Poincaré–Birkhoff–Witt isomorphism, Hochshild homology and Riemann–Roch theorem

Nikita Markarian

To Andrei Nikolaevich Tyurin in occasion of his 60-th birthday

Introduction

An idea of considering a Lie algebra as a supermanifold with a closed vector field goes back to D. Quillen. For a Lie algebra g the function ring is the Grassman algebra S^{∗}(g^{∗}[1]) and the vector field is given by the expression c_{ij}^{k}x^{i}x^{j}∂_{k}, where c_{ij}^{k} are the structure constants of the algebra. This point of view allows to translate many construction from Lie groups and algebras theory into the (super)geometrical language. For example, the structural constants may be treated as the Atiyah class of the tangent bundle (this analogy is exploited in [K]); function ring of the group provides a resolution of the diagonal. Another case where such supermanifolds (in other words, DG-algebras) arise is the case of complex non-affine manifolds. The function ring may be realized as the ring of (0, q)-forms, the vector field is \overline{∂}.

Inspired by Lie groups and algebras theory we introduce in section 1 analogs of Poincaré–Birkhoff– Witt (PBW) isomorphism and of the exponential map for any smooth manifold (definition–propositions 1 and 2). Theorem 1 contains the main calculation of the paper. Under the analogy above, this result corresponds to the formula of left and right invariant vector fields on a Lie group in terms of exponential coordinates.

In section 2 we discuss the Riemann-Roch theorem. To write down the Riemann–Roch theorem one have to calculate a particular class from H^{∗}(Ω^{∗}), namely the Todd genus. In this calculation (lemma 11) theorem 1 is crucially used. Drawing the analogy, the Todd genus corresponds to the (left) invariant volume form written in exponential coordinates. In this section we also prove the holomorphic Lefschetz formula.

Section 3 is nearly independent of section 2 and partly duplicate results of it. The main character of this section is the duality isomorphism (44). Theorem 8 describes interrelation of this map with isomorphisms introduced in section 1. It seems that this result explains appearance of the Todd genus in the Riemann–Roch theorem and other related statements.

This paper would never have been written without numerous fruitful discussions with A. Beilinson, A. Bondal, B. Feigin and M. Kontsevich. It is pleasure for me to express them my deep gratitude. I am grateful to Max–Planck–Institut für Mathematik, Bonn for fine work conditions.

1 Poincaré–Birkhoff–Witt isomorphism

Notations Let M be a smooth complex manifold. By △ we denote the diagonal in the direct product M × M, by O_{△} denote the structure sheaf of the diagonal. By p_{1} and p_{2} denote projections on the first and the second factors.

The tangent and cotangent bundles on the manifold under consideration we shall denote by Ω^{1} and T. By S(Ω^{1}[1]) and S(T [−1]) denote correspondingly ⨁_{i} Ω^{i}[i] and ⨁_{i} Λ^{i}T [−i]. Objects p_{1}^{∗}(−) ⊗ O_{△} we shall denote by −_{△}, where − is T, Ω^{1} and so on. We shall identify normal bundle to the diagonal with T_{△}, conormal bundle with Ω_{△}^{1} and so on.

By D^{−}(M) denote the derived category of bounded above complexes of sheaves on M .

1 Atiyah class

Let O_{△^{2}} be the structure sheaf of the first neighborhood of the diagonal in M × M. It fits into an exact sequence

0 → Ω_{△}^{1} → O_{△^{2}} → O_{△} → 0,  (1)

where the last arrow is a restriction map.

Take any object E ∈ D^{−}(M) and apply the functor Rp_{2,∗}(− ⊗^{L} p_{1}^{∗}(E)) to the exact sequence (1). In the resulting triangle

Ω^{1} ⊗ E → Rp_{2∗}(E ⊗^{L} p_{1}^{∗}(E)) → E →^{+1}  (2)

the middle term is called the object of first jets of E and is denoted by j^{1}(E).

Definition 1 (see [A]) For E ∈ D^{−}(M) morphism α(E) ∈ Ext^{1}(E, E ⊗ Ω^{1}) from the triangle (2) is called the Atiyah class of E.

The following lemma is needed for the sequel.

Lemma 1 Let (E, d) be a complex of locally free quasicoherent sheaves on M every component of which is supplied with a connection ∇. Then the Atiyah class α(E) is represented by ∇ · d + d · ∇.

