} > is clearly maximal isotropic subspace of Λ and I_{iφ_{2}} -invariant. This proves the proposition. □

9.3.3. Let A be an abelian variety. Recall that the total cohomology H^{∗}(A, ℚ) is a representation space of the ℚ –algebraic group Hdg_{A,ℚ} × \overline{Spin(A)} (7.1.2).

Proposition. Let algebraic pairs (A, ω_{A}) and (Bω_{B}) be mirror symmetric. As in 9.3 above consider the canonical (up to ± ) isomorphism of Cl(Λ, Q) –modules

β : H^{∗}(A, ℤ) ⟶^{∼} H^{∗}(B, ℤ).

Then β_{ℚ} induces the inclusions of algebraic groups

Hdg_{A,ℚ} ⊆ \overline{Spin(B)}, Hdg_{B,ℚ} ⊆ \overline{Spin(A)}.

Proof. Let us identify H^{∗}(A, ℤ) and H^{∗}(B, ℤ) by means of β. It suffices to prove one inclusion Hdg_{A,ℚ} ⊂ \overline{Spin(B)}.

We know that the action of Hdg_{A,ℚ} on H^{∗}(A, ℚ) = Λ^{·}Γ_{Â,ℚ} is induced from its action on Γ_{Â,ℚ} by functoriality. It suffices to show that this action agrees with the spinorial representation of Spin(Λ_{Q}, Q_{ℚ}) when Hdg_{A,ℚ} is considered as a subgroup of SO(Λ_{Q}, Q_{ℚ}). This is done by identifying the weights of the Lie algebra 𝔰𝔩(Γ_{A,ℚ}) ⊂ 𝔰𝔬(Λ_{ℚ}, Q_{ℚ}) in the spinorial representation and the natural representation on Λ^{·}Γ_{Â,ℚ}. □

9.4. Let (A, ω) be a weak pair. Let I ∈ U_{A,ℚ}(ℝ), I^{2} = −1. Then I and J_{A×Â} ∈ Hdg_{A,ℚ}(ℝ) are two complex structures on the space

Λ_{ℝ} = V_{A} ⊕ VÂ.

They commute and preserve the bilinear form Q_{A,ℝ} . Denote by c the product J_{A×Â}I . The operator c preserves the bilinear form Q_{A,ℝ} and c^{2} = 1 . Hence, the bilinear form Q_{A,ℝ}(c(·), ·) is symmetric. Denote by E_{A,I} the corresponding quadratic form. In case I = I_{ω} for some ω ∈ NS_{A}(ℂ)^{0} we denote E_{A,I_{ω}} = E_{A,ω}.

9.4.1 Lemma. Let A be a complex torus. Suppose I ∈ U_{A,ℚ}(ℝ), I^{2} = −1 and I has the form ( \begin{matrix}I_{11} & I_{12} \\ I_{21} & I_{22} \end{matrix} ) . The following conditions are equivalent:

1) I = I_{ω} for some ω ∈ NS_{A}(ℂ)^{0}.
2) I_{12} : VÂ ⟶ V_{A} is invertible.
3) The restriction of bilinear form Q_{A,ℝ}(c·, ·) on VÂ is non-degenerate.

Proof. 1) ⇒ 2) This immediately follows from the formula (14) for I_{ω} in 8.4.

2) ⇔ 3) Let x_{1}, x_{2} ∈ VÂ . We have

Q_{A,ℝ}(c(0, x_{1}), (0, x_{2})) = x_{2}(J_{A}I_{12}(x_{1})).

Thus, the restriction of Q_{A,ℝ}(c(·), ·) to VÂ is non-degenerate iff I_{12} is invertible.

2) ⇒ 1) Let I_{12} is invertible. Since I^{2} = −1 and I ∈ U_{A,ℚ}(ℝ) we have equalities

Î_{12} = I_{12}, Î_{21} = I_{21}, Î_{11} = −I_{22}

Put φ_{2} = −I_{12}^{−1} and φ_{1} = I_{22}I_{12}^{−1} . Take ω = φ_{1} + iφ_{2} . It is easy to see that I_{ω} = I . □

9.4.2 Lemma. Let A be a complex torus and let I ∈ U_{A,ℚ}(ℝ), I^{2} = −1. Then I coincides with I_{ω} for some ω ∈ C_{A}^{+} ( resp. C_{A}^{−} ) and, consequently, A is an abelian variety iff the quadratic form E_{A,I} is positive (resp. negative) definite.

Proof. ⇒ Suppose I = I_{ω} with ω = φ_{1} + iφ_{2} ∈ NS_{A}(ℂ)^{0} . It is not hard to check that I_{ω} = g_{φ_{1}}I_{iφ_{2}}g_{φ_{1}}^{−1} , where g_{φ_{1}} = ( \begin{matrix}1 & 0 \\ φ_{1} & 1 \end{matrix} ) . Therefore, we have an equality

E_{A,I}(λ) = Q_{A,ℝ}(cλ, λ) = Q_{A,ℝ}(I_{iφ_{2}}J_{A}(λ′), λ′),

where λ′ = g_{φ_{1}}^{−1}(λ) .

Taking λ′ = (l, x) with l ∈ V_{A} and x ∈ VÂ , we obtain

E_{A,I}(λ) = Q_{A,ℝ}((−φ_{2}^{−1}(JÂx), φ_{2}(J_{A}l)), (l, x)) = φ_{2}(J_{A}l, l) − x(φ_{2}^{−1}(JÂx)) = φ_{2}(J_{A}l, l) − x(J_{A}φ_{2}^{−1}(x)).

Denoting φ_{2}^{−1}(x) by m in the last expression, we get

E_{A,I}(λ) = −φ_{2}(m, J_{A}m) + φ_{2}(J_{A}l, l) = φ_{2}(J_{A}m, m) + φ_{2}(J_{A}l, l)

This shows that the quadratic form E_{A,J} is positive (resp. negative) definite on Λ_{A,ℝ} iff the form φ_{2}(J_{A}·, ·) is positive (resp. negative) definite on V_{A} , which is equivalent to ω ∈ C_{A}^{+} (resp. ω ∈ C_{A}^{−} ).

⇐ Assume that the form E_{A,I} is definite. Let I = ( \begin{matrix}I_{11} & I_{12} \\ I_{21} & I_{22} \end{matrix} ) . If I_{12} is degenerate , then there is 0 ≠ x ∈ VÂ such that I_{12}(x) = 0 . Hence

E_{A,I}((0, x)) = Q_{A,ℝ}((J_{A}I_{12}(x), JÂI_{22}(x)), (0, x)) = Q_{A,ℝ}((0, x′), (0, x)) = 0

This contradicts the definitness of E_{A,I} . By Lemma 9.4.1, as I_{12} is invertible, I = I_{ω} for some ω ∈ NS_{A}(ℂ)^{0} . Then the previous argument shows that ω belongs to C_{A} . Lemma is proved. □

9.4.3 Proposition. Let weak pairs (A, ω_{A}), (B, ω_{B}) be mirror symmetric. Suppose that the weak pair (A, ω_{A}) is in fact an algebraic pair. Then (B, ω_{B}) is an algebraic pair too.

Proof. Since (A, ω_{A}) is an algebraic pair we have ω_{A} ∈ C_{A} . Hence, by Lemma 9.4.2, the quadratic form E_{A,ω_{A}} (9.4) is definite. Since the weak pairs (A, ω_{A}) and (B, ω_{B}) are mirror symmetric the quadratic forms E_{A,ω_{A}} and E_{B,ω_{B}} coincide under an identification α . Therefore, the quadratic form E_{B,ω_{B}} is definite. Lemma 9.4.2 implies that ω_{B} ∈ C_{B} . Thus B is an abelian variety and the weak pair (B, ω_{B}) is an algebraic pair. □

9.4.4 Lemma. Let (A, ω_{A}) be an algebraic pair and B be a complex torus. Let α : Γ_{A} ⊕ ΓÂ ⟶^{∼} Γ_{B} ⊕ ΓB̂ be an isomorphism which identifies the forms Q_{A} and Q_{B} and α_{ℝ} · I_{ω_{A}} = J_{B×B̂} · α_{ℝ}. Then there exists ω_{B} ∈ C_{B} (in particular B is an abelian variety) such that α_{ℝ} · J_{A×Â} = I_{ω_{B}} · α_{ℝ}. That is α establishes the mirror symmetry of algebraic pairs (A, ω_{A}), (B, ω_{B}) .

Proof. Since (A, ω_{A}) is an algebraic pair the form E_{A,ω_{A}} on V_{A} ⊕ VÂ is definite (9.4.2). Identify the spaces V_{A} ⊕ VÂ and V_{B} ⊕ VB̂ with the forms Q_{A,ℝ} and Q_{B,ℝ} by means of α_{ℝ} . Thus

I := J_{A×Â} ∈ Hdg_{A,ℚ}(ℝ)

and

I_{ω_{A}} = J_{B×B̂} ∈ Hdg_{B,ℚ}(ℝ).

Since the ℚ -closures of J_{A×Â} and I_{ω_{A}} commute (5.2,4)) we find that I ∈ U_{B,ℚ}(ℝ) (5.2,4)). Thus we may apply Lemma 9.4.2 to I and B . Note that α_{ℝ} identifies the quadratic forms E_{A,ω_{A}} and E_{B,I} . Since the first one is definite it follows from Lemma 9.4.2 that I = I_{ω} for some ω = ω_{B} ∈ C_{B} . This proves the lemma. □

9.4.5 Lemma. Let Λ = Γ ⊕ Γ^{∗} be a lattice with canonical symmetric bilinear form Q (3.1). Let there be given an isotropic decomposition Λ = Λ_{1} ⊕ Λ_{2} . Suppose I ∈ GL(Λ_{ℝ}) is a complex structure on Λ_{ℝ} , i.e. I^{2} = −1 , that satisfies the following assumptions

\begin{matrix}a) & I(Λ_{i,ℝ}) = Λ_{i,ℝ}, i = 1, 2 \\ b) & I ∈ O(Λ_{ℝ}, Q_{ℝ}) \end{matrix}

Then the complex tori

B_{1} := (Λ_{1,ℝ}/Λ_{1}, I_{1}), B_{2} := (Λ_{2,ℝ}/Λ_{2}, I_{2}),

where I_{i} are the restriction of I on Λ_{i,ℝ} , are dual to each other, i.e. B_{2} ≅ B̂_{1} .

Proof. The form Q induces the isomorphism t : Λ_{2} ⟶ Λ_{1}^{∗} that takes λ_{2} ∈ Λ_{2} to linear functional Q(λ_{2}, ·) . It remains to show that for any v_{1} ∈ Λ_{1,ℝ}, v_{2} ∈ Λ_{2,ℝ} the following equality holds

t(I_{2}(v_{2}))(v_{1}) = t(v_{2})(−I_{1}(v_{1}))

(3.1). This is straightforward using the inclusion J ∈ O(Λ_{ℝ}, Q_{ℝ}) .

t(I_{2}(v_{2}))(v_{1}) = Q_{ℝ}(I_{2}(v_{2}), v_{1}) = Q_{ℝ}(I_{2}^{2}(v_{2}), I_{1}(v_{1})) = Q_{ℝ}(−v_{2}, I_{1}(v_{1})) = t(v_{2})(−I_{1}(v_{1})).

Lemma is proved. □

9.4.6 Summary. Let (A, ω_{A}) be an algebraic pair. The last two lemmas give us a “method” for construction of a mirror symmetric pair. Namely, put Λ = Γ_{A} ⊕ΓÂ . It suffices to find a Q_{A} –isotropic decomposition Λ = Λ_{1} ⊕ Λ_{2} such that Λ_{1ℝ} , Λ_{2ℝ} are preserved by I_{ω_{A}} . Then by Lemma 9.4.5 the tori B := (Λ_{1ℝ}/Λ_{1}, I_{ω_{A}}) and B̂ := (Λ_{2ℝ}/Λ_{2}, I_{ω_{A}}) are dual. Moreover, the natural identification

α = id : Γ_{A} ⊕ ΓÂ ⟶^{∼} Γ_{B} ⊕ ΓB̂

( Γ_{B} = Λ_{1} , ΓB̂ = Λ_{2} ) identifies the forms Q_{A} and Q_{B} . Thus by Lemma 9.4.4 there exists ω_{B} ∈ C_{B} such that the pairs (A, ω_{A}) , (B, ω_{B}) are mirror symmetric.

9.5. It is not true that for every pair (A, ω_{A}) there exists a mirror symmetric pair (B, ω_{B}) . The problem may occur if the group U_{A,ℚ} is too big and ω_{A} ∈ C_{A} is chosen too general.

9.5.1 Counterexample. Let E be an elliptic curve with complex multiplication. We have e_{0} = 1, d = 1 (1.8) and U_{E×E,ℚ}(ℝ) = U(2, 2) (5.3.2). Therefore this group acts irreducibly on V_{E×E} ⊕ V{ \atop E×E} . Assume that for all ω ∈ C_{E×E} there exists a mirror symmetric pair (B, ω′) , i.e. there exists α as in 9.2 with the corresponding properties. Let us use α to identify

Γ_{E×E} ⊕ Γ{ \atop E×E} = Γ_{B} ⊕ ΓB̂.

The Hodge group Hdg_{B,ℚ} is the ℚ –closure of the compact torus μ_{ω}(S^{1}) ⊂ Aut(V_{B} ⊕ VB̂) . For a general ω this ℚ –closure is the group SU(2, 2) ⊂ U_{E×E,ℚ}(ℝ) . But the group Hdg_{B,ℚ}(ℝ) acts faithfully on V_{B} ⊕ VB̂ and preserves each summand. This is a contradiction.

9.6. However, each torus A with a nonempty NS_{A}^{0} has an ω_{A} ∈ NS_{A}(ℂ)^{0} such that the pair (A, ω_{A}) has a mirror symmetric pair. In particular for every abelian variety A there exists ω_{A} ∈ NS_{A}(ℂ)^{0} (even ω_{A} ∈ C_{A} ) such that the pair (A, ω_{A}) has a mirror symmetric one. This follows from the next proposition.

9.6.1 Proposition. Let A be a complex torus of dimension n . Let φ ∈ NS_{A}^{0} , i.e. φ ∈ Hom(A, Â) is an isogeny. Let τ = a+ib ∈ ℂ , b ≠ 0 . Consider the element ω_{A} := τφ ∈ NS_{A}(ℂ)^{0} and the weak pair (A, ω_{A}) . Then there exist isogeneous elliptic curves E_{1}, ..., E_{n} and an element ω_{E} ∈ NS_{E}(ℂ)^{0} , where E = E_{1} × ... × E_{n} , such that the weak pair (E, ω_{E}) is mirror symmetric to the weak pair (A, ω_{A}) .

Proof. There exists a basis e_{1}, ..., e_{n}, e_{−1}, ..., e_{−n} of Γ_{A} in which the bilinear form φ has a matrix

( \begin{matrix}0 & Δ \\ −Δ & 0 \end{matrix} ) , where Δ = ( \begin{matrix}δ_{1} & & 0 \\ & ... \\ 0 & & δ_{n} \end{matrix} ) , δ_{i} ∈ ℤ, δ_{1}|δ_{2}|δ_{3}...

Let e_{1}^{∗}, ...e_{n}^{∗}, e_{−1}^{∗}, ..., e_{−n}^{∗} be the dual basis of ΓÂ. Then the map φ : Γ_{A} ⟶ ΓÂ is

φ : e_{i} ↦ δ_{i}e_{−i}^{∗}, e_{−i} ↦ −δ_{i}e_{i}^{∗}.

Put

Γ_{i} := ℤe_{i} ⊕ ℤe_{−i}^{∗}, Γ_{i}^{∗} := ℤe_{−i} ⊕ ℤe_{i}^{∗}, i = 1, ..., n.

Clearly the subgroups Λ_{1} := ⊕Γ_{i} , Λ_{2} := ⊕Γ_{i}^{∗} are isotropic. We have

I_{ω_{A}} = ( \begin{matrix}b^{−1}a & −b^{−1}φ^{−1} \\ (b + ab^{−1}a)φ & −ab^{−1} \end{matrix} ) .

This operator preserves each subspace Γ_{iℝ}, Γ_{iℝ}^{∗} . In particular it preserves Λ_{1,ℝ}, Λ_{2,ℝ} . Thus by Lemma 9.4.5 the tori (Λ_{1,ℝ}/Λ_{1}, I_{ω_{A}}), (Λ_{2,ℝ}/Λ_{2}, I_{ω_{A}}) are dual. Moreover, it is clear that the torus (Λ_{1,ℝ}/Λ_{1}, I_{ω_{A}}) is a product of elliptic curves E_{i} := (Γ_{iℝ}/Γ_{i}, I_{ω}), i = 1, ..., n .

For each i the map

Γ_{1} ⟶ Γ_{i}, e_{1} ↦ e_{i}, e_{−1}^{∗} ↦ δ_{1}^{−1}δ_{i}e_{−i}^{∗}

commutes with I_{ω_{A}} and so is an isogeny of elliptic curves E_{1} and E_{i} .

If the weak pair (A, ω_{A}) were an algebraic pair, then by Lemma 9.4.4 there would exist an element ω_{E} ∈ C_{E} such that the pairs (A, ω_{A}) and (E, ω_{E}) are mirror symmetric. Thus we are done in the algebraic case. In general there may not exist the desired ω_{E} ∈ NS_{E}(ℂ)^{0} , so it may be necessary to choose a different decomposition Λ = Λ_{1} ⊕ Λ_{2}.