Proof Left to reader. □

2 Poincaré–Birkhoff–Witt isomorphism

Consider the class

α ∈ Ext^{1}(O_{△}, Ω_{△}^{1})

of extension (1). Define an exponent of this class as

e^{α} = 1 + α +\frac{α^{2}}{2} + … ∈ Hom(O_{△}, S(Ω^{1}[1])_{△}).  (3)

Definition-proposition 1 Let ⊢: S(T [−1])_{△} ⊗ S(Ω^{1}[1])_{△} → O_{△} be the contraction map. Then the composition of e^{α} and ⊢ gives an isomorphism between S(T [−1])_{△} and Ext(O_{△}, O_{△}). We refer to this isomorphism as the Poincaré–Birkoff–Witt (PBW) isomorphism and denote it by I_{P BW}.

Proof Local check is enough. □

Corollary Ext^{n}(O_{△}, O_{△}) = ⨁_{i+j=n} H_{M}^{i}(Λ^{j}T).

This isomorphism we also denote by I_{P BW}.

3 Ext^{∗}(O_{△}, O_{△}) as Ext^{∗}(id, id)

For any object f ∈ D^{−}(M × M) define a functor Φ_{f} from the category D^{−}(M) to itself by the formula

Φ_{f}(−) = Rp_{2∗}(f ⊗^{L} p_{1}^{∗}(−)).  (4)

Any morphism of objects x ∈ Hom_{M×M}(g, h) gives us a morphism of corresponding functors. We denote it by Θ_{x} : Φ_{g} → Φ_{h}.

An example where these notions naturally arise is related with the Atiyah class (see subsection 1). For any object E ∈ D^{−}(M) the Atiyah class defines a morphism α(E) ∈ Ext^{1}(E, E ⊗ Ω^{1}). One may easily check that this is a morphism from identity functor to multiplication by cotangent bundle functor shifted by one. We denote this morphism by A ∈ Ext^{1}(id, − ⊗ Ω^{1}).

Lemma 2

A = Θ_{α}

where α ∈ Ext^{1}(O_{△}, Ω_{△}^{1}) corresponds to extension (1).

Proof By definition. □

As functor Φ_{O_{△}} is identity, any element a ∈ Ext^{i}(O_{△}, O_{△}) defines a morphism of functors Θ_{a} ∈ Ext^{i}(id, id). This point of view on higher automorphisms of a derived category is outlined in [B].

Lemma 3 Morphism of functors Θ_{I_{P BW} (p)} for p ∈ H^{i}(Λ^{j}T) acts on an element E ∈ D^{−}(M) as follows

Θ_{I_{P BW} (p)}(E) = e^{α(E)} ⊢ p ∈ Ext^{i+j}(E, E),  (5)

where e^{α(E)} = 1 + α(E) + α(E)^{2}/2 + … ∈ ⨁_{i} Ext^{i}(E, E ⊗ Ω^{i}) and ⊢ is the contraction map.

Proof The lemma directly follows from the construction of PBW isomorphism and lemma 2. □ The reader may deduce commutativity of the diagram

E  ⟶^{x}  F
↓ Θ_{I_{P BW} (p)}  ↓ Θ_{I_{P BW} (p)}
E  ⟶^{x}  F

for any E, F and x immediately from properties of the Atiyah class.

4 Exponential map

When treating the object Ext(O_{△}, O_{△}) it is useful to consider its action on the torsion product O_{△}⊗^{L}O_{△}.

Introduce three natural maps connected with O_{△} ⊗^{L} O_{△}:

\begin{aligned}S & : & O_{△} ⊗^{L} O_{△} → O_{△} ⊗^{L} O_{△} \\ 1 ⊗ − & : & O_{△} → O_{△} ⊗^{L} O_{△} \\ μ & : & O_{△} ⊗^{L} O_{△} → O_{△} ⊗ O_{△} = O_{△} \end{aligned}  (6)

Here the first map is permutation of factors and the third map is a natural morphism from a left derived functor to the initial one.

Analog of the PBW isomorphism for O_{△} ⊗^{L} O_{△} is the next statement.

Definition-proposition 2 Composition of maps

O_{△} ⊗^{L} O_{△} ⟶^{id⊗e^{α}} O_{△} ⊗^{L} S(Ω^{1}[1])_{△} ⟶^{μ} S(Ω^{1}[1])_{△}  (7)

provides an isomorphism between O_{△} ⊗^{L} O_{△} and S(Ω^{1}[1])_{△}. We refer to this isomorphism as the right exponential map and denote it by exp_{R}. By definition, the left exponential map exp_{L} equals exp_{R} ∘S.

Proof Local check is enough. □ Corollary Tor^{n}(O_{△}, O_{△}) = ⨁_{i−j=n} H_{M}^{i}(Ω^{j}). Remark One may derive the PBW isomorphism from the right exponential map by composing with 1 ⊗ −.