By Lemma 9.4.1 in order for ω_{E} to exist the symmetric bilinear form Q_{ℝ}(I_{ω_{A}}J_{A×Â}(·), ·) must be nondegenerate on Λ_{2,ℝ} . This is equivalent to the statement that J_{A×Â}Λ_{2,ℝ} ∩ Λ_{2,ℝ} = 0 . Put W := ⊕ℝe_{−i}. Then since J_{A×Â} preserves the form Q_{ℝ} the last equality is equivalent to

J_{A×Â}W ∩ W = 0. (∗)

We are free to apply elements of the symplectic group Sp(Γ_{A}, φ; ℤ) to the sublattice ⊕ℤe_{−i} to achieve the transversality condition (*) above. Since this discrete group is Zariski dense in the corresponding group of ℝ –points Sp(V_{A}, φ; ℝ) and since Sp(V_{A}, φ; ℝ) acts transitively on the collection of maximal φ –isotropic subspaces of V_{A} it suffices to prove the following lemma

9.6.2 Lemma. Let V ≅ ℝ^{2n} be a real vector space with a nondegenerate symplectic form φ . Let J ∈ End(V) be a complex structure on V , which preserves φ . Then there exists a maximal φ – isotropic subspace W ⊂ V such that

JW ∩ W = 0.

Proof. Note that the form φ(J·, ·) on V is symmetric and nondegenerate.

We will choose elements x_{1}, ..., x_{n} ∈ V such that W := ⊕ℝx_{i} has the desired properties by induction on i . Choose x_{1} such that φ(Jx_{1}, x_{1}) ≠ 0. Let V =< x_{1}, Jx_{1} > ⊕V′ be a φ - orthogonal decomposition. Note that V′ is J –invariant. Hence we may replace V by V′ and choose x_{2} ∈ V′ s.t. φ(Jx_{2}, x_{2}) ≠ 0. And so on. Cleary the resulting space W = ⊕ℝx_{i} is maximal φ -isotropic and JW ∩ W = 0. This proves the lemma and the Proposition 9.6.1. □

9.6.3 Corollary. Let A be an abelian variety with NS(A) ≅ ℤ. Then C_{A} = NS_{A}(ℂ)^{0} and for any ω_{A} ∈ C_{A} the algebraic pair (A, ω_{A}) has a mirror symmetric one (E, ω_{E}) which is like in Proposition 9.6.1 above.

Proof. The first assertion is obvious. The second is a direct consequence of Proposition 9.6.1. □

10. The G–construction and its application to mirror symmetry.

10.1. Let A be an abelian variety. The ring R = End(A) is finite over ℤ such that D := End^{0}(A) = R ⊗_{ℤ} ℚ is semisimple with a positive definite anti-involution ∗ : D → D^{op} . The lattice Γ = H_{1}(A, ℤ) carries a natural structure of a faithful left R -module.

For a given D with an anti-involution ∗ the construction by Gerritzen [5] gives an abelian variety X with End^{0}(X) ≅ D . Furthermore, it was proved in [22] that any finite dimensional algebra over ℚ can be realized as End^{0}(T) for some complex torus T .

Now we are going to describe these constructions with some corrections and modifications related to the specifics of our situation being that we work with R and not with D .

10.1.1. Let R be an order in ℚ -algebra D of finite dimension over ℚ , i.e. D = R⊗_{ℤ} ℚ . Denote by Γ a left R -module that satisfies the following conditions:

1. Γ is a lattice of even dimension, i.e. Γ = ℤ^{2n} as abelian group.
2. Γ is a faithful module, i.e. the homomorphism R ⟶ End_{ℤ}(Γ) is an embedding.
3. The order O(Γ) := { a ∈ D | a(Γ) ⊂ Γ } coincides with R.
4. Γ is a direct sum of R-modules Γ_{1} and Γ_{2} such that there exists an isomorphism

e : M_{1} →^{∼} M_{2} of D–modules M_{i} = Γ_{i} ⊗_{ℤ} ℚ, (i = 1, 2).

(15)

Let us fix an isomorphism e and, henceforth, we will often identify D –modules M_{1} and M_{2} with respect to e and will use notation M .

Denote by C the algebra End_{D}(M) . The algebras D and C are two subalgebras of End_{ℚ}(M) ≅ M(n, ℚ) . Moreover C is a centralizer of D in M(n, ℚ) . On the other side, it is true that the centralizer of C contains D but does not necessarily coincide with D .

Let J be an element of M(2, C ⊗ ℝ) = (End_{D}M^{⊕2}) ⊗_{ℚ} ℝ such that J^{2} = −Id . Any such J defines a complex structure on the real space Γ_{ℝ} = (M_{1} ⊕M_{2})_{ℝ} and, as consequence, defines a complex torus

A_{J} := (Γ_{ℝ}/Γ, J).  (16)

10.1.2 Lemma. For any complex torus A_{J} , defined above, the endomorphism ring End(A_{J}) contains R as subring. Proof. It is clear, because End(A_{J}) coincides with maximal subring of End_{ℤ}(Γ) that commutes with J . And each r ∈ R ⊂ D commutes with any element from M(2, C ⊗ ℝ) . □

10.1.3. The set of operators J ∈ M(2, C ⊗ ℝ) with J^{2} = −Id is not empty. Actually, any element q = q_{1} + iq_{2} ∈ C ⊗_{ℚ} ℂ with nondegenerate q_{2} defines the operator

J(q) = (\begin{matrix}q_{2}^{−1}q_{1} & −q_{2}^{−1} \\ q_{2} + q_{1}q_{2}^{−1}q_{1} & −q_{1}q_{2}^{−1} \end{matrix}) = (\begin{matrix}1 & 0 \\ q_{1} & 1 \end{matrix}) (\begin{matrix}0 & −q_{2}^{−1} \\ q_{2} & 0 \end{matrix}) (\begin{matrix}1 & 0 \\ −q_{1} & 1 \end{matrix})  (17)

It is easy to check that J(q)^{2} = −Id .

10.1.4 Proposition. Suppose that the centralizer of C in M(n, ℚ) coincides with D . Then there exists q ∈ C ⊗ ℂ such that End(A_{J(q)}) = R .

Proof. Let {e_{1} = 1, ..., e_{t}} be a basis of ℚ -vector space C . Set q_{1} = 0 and q_{2} = ∑_{i=1}^{t} r_{i}e_{i} , where

r_{i} ∈ ℝ .

Let us describe the algebra End^{0}(A_{J(q)}) . It consists of elements of End_{ℚ}(M^{⊕2}) that commute with J . Each such element is a 2 × 2 matrix that satisfies the condition

(\begin{matrix}0 & q_{2}^{−1} \\ −q_{2} & 0 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) (\begin{matrix}0 & −q_{2}^{−1} \\ q_{2} & 0 \end{matrix}) = (\begin{matrix}q_{2}^{−1}dq_{2} & −q_{2}^{−1}cq_{2}^{−1} \\ −q_{2}bq_{2} & q_{2}aq_{2}^{−1} \end{matrix}) = (\begin{matrix}a & b \\ c & d \end{matrix})

This implies that c = −q_{2}bq_{2} and q_{2}a = dq_{2} .

Let us take q_{2} such that (r_{1}, ..., r_{t}) are algebraic independent over ℚ . For such q_{2} we obtain c = 0 = b and ae_{i} = e_{i}d for all i . As e_{1} = 1 we have a = d and a commutes with any element of C . It follows that a belongs to centralizer of C which coincides with D . Thus for general q_{2} the algebra End^{0}(A(q)) is isomorphic to D .

Now, the algebra End(A_{J(q)}) is an order in End^{0}(A_{J(q)}) = D that takes the lattice Γ to itself. By condition 3) of (15) it coincides with R . □

10.1.5 Remark. Note that if Γ_{i} ≅ R as left R -modules for i = 1, 2 then the lattice Γ = Γ_{1} ⊕ Γ_{2} satisfies the condition (15). In this case C ≅ D^{op} and, in addition, the centralizer of C coincides with D . This way by Proposition 10.1.4 for any R there exists a complex torus with R as the endomorphism ring.

10.2. Let Γ be a lattice as in (15). Suppose that, in addition, Γ satisfies the following extra condition:

5. There is a rational symmetric form g : M × M → ℚ such that the algebra D

is invariant with respect to an anti-involution ∼ on End_{ℚ}(M), defined by the rule

α̃ = g^{−1}α^{t}g. (18)

Consider a rational skew-symmetric form s on Γ_{ℚ} given as

s⟨(m_{1}, m_{2}), (n_{1}, n_{2})⟩ = g(m_{2}, e(n_{1})) − g(e(m_{1}), n_{2})  (19)

Both subspaces M_{1} = Γ_{1} ⊗ ℚ and M_{2} = Γ_{2} ⊗ ℚ are isotropic with respect to s . Since C is a centralizer of D in End_{ℚ}(M) it is invariant with respect to ∼ too. Denote by S(D) and S(C) the subspaces of symmetric elements α̃ = α of D and C respectively.

Take an element q = q_{1} + iq_{2} ∈ S(C) ⊗ ℂ with nondegenerate q_{2} and consider the complex structure J(q) defined by the formula (17). Since q is symmetric element a computation shows that the form s_{ℝ} is J(q) -invariant. A some multiple of s is integral. Hence it determines a line bundle on the complex torus A_{J(q)} = (Γ_{ℝ}/Γ, J(q)) . Further, any element d ∈ S(D) defines the rational form

s⟨(dm_{1}, dm_{2}), (n_{1}, n_{2})⟩

that is skew-symmetric. This correspondence gives the embedding i_{s} : S(D) ↪ NS_{A}(ℚ) .

10.2.1 Definition. We say that a weak pair (A_{J(q)}, ω) is obtained by G–construction if the complex torus A_{J(q)} is (Γ_{ℝ}/Γ, J(q)) , where Γ satisfies contions (15) and (18), q ∈ S(C) ⊗ ℂ , and the form ω belongs to i_{s}(S(D)) ⊗ ℂ .

Now we show that under some assumption we can guarantee the existence of such a form s and moreover the complex torus A_{J(q)} would be an abelian variety.

10.2.2 Lemma. Let the algebra D be semi-simple with a positive definite anti-involution ∗ : D → D^{op} . Then for any D -module M there exists a positive definite symmetric form g : M × M ⟶ ℚ such that the anti-involution ∼ on End_{ℚ}(M) , defined by the rule ã = g^{−1}a^{t}g , under the restriction on D coincides with ∗ . Proof. Actually, since D is semi-simple each module is projective. Hence M can be considered as submodule of left free module D^{⊕k} for some natural number k . Let us define a bilinear form u : D^{⊕k} × D^{⊕k} ⟶ ℚ by formula:

u(x, y) := ∑_{i=1}^{k} [T r(x_{i}^{∗}y_{i}) + T r(y_{i}^{∗}x_{i})] where x = (x_{1}, ..., x_{k}), y = (y_{1}, ..., y_{k}) ∈ D^{⊕k}

The bilinear form u is symmetric and positive definite. Denote by g the restriction of u on M . It is a positive definite symmetric bilinear form on M . For any endomorphism a ∈ End_{ℚ}(M) there exists an adjoint with respect to g endomorphism α̃ given by rule

g(m, a(n)) = g(ã(m), n), i.e. ã = g^{−1}a^{t}g

It is clear that ∼ is anti-involution on End_{ℚ}(W) and d̃= d^{∗} for any d ∈ D . □

10.2.3 Proposition. Let a lattice Γ be as in (15) and the algebra D be a semi-simple with a positive definite anti-involution ∗ . Then there exists a neighbourhood W ⊂ S(C) ⊗ ℂ of the point i such that for any q ∈ W the complex torus A_{J(q)} is an abelian variety.

If, in addition, the algebra C is symmetrically generated (i.e. is generated by the subspace S(C) ). Then there exists q ∈ W such that End(A_{J(q)}) = R . Proof. By Lemma 1.1.2 there is a positive definite symmetric form g on M . consider the skewsymmetric form s defined as in (19). The form s_{ℝ} on M_{ℝ}^{⊕2} is invariant under the action of J(q) , and, consequently, a some multiple of s gives a line bundle on A_{J(q)} . If this line bundle is ample, then A_{J(q)} is an abelian variety. The ampleness is equivalent to positive definiteness of the symmetric form s_{ℝ}⟨J(q)(·), ·⟩ . For q_{1} = 0 and q_{2} = 1 we have

s_{ℝ}⟨J(q)(m_{1}, m_{2}), (n_{1}, n_{2})⟩ = s_{ℝ}⟨(−m_{2}, m_{1}), (n_{1}, n_{2})⟩ = g(m_{1}, n_{1}) + g(m_{2}, n_{2})

Hence the positive definiteness of g implies the positive definiteness of the form s_{ℝ}⟨J(q)(·), ·⟩ for q = i and, consequently, for any q from some neighbourhood W of i .

Further, since D is a semi-simple algebra the cenralizer of C coincides with D . By assumption C is generated by S(C) and , consequently, D is the centralizer of S(C) . Now the existence of q for which EndA_{J(q)} = R is proved the same way as in Proposition 10.1.4. □

10.2.4 Remark. Note that if the endomorphism ring EndA_{J(q)} coincides with R then the embedding i_{s} : S(D) ↪ NS_{A_{J}} (ℚ) is an isomorphism.

10.3. Now we are going to give an alternative description of weak pairs which are obtained by the G–construction.

10.3.1 Definition. We say that a weak pair (A, ω) is well-becoming, if there is a decomposition Γ = H_{1}(A, ℤ) = Γ_{1} ⊕ Γ_{2} such that the following conditions hold:

1. The subspaces Γ_{kℝ}(k = 1, 2) are isotropic with respect to ω = φ_{1} + iφ_{2} , i.e. ω |_{Γ_{kℝ}}≡ 0 .

2. Maps J_{21} : Γ_{1ℝ} → Γ_{2ℝ} and J_{12} : Γ_{2ℝ} → Γ_{1ℝ} , which are the components of the complex structure J = ( \begin{matrix}J_{11} & J_{12} \\ J_{21} & J_{22} \end{matrix} ) on A , are invertible.

10.3.2 Remark. The second condition of the previous definition is equivalent to saying that the restrictions of holomorphic n -form to the tori Γ_{kℝ}/Γ_{k} are non–zero. Moreover, these tori are special Lagragian submanifolds of A .

The following technical results are needed in the sequel.

10.3.3 Definition. Let M be a vector space over ℚ and let r_{1}, …, r_{k} be elements of real space M_{ℝ} . The minimal vector subspace W ⊂ M sucht that r_{1}, …, r_{k} ∈ W_{ℝ} is called ℚ –envelope of the set {r_{1}, …, r_{k}} .

10.3.4 Lemma. Let (A, ω) be a weak pair. Suppose there is a decomposition Γ = H_{1}(A, ℤ) = Γ_{1}⊕Γ_{2} such that the subspaces Γ_{kℝ} are ω –isotropic and the map J_{12} : Γ_{2ℝ} ⟶ Γ_{1ℝ} is invertible. Then the pair (A, ω) is well-becoming. Proof. Consider the subspace U ∈ Hom_{ℚ}(Γ_{1ℚ}, Γ_{2ℚ}) consisting of such maps that satisfy the condition s′⟨l, e(m)⟩ = s′⟨m, e(l)⟩ with l, m ∈ Γ_{1ℚ} for any s′ ∈ NS_{A} ⊗ ℚ . One can see that J_{12}^{−1} belongs to U_{ℝ} . Hence, there is invertible e ∈ U . Fix one such e .

Take a sublattice Γ′_{1} = (γ_{1}, Ne(γ_{1})) for sufficient large N , where γ_{1} ∈ Γ_{1} . Consider the decomposition Γ = Γ′_{1} ⊕ Γ_{2} . It easy to check that Γ′_{1ℝ} is isotropic with respect to ω . In addition, maps J′_{21} : Γ′_{1ℝ} → Γ_{2ℝ} and J′_{12} : Γ_{2ℝ} → Γ′_{1ℝ} will be invertible. Hence the pair (A_{J(q)}, ω) is well-becoming. □

10.3.5 Proposition. The weak pair (A, ω) is well-becoming if and only if it can be obtained by the G–construction. Proof. ⇐ Suppose that a weak pair (A_{J(q)}, ω) is obtained by the G–construction. The lattice H_{1}(A_{J(q)}, ℤ) = Γ = Γ_{1} ⊕ Γ_{2} satisfies the conditions (15). Moreover, there is a skew-symmetric form s with isotropic sublattices Γ_{i} for i = 1, 2 . This imples that Γ_{iℚ} are isotropic with respect to any element from i_{s}(S(D)) . As ω belongs to i_{s}(S(D)) ⊗ ℂ , the real spaces Γ_{iℝ} are isotropic with respect to ω . Further, it follows from the formula (17) for J(q) that the map J_{21} is invertible. Hence, by Lemma 10.3.4 the pair (A_{J(q)}, ω) is well-becoming.

⇒ Set ω = φ_{1} + φ_{2} . Denote by W the ℚ -envelope of {φ_{1}, φ_{2}} . Let D ⊂ End(Γ_{ℚ}) be a ℚ -algebra that is generated by all elements of the form s_{2}^{−1}s_{1} with s_{1}, s_{2} ∈ W . Any element s_{2}^{−1}s_{1} sends the space Γ_{iℚ} to itself, because Γ_{iℚ} are maximal isotropic with respect to any nondegenerate s_{2} ∈ W . Therefore the rational spaces Γ_{iℚ} are modules over D . Take some integral element s ∈ W ⋂ NS_{A}^{0} . Define an anti-involution ∼ on D by rule d̃ = s^{−1}d^{t}s . It is clear that an element s^{−1}s′ belongs to the subspace of symmetric elements S(D) for any s′ . As in previous lemma let us consider the subspace U ∈ Hom_{ℚ}(Γ_{1ℚ}, Γ_{2ℚ}) consisting of such maps that satisfy the condition s′⟨l, e(m)⟩ = s′⟨l, e(m)⟩ with l, m ∈ Γ_{1ℚ} for any s′ ∈ NS_{A} ⊗ ℚ . Since J_{12}^{−1} belongs to U_{ℝ} , there is an invertible e ∈ U . It follows from definition of D that e gives an isomorphism between D -modules Γ_{1ℚ} and Γ_{2ℚ} .