5 Standard resolution

Let A be the algebra of functions on the manifold M (for non-affine manifold one should consider the formal neighborhood of the diagonal). By definition, the standard resolution (see [NB, §3 , n^{o} 8]) of O_{△} on M × M, that is a resolution p: B(A) → A of A considered as a A-A-bimodule looks as follows:

B_{−n}(A) = \underbrace{A ⊗ A ⊗ ⋯ ⊗ A}_{n+2 factors}, n ≥ 0
d(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯) = a_{0}a_{1} ⊗ a_{2} ⊗ ⋯ − a_{0} ⊗ a_{1}a_{2} ⊗ ⋯ + ….

Here B_{n}(A) is equipped with a structure of left (right) free A-module by the action of A on the first (the last) factor.

Describe the Atiyah class of the diagonal in terms of the standard resolution.

Lemma 4 The Atiyah class of the diagonal α(O_{△}) ∈ Hom(B(A), B(A) ⊗ Ω^{1}[1]) is given by the formula

\begin{matrix}α(O_{△}): a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i} ⊗ a_{i+1} ↦ \\ a_{0}da_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i} ⊗ a_{i+1} + (−1)^{i}a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i+1}da_{i}. \end{matrix}  (8)

Proof Complex B(A) is equipped with an obvious flat connection. Applying lemma 1 we get the Atiyah class of B(A). □

The Atiyah class α(O_{△}) nearly coincides with class α.

Lemma 5 Class α of the extension (1) acts on the standard resolution as follows

α: a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i} ⊗ a_{i+1} ↦ a_{0}da_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{i} ⊗ a_{i+1}.  (9)

Proof Consider the decomposition Ω_{M×M}^{1} = p_{1}^{∗}Ω_{M}^{1} ⊕ p_{2}^{∗}Ω_{M}^{1}. Denote by α_{1,2}(O_{△}) the components of α(O_{△}) under this decomposition (they corresponds to the first and the second summands in (8)). One may easily shows, that each of this component is equal to class α of the extension (1) under a proper isomorphism between the conormal bundle and p_{1,2}^{∗}Ω^{1}. We identify conormal bundle with p_{1}^{∗}Ω^{1}, so α = α_{1}(O_{△}). □

Substituting the lemma in the definition of PBW isomorphism we get the next statement. Corollary PBW isomorphism I_{P BW}: ⨁_{i} Λ^{i}T_{△}[−i] → Hom(B(A), A) looks as follows

\begin{matrix}I_{P BW} (∂_{1} ∧ ∂_{2} ∧ … ∧ ∂_{k}): \\ a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{k+1} ↦ \frac{1}{k!} ∑_{σ∈Σ_{k}}(−1)^{sgn σ}a_{0}a_{k+1}∂_{σ(1)}a_{1} ⋯ ∂_{σ(k)}a_{k}, \end{matrix}  (10)

where ∂_{i} are vector fields.

Thus our definition of PBW isomorphism for an affine manifold coincides with the Hochshild– Kostant–Rosenberg isomorphism (see e. g. [Ko]).

Complexes B(A)⊗B(A) and A⊗B(A) represent one and the same object O_{△} ⊗^{L}O_{△} ∈ D^{−}(M×M), as a homotopical equivalence is defined by the map id ⊗ p. Describe explicitly an inverse map.

Lemma 6 A map

γ: a_{0} ⊗ (a_{1} ⊗ ⋯ ⊗ a_{n}) ↦ ∑_{i=0}^{n} a_{0} ⊗ (a_{1} ⊗ ⋯ ⊗ a_{i}) ⊗ (a_{i+1} ⊗ ⋯ ⊗ a_{n})

is a morphism of complexes γ: A ⊗ B(A) → B(A) ⊗ B(A) and is right-inverse to id ⊗ p.

Proof Straightforward check. □

6 Hochshild complex

Using the standard resolution we see that the direct image p_{1∗}(O_{△} ⊗^{L} O_{△}) is represented by the complex of A-modules H

H_{−n} = \underbrace{A ⊗ A ⊗ ⋯ ⊗ A}_{n+1 factors}, n ≥ 0;
d(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1} ⊗ a_{n})
= a_{0}a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1} ⊗ a_{n} − a_{0} ⊗ a_{1}a_{2} ⊗ ⋯ ⊗ + …
−(−1)^{n}a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1}a_{n} + (−1)^{n}a_{0}a_{n} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1}

known as the Hochshild complex.