Let R be a maximal order in D that takes Γ to itself. It follows from above that Γ = Γ_{1} ⊕ Γ_{2} satisfies the conditions (15). By construction, Γ_{1} and Γ_{2} are isotropic by means of s . Identifying Γ_{1ℚ} and Γ_{2ℚ} with respect to e , we can consider J_{A} as element of M(2, C ⊗ ℝ) , where C is a centralizer of D . In addition, J_{A} preserves the form s . Let it has a form ( \begin{matrix}J_{11} & J_{12} \\ J_{21} & J_{22} \end{matrix} ) with respect to decomposition Γ = Γ_{1} ⊕ Γ_{2} . As in the proof of Lemma 9.4.1, we have

J̃_{21} = J_{21}, J̃_{12} = J_{12}, J̃_{11} = −J_{22}.

Put q_{2} = −e^{−1}J_{12}^{−1} and q_{1} = e^{−1}J_{22}J_{12}^{−1} . It is easy to see that q = q_{1} + iq_{2} belongs to S(C) ⊗ ℂ and J_{A} = J(q) , where J(q) is defined by the formula (17).

Finally, any ω′ ∈ W_{ℂ} belongs to i_{s}(S(D))⊗ℂ ⊆ NS_{A}(ℂ) because W ⊆ i_{s}(S(D)) . Thus (A, ω′) is obtained by G–consruction. Proposition is proved. □

We claim that any well-becoming pair has a mirror symmetric pair. First, let us give a construction.

10.4. Construction. Let (A_{J}, ω) be a well-becoming pair. As above by Λ denote H_{1}(A_{J} ×Â_{J}, ℤ) . There is a decomposition

Λ = Γ ⊕ Γ^{∗} = Γ_{1} ⊕ Γ_{2} ⊕ Γ_{1}^{∗} ⊕ Γ_{2}^{∗}.

The form ω defines an element I_{ω} ∈ SO_{ℚ}(Q, ℝ) by formula (14), where Q is canonical symmetric bilinear form on Λ as in (3.1). By definition the subspaces Γ_{k,ℝ} are isotropic with respect to a ω . This implies that (Γ_{1}^{∗}⊕Γ_{2})_{ℝ} and (Γ_{1}⊕Γ_{2}^{∗})_{ℝ} are the I_{ω} –invariant subspaces. By I and I′ denote the restriction of the linear operator I_{ω} on the subspaces (Γ_{1}^{∗} ⊕ Γ_{2})_{ℝ} and (Γ_{1} ⊕ Γ_{2}^{∗})_{ℝ} respectively. Denote the sublattice (Γ_{1}^{∗} ⊕ Γ_{2}) by Σ . Define a complex torus B_{I} by the rule B_{I} = (Σ_{ℝ}/Σ, I) .

10.4.1 Theorem. Any well-becoming pair (A_{J}, ω) has a mirror symmetric pair (B_{I}, θ) , where the complex torus B_{I} is constructed above. Moreover, the pair (B_{I}, θ) is well-becoming too. Proof. First, take Σ^{∗} = Γ_{1} ⊕ Γ_{2}^{∗} and consider the complex torus

C_{I′} = (Σ_{ℝ}^{∗}/Σ^{∗}, I′)

By Lemma 9.4.5, the torus C_{I′} is isomorphic to B̂_{I} . Therefore, there is a canonical identification of lattices H_{1}(B_{I} × B̂_{I}, ℤ) and Λ . Moreover, by construction, the element J_{B×B̂} coincides with I_{ω} . Hence, to define mirror symmetric pair we have to find θ ∈ NS_{B_{I}} (ℂ)^{0} such that I_{θ} = J_{A×Â} .

Denote by ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) a matrix of J_{A×Â} with respect to the decomposition Λ = Σ ⊕ Σ^{∗} . Since

J_{A} = ( \begin{matrix}J_{11} & J_{12} \\ J_{21} & J_{22} \end{matrix} ) the map β sends (a, b) ∈ Σ_{ℝ}^{∗} = (Γ_{1} ⊕Γ_{2}^{∗})_{ℝ} to (−J_{21}^{∗}(b), J_{21}(a)) ∈ Σ_{ℝ} = (Γ_{1}^{∗} ⊕Γ_{2}^{∗})_{ℝ} .

As J_{21} is invertible β is invertible too. Hence, by Lemma 9.4.1, there is θ = ψ_{1} + iψ_{2} ∈ NS_{B_{I}} (ℂ)^{0} such that I_{θ} = J_{A×Â} . Thus, there is a mirror symmetric pair (B_{I}, θ) .

Further, since ψ_{2} = −β^{−1} and ψ_{1} = δβ^{−1} we have that ψ_{k}(Γ_{2ℝ}) = Γ_{1ℝ} and ψ_{k}(Γ_{1ℝ}^{∗}) = Γ_{2ℝ}^{∗} , for k = 1, 2 . Hence the subspaces Γ_{1ℝ}, Γ_{2ℝ}^{∗} are isotropic with respect to θ . In addition, the map I_{21} : Γ_{1ℝ}^{∗} → Γ_{2ℝ} coincides with the restriction on Γ_{1ℝ} of the map −φ_{2}^{−1} . Hence I_{21} is invertible and the pair (B_{I}, θ) is well-becoming by Lemma 10.11. This completes the proof. □

10.4.2 Corollary. Let A_{J(q)} = (Γ_{ℝ}/Γ, J(q)) be a complex torus obtained by G–constriction with R = EndA_{J(q)} . Then for any ω = φ_{1}+iφ_{2} ∈ NS_{A_{J(q)}}(ℂ) with non-degenerate φ_{2} the pair (A_{J(q)}, ω) has a mirror symmetric pair. Proof. Since EndA_{J(q)} = R the map i_{s} : S(D) ⟶ NS_{A_{J(q)}}(ℚ) is an isomorphism. Hence, by Lemma 10.3.5, any weak pair (A_{J(q)}, ω) is well-becoming. By Theorem 10.4.1, it has a mirror symmetric pair (B_{I}, θ) . □

10.4.3 Corollary. For any abelian variety A there exists an element ω ∈ C_{A} such that the pair (A, ω) has a mirror symmetric pair. Proof. By Theorem 10.4.1, it is sufficient to show that there is ω such that the pair (A, ω) is well-becoming. Since A is an abelian variety there is a non-degenerate skew-symmetric form s on Γ = H_{1}(A, ℤ) such that the form s⟨J(·), ·⟩ is positive definite. There exists isotropic with respect to s decomposition Γ = Γ_{1} ⊕ Γ_{2} . Since the form s⟨J(·), ·⟩ is positive definite the linear maps J_{12} and J_{21} are invertible. Thus, the pair (A, ω) with ω = cs is well-becoming for any c ∈ ℂ^{∗} . Therefore, it has a mirror symmetric pair. □

Notice that this corollary is a particular case of (9.6.1).

10.5 Theorem. Let A be an abelian variety of dimension n and ω = φ_{1} + iφ_{2} ∈ C_{A} . Let W ⊂ NS_{A}(ℚ) be an ℚ –envelope of {φ_{1}, φ_{2}} . Suppose that algebraic pair (A, tω) has a mirror symmetric pair for any t ∈ ℝ^{∗} . Then for any ω′ ∈ C_{A} ⋂ W_{ℂ} the pair (A, ω′) is well-becoming. Proof. As above, consider operators

I_{tω} := (\begin{matrix}φ_{2}^{−1}φ_{1} & −t^{−1}φ_{2}^{−1} \\ t(φ_{2} + φ_{1}φ_{2}^{−1}φ_{1}) & −φ_{1}φ_{2}^{−1} \end{matrix})

that act on the real space Λ_{ℝ} = (Γ ⊕ Γ^{∗})_{ℝ} . By assumption, any pair (A, tω) has a mirror symmetric one. Therefore there is a decomposition Λ = Σ_{1t} ⊕ Σ_{2t} for any t ∈ ℝ^{∗} such that both Σ_{kt} and are Q –isotropic and Σ_{ktℝ} are I_{tω} –invariant ( k = 1, 2 ). Since the set of all sublattices in a lattice is countable there is a decomposition Λ = Σ_{1} ⊕ Σ_{2} such that Σ_{kℝ} are I_{tω} –invariant for infinite set of t . Hence these subspaces are invariant with respect to the operators

γ_{1} = (\begin{matrix}0 & φ_{2}^{−1} \\ 0 & 0 \end{matrix}) ,  γ_{2} = (\begin{matrix}φ_{2}^{−1}φ_{1} & 0 \\ 0 & −φ_{1}φ_{2}^{−1} \end{matrix}) ,  γ_{3} = (\begin{matrix}0 & 0 \\ φ_{2} + φ_{1}φ_{2}^{−1}φ_{1} & 0 \end{matrix}) .

It follows from Lemmas 9.4.1 and 9.4.2 that the map ψ = φ_{2}+φ_{1}φ_{2}^{−1}φ_{1} from Γ_{ℝ} to Γ_{ℝ}^{∗} is invertible.

Denote by Π and Ξ the projections of sublattice Σ_{1} on Γ and Γ^{∗} respectively. It is clear that Σ_{1} ⊆ Π⊕Ξ . Since Σ_{1} is invariant under the action of γ_{1} and γ_{3} we have γ_{1}γ_{3}(Σ_{1ℝ}) ⊂ Σ_{1ℝ} . One can see that γ_{1}γ_{3}(Σ_{1ℝ}) = (ψφ_{2}^{−1}(Π_{ℝ}), 0) This imples that ψφ_{2}^{−1}(Π_{ℝ}) ⊆ Π_{ℝ} . As ψ is invertible, we have that ψφ_{2}^{−1}(Π_{ℝ}) = Π_{ℝ} and (Π_{ℝ}, 0) ⊂ Σ_{1ℝ} . By the same argument, (0, Ξ_{ℝ}) ⊂ Σ_{1ℝ} . Hence Σ_{1ℝ} = Π_{ℝ} ⊕ Ξ_{ℝ} and, consequently, Σ_{1} is a sublattice of finite index in Π ⊕ Ξ . But, as the sublattice Σ_{1} is a direct summand of Λ it coincides with Π ⊕ Ξ . Moreover, it is easy to see that dim Π = dim Ξ = n , because there are inclusions ψ(Π_{ℝ}) ⊆ Ξ_{ℝ} and φ^{−1}(Ξ_{ℝ}) ⊆ Π_{ℝ} which are actually equalities.

Futher, take x, y ∈ Π_{ℝ} . There is l ∈ Ξ_{ℝ} such that x = φ_{2}^{−1}(l) . Since Σ_{1} is isotropic with respect to the form Q we obtain that φ_{2}(x, y) = l(y) = 0 . Similarly, we have φ_{1}(x, y) = φ_{1}φ_{2}^{−1}(l)(y) = 0 . Hence Π_{ℝ} is isotropic with respect to φ_{2} and φ_{1} . This yields that the lattice Π is isotropic with respect to any element from W that is the ℚ –envelope of {φ_{1}, φ_{2}} .

Now put Γ_{1} := Π . Produce a sublattice Γ_{2} ⊂ Γ from Σ_{2} in the same way as Γ_{1} from Σ_{1} . It is clear that Γ = Γ_{1} ⊕ Γ_{2} . By construction, Γ_{1} and Γ_{2} are isotropic with respect to any element of W . Let s ∈ W ⋂ C_{A} be a form that corresponding to an ample line bundle on A . Hence the operator J_{A} preserves s_{ℝ} and the symmetric form s⟨J(·), ·⟩ is positive definite. It can be checked as in Lemma 9.4.2 that J_{12}, J_{21} are invertible. Thus (A, ω′) is well-becoming. □

10.6. Let (A, ω) and (B, θ) be two mirror symmetric well–becoming pairs, as in Theorem 10.4.1. Then H_{1}(A, ℤ) = Γ = Γ_{1} ⊕ Γ_{2} and H_{1}(B, ℤ) = Σ = Γ_{1}^{∗} ⊕ Γ_{2} . According to (3.7), both cohomology lattices H^{∗}(A, ℤ) and H^{∗}(B, ℤ) carry the structure of Cl(Λ, Q) –modules.

Recall that by Proposition 9.3.3 there exists a unique (up to ±1 ) isomorphism of Cl(Λ, Q) –modules β : H^{∗}(A, ℤ) ⟶^{∼} H^{∗}(B, ℤ) . It is clear that any such isomorphism can be represented as

v_{ξ}(·) = p_{2∗}(ξ ∪ p_{1}^{∗}(·))  (20)

for some class ξ ∈ H^{∗}(A × B, ℤ).

We want to show that this class ξ is in fact the Chern character of some complex line bundle on a real subtorus of the product A × B. As we shall see, this torus is of real dimension 3n, where n = dim_{ℂ} A = dim_{ℂ} B.

10.6.1. Let us fix bases ⟨l_{1}, …, l_{n}⟩ of Γ_{1} and ⟨l_{n+1}, …, l_{2n}⟩ of Γ_{2}. Let the dual bases of Γ_{1}^{∗} and Γ_{2}^{∗} be ⟨x_{1}, …, x_{n}⟩ and ⟨x_{n+1}, …, x_{2n}⟩, respectively.

Let, as in (3.2), I_{A} and I_{B} be the following left ideals of Cl(Λ, Q) :

I_{A} = Cl(Λ, Q) · l_{1} ⋯ l_{2n}, I_{B} = Cl(Λ, Q) · l_{n+1} ⋯ l_{2n}x_{1} ⋯ x_{n}.

Since H^{1}(A, ℤ) ≅ Γ^{∗}, we identify H^{∗}(A, ℤ) with the ideal I_{A} as Cl(Λ, Q) –modules by the following rule: x_{i_{1}} ∪ ⋯ ∪ x_{i_{k}} ∈ H^{k}(A, ℤ) goes to x_{i_{1}} ⋯ x_{i_{k}} · l_{1} ⋯ l_{2n} ∈ I_{A}. In the same manner, since H^{1}(B, ℤ) ≅ Σ^{∗} = Γ_{1} ⊕ Γ_{2}^{∗}, we identify H^{∗}(B, ℤ) with the ideal I_{B} as Cl(Λ, Q) –modules.

Both I_{A} and I_{B} are isomorphic irreducible Cl(Λ, Q) –modules. Therefore, there is a unique (up to ±1 ) isomorphism between them. It is given by the right multiplication I_{A} · x_{1} ⋯ x_{n} = I_{B}.

All the above gives us a formula for cohomology map β in terms of the bases chosen above:

x_{S} ∪ x_{R} ↦ (−1)^{ϵ}x_{R} ∪ l\overline{S},  (21)

with

ϵ = |S| · |R| + ∑_{i∈S}(i − 1),  (22)

Here S ⊂ \{1, …, n\}, R ⊂ \{n + 1, …, 2n\} , \overline{S} is the complement subset, and x_{S} = x_{i_{1}} ∪ ⋯ ∪ x_{i_{|S|}} , where i_{1} < i_{2} < ⋯ < i_{|S|} are the elements of S, etc.

10.7. Now that we have an explicit formula for the map β in the chosen bases, our next step is to compute the class ξ ∈ H^{∗}(A × B, ℤ), that represents β in the form (20). In fact, we will define a real subtorus in A × B with a complex line bundle on it, and then show that the direct image of the Chern character of this line bundle is the desired ξ. First, take this real torus to be T := Π_{ℝ}/Π, where Π = Γ_{1} ⊕ Γ_{2} ⊕ Γ_{1}^{∗}. Clearly, since Π is a sublattice of Λ one can consider T as a subtorus both of A × Â and B × B̂, because the first homology lattices of these two are identified with Λ.

10.7.1 Lemma. The restrictions P_{A} |_{T} and P_{B}^{−1} |_{T} are isomorphic as complex line bundles on T . Proof. It follows from the well-known fact that the complex line bundle is determined by its first Chern class (see for example [10] or [11]). Using Lemma 4.2.3.1, it is easy to check that the first Chern classes of these complex line bundles coincide. □

10.7.2. Put L := P_{A}|_{T} . Lemma 4.2.3.1 gives us the formula for c_{1}(P_{A}) . So we have

c_{1}(L) = ∑_{i=1}^{n} x_{i} ∪ l_{i},

where x_{i} and l_{i} are elements of H^{1}(T, ℤ) ≅ Π^{∗} = Γ_{1}^{∗} ⊕ Γ_{2}^{∗} ⊕ Γ_{1} .

Note that T has projections onto A and B and thus is naturally embedded into the product j : T ↪ A × B.

10.7.3. Let now τ be a class in H^{n}(A × B, ℤ) such that if we evaluate the formula (20) on the class τ, the resulting transformation sends the monomials of the form x_{R} ∪ x_{n} ∪ ⋯ ∪ x_{1} ∈ H^{∗}(A, ℤ) to x_{R} ∈ H^{∗}(B, ℤ), and any other monomials to zero (here R ⊂ {n + 1, …, 2n}). The class τ corresponds to the homology class [T] in H_{3n}(A × B, ℤ) under Poincare isomorphism and suitable choice of an orientation on T .

Any class in H^{∗}(T, ℤ) is the restriction of a class from H^{∗}(A × B, ℤ). In particular, c_{1}(L) is the restriction of the class D ∈ H^{2}(A × B, ℤ) given by the following formula:

D = ∑_{i=1}^{n} p_{1}^{∗}(x_{i}) ∪ p_{2}^{∗}(l_{i}).

Therefore, one has

j_{∗}(ch(L)) = j_{∗}j^{∗}(exp(D)) = τ ∪ exp(D),

where j_{∗} is map from H^{∗}(T, ℤ) to H^{∗+n}(A × B, ℤ) that is obtained from evident homology map H_{∗}(T, ℤ) ⟶ H_{∗}(A × B, ℤ) under Poincare isomorphism.

10.8 Proposition. The map β coincides with the map given by the formula (20) with ξ = j_{∗}(ch(L^{(−1)^{(n−1)}}) .

Proof. Let S and R be index subsets of {1, …, n} and {n + 1, …, 2n}, respectively. Take a monomial x_{S} ∪ x_{R} in the cohomologies of A. We already know the image of this element under β (see (21)). Now we are going to compute its image under the map given by v_{ξ} .

v_{ξ}(x_{S} ∪ x_{R}) = p_{2∗}(τ ∪ exp((−1)^{n−1}D) ∪ p_{2}^{∗}(x_{S} ∪ x_{R})).  (23)

Computing the Chern character exp((−1)^{n−1}D) analogously to Lemma 4.2.3.2, we see that

exp((−1)^{n−1}D) = ∑_{Q⊂{1,··· ,n}}(−1)^{(n−1)|Q|}p_{1}^{∗}(_{Q}x) ∪ p_{2}^{∗}(l_{Q})

where _{Q}x is a product x_{i_{|Q|}} ∪ ⋯ ∪ x_{i_{1}} by set Q = {i_{1}, …, i_{|Q|}} in the descending order. Therefore, the entire expression on the right–hand side of (23) rewrites as

\begin{matrix}v_{ξ}(x_{S} ∪ x_{R}) = (−1)^{(n−1)|\overline{S}|} & p_{2∗}(τ ∪ p_{1}^{∗}(x_{S}) ∪ p_{1}^{∗}(x_{R}) ∪ p_{1}^{∗}(\overline{S}x) ∪ p_{2}^{∗}(l\overline{S})) = \\ (−1)^{(n−1)|\overline{S}|+|S||R|} & p_{2∗}(τ ∪ p_{1}^{∗}(x_{R} ∪ x_{S} ∪ \overline{S}x)) ∪ p_{2}^{∗}(l\overline{S})) = \\ (−1)^{(n−1)|\overline{S}|+|S||R|+ ∑_{i∈S}(n−i)} & p_{2∗}(τ ∪ p_{1}^{∗}(x_{R} ∪ x_{n} ∪ ⋯ ∪ x_{1}) ∪ p_{2}^{∗}(l\overline{S})) = \\ (−1)^{(n−1)(|\overline{S}|+|S|)+|S||R|+ ∑_{i∈S}(i−1)} & p_{2∗}(τ ∪ p_{1}^{∗}(x_{R} ∪ x_{n} ∪ ⋯ ∪ x_{1})) ∪ l\overline{S} = \\ (−1)^{(n−1)n+ϵ} & x_{R} ∪ l\overline{S} = \\ (−1)^{ϵ} & x_{R} ∪ l\overline{S} \end{matrix}

in H^{∗}(B, ℤ) and ϵ was defined by formula (22). Thus, the maps v_{ξ} and β coincide. □

APPENDIX (the proof of Theorem 8.2).

A.1 Theorem. Let A be an abelian variety.

a) The action of U_{A,ℚ}(ℝ) on C_{A} is well defined

b) C_{A}^{+} and C_{A}^{−} are single orbits under this action.

c) The stabilizer of a point in C_{A} is a maximal compact subgroup of U_{A,ℚ}(ℝ)

Proof. Let us prove the assertion about C_{A}^{+} . The case of C_{A}^{−} is similar. It will be more convenient to work with a different open subset of NS_{A}(ℂ). Namely, recall that every element z ∈ NS_{A}(ℝ)^{0} defines the corresponding Rosati involution on End(A) ⊗ ℝ :

a ↦ a′ := z^{−1}âz.

Put NS_{A}(ℝ)^{+} := {z ∈ NS_{A}(ℝ) | ∀a ∈ End(A) ⊗ ℝ T r(aa′) > 0}. That is NS_{A}(ℝ)^{+} is an open subset of NS_{A}(ℝ) which consists of elements z that define a positive Rosati involution.

A.2 Lemma. The ample cone C_{A}^{a} is a connected component of NS_{A}(ℝ)^{+}. Hence C_{A}^{+} is a connected component of NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} .

Proof. We know [Mum1] that an ample class z ∈ NS_{A} belongs to NS_{A}(ℝ)^{+}. Hence C_{A}^{a} ⊂ NS_{A}^{+}. Every z ∈ NS_{A}(ℝ) defines a symmetric bilinear form b_{z} on V_{A} as follows

b_{z}(x, y) := z(J_{A}x)(y).

By the Lefschetz theorem ([Mum1]) a class z ∈ NS_{A} is ample iff b_{z} is positive definite. Thus we obtain another characterization of the ample cone

C_{A}^{a} = {z ∈ NS_{A}(ℝ) | b_{z} is positive definite}.

It follows that for z ∈ ∂C_{A}^{a} the form b_{z} is degenerate, i.e. the map z has nontrivial kernel. Thus ∂C_{A}^{a} ⊂ ∂NS_{A}(ℝ)^{+}. Therefore, C_{A}^{a} is a connected component of NS_{A}(ℝ)^{+}. □

In the next few lemmas we study the U_{A,ℚ}(ℝ) -action on the set NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} .

A.3 Lemma. Let g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ U_{A,ℚ}(ℝ), ω ∈ NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}. Then the endomorphism (a + bω) ∈ End(V_{A}) is invertible, i.e. the element gω ∈ NS_{A}(ℂ) is well defined.

Proof. Let ω = η + iz, η ∈ NS_{A}(ℝ), z ∈ NS_{A}(ℝ)^{+}. The lemma follows from the following assertion: for every nonzero x ∈ End(V_{A} ⊗_{ℝ} ℂ)

Im(T r(z^{−1}ˆx̄(ĉ+ ˆ ω̄d̂)(a + bω)x)) < 0.

The assertion is proved by a straightforward calculation which we omit .

A.4 Remark. Let ω = η + iz ∈ NS_{A}(ℂ). Then ( \begin{matrix}1 & 0 \\ −η & 1 \end{matrix} ) ∈ U_{A,ℚ}(ℝ) and ( \begin{matrix}1 & 0 \\ −η & 1 \end{matrix} ) ω = iz. Therefore all U_{A,ℚ}(ℝ) -orbits in NS_{A}(ℂ) are invariant under translation along NS_{A}(ℝ).

A.5 Lemma. Let ω ∈ NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}. Let K_{ω} be its stabilizer in U_{A,ℚ}(ℝ). Then K_{ω} is a maximal compact subgroup of U_{A,ℚ}(ℝ).

Proof. By the previous remark we may assume that ω = iz. Denote by ′ the Rosati involution defined by z. We have

K_{ω} = \{(\begin{matrix}a & b \\ −zbz & zaz^{−1} \end{matrix}) ∈ U_{A,ℚ}(ℝ)| z^{−1}âza + b̂zbz = 1, z^{−1}âzb = b̂zaz^{−1}\}

Consider a map ϑ : U_{A,ℚ}(ℝ) → M_{2}(End(A) ⊗ ℝ)^{∗}, given as ϑ (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}a & bz \\ z^{−1}c & z^{−1}dz \end{matrix}) .

Then

Imϑ = \{(\begin{matrix}α & β \\ γ & δ \end{matrix})| (\begin{matrix}α & β \\ γ & δ \end{matrix})^{−1} = (\begin{matrix}δ′ & −β′ \\ −γ′ & α′ \end{matrix})\} and ϑ(K_{ω}) = \{(\begin{matrix}α & β \\ −β & α \end{matrix})| \begin{matrix}α′α + β′β = Id, \\ α′β = β′α \end{matrix} \}

Extend the Rosati involution ′ to M_{2}(End(A)) ⊗ ℝ by the formula

( \begin{matrix}α & β \\ γ & δ \end{matrix} )′ = ( \begin{matrix}α′ & γ′ \\ β′ & δ′ \end{matrix} )

and consider the symmetric bilinear form on M_{2}(End(A)) ⊗ ℝ

Ψ(X, Y) = T r(X′Y), i.e. Ψ((\begin{matrix}α & β \\ γ & δ \end{matrix}) , (\begin{matrix}a & b \\ c & d \end{matrix})) = T r((\begin{matrix}α′ & γ′ \\ β′ & δ′ \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix})) = T r(α′a+γ′c+β′b+δ′d).

This form is positive definite.

Consider the action of the group M_{2}(End(A) ⊗ ℝ)^{∗} on M_{2}(End(A) ⊗ ℝ) by left multiplication. Let U(Ψ) ⊂ M_{2}(End(A) ⊗ ℝ)^{∗} be the subgroup which preserves the form Ψ. We claim that

ϑ(U_{A,ℚ}(ℝ)) ∩ U(Ψ) = ϑ(K_{ω}).

Indeed,

U(Ψ) = {u ∈ M_{2}(End(A) ⊗ ℝ)^{∗} | u′u = Id}.

Thus ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ϑ(U_{A,ℚ}(ℝ)) ∩ U(Ψ) iff ( \begin{matrix}a′ & b′ \\ c′ & d′ \end{matrix} ) = ( \begin{matrix}d′ & −b′ \\ −c′ & a′ \end{matrix} ) . This means that a = d, b = −c,

a′a + b′b = 1, a′b = b′a, which in turn means that ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ ϑ(K_{ω}). Thus K_{ω} is compact.

Notice that the adjoint operator for the left multiplication by m ∈ M_{2}(End(A) ⊗ ℝ) with respect to Ψ is the left multiplication by m′. The group ϑ(U_{A,ℚ}(ℝ)) is self adjoint, i.e. t ∈ ϑ(U_{A,ℚ}(ℝ)) ⇒ t′ ∈ ϑ(U_{A,ℚ}(ℝ)). Put

P_{ω} = {g ∈ U_{A,ℚ}(ℝ) | ϑ(g) is positive self adjoint}.

The lemma now follows from the following claim.

A.6 Claim.

a) The multiplication map K_{ω} × P_{ω} → U_{A,ℚ}(ℝ) , such that (b, p) ↦ bp , is a diffeomorphism

b) K_{ω} is a maximal compact subgroup of U_{A,ℚ}(ℝ).

Indeed, since ϑ(U_{A,ℚ}(ℝ)) is self-adjoint, a) follows from theorem 1 in sect.2, ch.5 in [OV]. The assertion b) follows from problem 2 in sect.2, ch.5 in [OV]. □

A.7 Lemma. The set NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} is U_{A,ℚ}(ℝ) -invariant.

Proof. Let ω ∈ NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}, g ∈ U_{A,ℚ}(ℝ). We need to show that gω ∈ NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}. By Remark A.4 above we may asssume that ω = iz, z ∈ NS_{A}(ℝ)^{+}, and gω = ip, p ∈ NS_{A}(ℝ). Then we need to show that

(i) p is invertible,

(ii) the Rosati involution defined by p is positive definite (i.e. p ∈ NS_{A}(ℝ)^{+}. )

Denote by ′ the Rosati involution defined by z. Put p = zk for k ∈ End(A) ⊗ ℝ. We have k′ = k. Let λ ∈ U_{A,ℚ}(ℝ) be such that λ(iz) = ip . Then λ has the form

λ = (\begin{matrix}a & b \\ −zkbz & zkaz^{−1} \end{matrix}) .

Under the map ϑ : U_{A,ℚ}(ℝ) → M_{2}(End(A) ⊗ ℝ), defined in the proof of Lemma A.5, λ goes to

ϑ(λ) = (\begin{matrix}α & β \\ −kβ & kα \end{matrix}) ,

where

1) αα′k + ββ′k = Id and 2) − αβ′ + βα′ = 0

1) implies that k is invertible , hence p is such. This proves (i). Let us prove (ii). Note that

(\begin{matrix}k & 0 \\ 0 & k̂^{−1} \end{matrix}) ∈ U_{A,ℚ}(ℝ) and (\begin{matrix}k & 0 \\ 0 & k̂^{−1} \end{matrix}) ip = izk^{−1}.

Replacing izk by izk^{−1} and applying the previous argument we find s, t ∈ End(A) ⊗ ℝ such that ss′k^{−1} + tt′k^{−1} = 1, −st′ + ts′ = 0, that is,

3) k = ss′ + tt′, and 4) − st′ + ts′ = 0

The Rosati involution defined by p is

a ↦ p^{−1}âp = k^{−1}a′k, a ∈ End(A) ⊗ ℝ.

We need to show that T r(k^{−1}a′ka) > 0 for all 0 ≠ a ∈ End(A) ⊗ ℝ. Or, equivalently, that the following quadratic form on M_{2}(End(A) ⊗ ℝ) is positive definite

T r((\begin{matrix}k^{−1} & 0 \\ 0 & k^{−1} \end{matrix}) X′ (\begin{matrix}k & 0 \\ 0 & k \end{matrix}) X), X ∈ M_{2}(End(A) ⊗ ℝ).

Put Y = ( \begin{matrix}α & β \\ −β & α \end{matrix} ) and Z = ( \begin{matrix}s & t \\ −t & s \end{matrix} ) . Then

1), 2) ⇔ Y Y′ = (\begin{matrix}k^{−1} & 0 \\ 0 & k^{−1} \end{matrix}) , and 3), 4) ⇔ ZZ′ = (\begin{matrix}k & 0 \\ 0 & k \end{matrix}) .

Hence

T r((\begin{matrix}k^{−1} & 0 \\ 0 & k^{−1} \end{matrix}) X′ (\begin{matrix}k & 0 \\ 0 & k \end{matrix}) X) = T r(Y Y′X′ZZ′X) = T r(Y′X′ZZ′XY) = T r((Z′XY)′(Z′XY)) > 0

This proves (ii) and the Lemma.

A.8 Lemma. The set NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} consists of finitely many U_{A,ℚ}(ℝ) -orbits.

A.9 Corollary. The U_{A,ℚ}(ℝ) -orbits in NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} coincide with the connected components of this set.

Proof of corollary. By the Corollary 5.3.5 the group U_{A,ℚ}(ℝ) is connected. Hence each U_{A,ℚ}(ℝ) - orbit is contained in a connected component of NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}. It follows from Lemma A.5 that all U_{A,ℚ}(ℝ) -orbits in NS_{A}(ℝ) + iNS_{A}(ℝ)^{+} are isomorphic (all maximal compact subgroups in a reductive Lie group are conjugate). Then by Lemma A.8 each U_{A,ℚ}(ℝ) -orbit is an open subset in NS_{A}(ℝ) + iNS_{A}(ℝ)^{+}, hence must coincide with a connected component of this set. This proves the corollary.

A.10 Remark. Theorem A.1 now follows from A.2, A.3, A.5 and A.9. So it remains to prove Lemma A.7.

Proof of Lemma A.7. By Remark A.4 above it suffices to show that the set iNS_{A}(ℝ)^{+} is contained in a finite number of U_{A,ℚ}(ℝ) -orbits.

Fix z ∈ NS_{A}(ℝ)^{+} and let ′ be the corresponding Rosati involution. Let l ∈ Aut_{A}(ℝ) := (End(A) ⊗ ℝ)^{∗}. Then iz and izl′l are U_{A,ℚ}(ℝ) –conjugate. Indeed,

(\begin{matrix}l^{−1} & 0 \\ 0 & l̂ \end{matrix}) iz = il̂zl = izl′l.

Consider the space End(A) ⊗ ℝ with the positive definite symmetric form ϕ(a, b) := T r(ab′) and the adjoint action of the group Aut_{A}(ℝ) : ad_{s}(a) := sas^{−1}.

The adjoint with respect to ϕ of the operator ad_{s} is ad_{s′}. Hence the following are equivalent (i) ad_{s} is self adjoint; (ii) s = s′c for some c in the center of Aut_{A}(ℝ). Let now izk ∈ iNS_{A}(ℝ)^{+} for some k ∈ Aut_{A}(ℝ).

A.11 Claim. There exist l ∈ Aut_{A}(ℝ) and c ∈ Z(Aut_{A}(ℝ)) such that c′ = c, c^{2} = 1 and k = l′lc.

Indeed, we have ẑk = zk, i.e. k′ = k. Thus ad_{k} is self adjoint with respect to ϕ. The positivity of the Rosati involution defined by zk

a ↦ k^{−1}a′k

is equivalent to the positivity of the self adjoint operator ad_{k}. Hence the operator ad_{k} has a square root (which is also positive self adjoint). But this square root is also in the image of the homomorphism ad (see ch. 5, sect 2, thm. 1 in [OV]). Thus there exists l ∈ Aut_{A}(ℝ) such that ad_{l} = ad_{l′} and (ad_{l})^{2} = ad_{k}. We get l′lc = k for some c ∈ Z(Aut_{A}(ℝ)). Moreover,

l′lc = k = k′ = (l′lc)′ = c′l′l.

Thus c′ = c.

It follows from the discussion of End(A) in section 1.8 that the ′ -invariant part of the center Z(End(A) ⊗ ℝ) is isomorphic to ℝ × ⋯ × ℝ. therefore, multiplying c by a positive real number r (and l by r^{−1/2} ) we may assume that c^{2} = 1. This proves the claim.

Note that there are finitely many c ∈ Z(Aut_{A}(ℝ)) such that c′ = c and c^{2} = 1, say, c_{1}, …, c_{m}. Then by the argument in the beginning of the proof of the lemma iNS_{A}(ℝ)^{+} is contained in the union of U_{A,ℚ}(ℝ) -orbits of izc_{1}, izc_{2}, …, izc_{m}. This proves Lemma A.8 and completes the proof of Theorem A.1.

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Institute for Problems of Information Transmission, Russian Academy of Sciences, 19 Bolshoi Karetnyi, Moscow 101447, Russia

   E-mail address: golyshev@mccme.ru

Department of Mathematics, Indiana University, Bloomington, IN 47405, USA

   E-mail address: vlunts@indiana.edu

Algebra Section, Steklov Mathematical Institute, Russian Academy of Sciences, 8 Gubkin str., GSP-1, Moscow 117966, Russia

   E-mail address: orlov@mi.ras.ru