The Hochshild complex is equipped with the following important operations

\begin{aligned}S & : & H → H \\ Δ & : & H → H ⊗ H \\ × & : & H ⊗ H → H \end{aligned}  (11)

(all morphisms and tensor products are over A!) given by the formulas

S(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1} ⊗ a_{n})

= (−1)^{n}a_{0} ⊗ a_{n} ⊗ a_{n−1} ⊗ ⋯ ⊗ a_{2} ⊗ a_{1}

Δ(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n−1} ⊗ a_{n})

= ∑_{i=0}^{n} a_{0} ⊗ (a_{1} ⊗ ⋯ ⊗ a_{i}) ⊗ (a_{i+1} ⊗ ⋯ a_{n})

a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{p} × a_{0} ⊗ a_{p+1} ⊗ ⋯ ⊗ a_{p+q}

= ∑_{σ∈Σ_{p+q} is p-q-shuffle} a_{0}a′_{0} ⊗ a_{σ(1)} ⊗ ⋯ ⊗ a_{σ(p)} ⊗ a_{σ(p+1)} ⊗ ⋯ ⊗ a_{σ(p+q)}

The map S is the direct image of S from (6), the map Δ is closely connected with 1 ⊗ − by lemma 6, the map × is closely connected with μ. These operations provide the Hochshild complex with the structure of Hopf algebra over A.

By definition-proposition 2, the Hochshild complex is quasi-isomorphic to S(Ω^{1}[1]). The corollary of lemma 4 gives us an explicit form of this equivalence defining by the following well-known formula:

exp_{∗}: H ∋ a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} ↦ a_{0} da_{1} ∧ … ∧ da_{n} ∈ Ω^{n}.  (12)

(We do not make difference between left and right exponential map because they coincide up to sign.) It is well known and easy to show that the following diagram is commutative.

H ⊗ H  ⟶^{×}  H
↓ exp_{∗} ⊗ exp_{∗}  ↓ exp_{∗}
S(Ω^{1}[1]) ⊗ S(Ω^{1}[1]) ⟶^{∧} S(Ω^{1}[1])

(13)

PBW isomorphism supplies an isomorphism p_{1∗} Ext(O_{△}, O_{△}) = S(T [−1]). The direct image of

action of Ext(O_{△}, O_{△}) on O_{△} ⊗^{L} O_{△} by acting on the second factor gives us an action of S(T [−1]) on S(Ω^{1}[1]). To describe this action it is enough to describe the direct image of the map 1 ⊗ α. Below we describe this morphism using the Hochshild complex. The reader may try to translate our arguments into resolution-independent language.

Theorem 1 Let α_{L} ∈ Hom(H, H ⊗ Ω^{1}) acts on the Hochshild complex as follows

α_{L}(a_{0} ⊗ a_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n}) = a_{0}da_{1} ⊗ a_{2} ⊗ ⋯ ⊗ a_{n}

(by lemma 5, α_{L} represents the direct image of the map 1 ⊗ α).

Let

\begin{aligned}α(Ω^{1}) & ∈ & H^{1}(End(Ω^{1}) ⊗ Ω^{1}) \\ ad & ∈ & H^{1}(End(S(Ω^{1}[1])) ⊗ Ω^{1}) \end{aligned}

be the Atiyah classes of Ω^{1} and S(Ω^{1}[1]), let e^{α(Ω^{1})} ∈ H^{∗}(End(Ω^{1}) ⊗ S(Ω^{1}[1])) be an exponent of the Atiyah class. Let

ı : S(Ω^{1}[1]) → End(S(Ω^{1}[1]))

be the map that takes an element of the Grassmann algebra to the multiplication by this element. Let End(Ω^{1}) ⊗ End(S(Ω^{1}[1])) acts on End(S(Ω^{1}[1])) ⊗ Ω^{1} as follows: the first factor acts on the second one and vice versa. Put

L_{L} = (ı e^{α(Ω^{1})} − id ⊗ id)^{−1} · ad ∈ H^{∗}(End(S(Ω^{1}[1])) ⊗ Ω^{1}).  (14)

(The reader will have no difficulty in showing that this expression is well defined.)

Then the following diagram is commutative.

H  ⟶^{α_{L}}  H ⊗ Ω^{1}
↓ exp_{∗}  ↓ exp_{∗} ⊗id
S(Ω^{1}[1]) ⟶^{L_{L}} S(Ω^{1}[1]) ⊗ Ω^{1}

Proof First let us introduce some notation. Put α_{R} = −S ∘ α_{L} ∘ S. By α̃ denote a map H → Ω^{1} that is equal to α̃ on H_{−1} and is equal to zero on other terms of the complex. Then

\begin{matrix}α_{L} = (α̃ ⊗ id) ∘ Δ \\ α_{R} = (id ⊗ α̃) ∘ Δ \end{matrix}  (15)

(in the second formula one should be careful with signs, α̃ is considered as an odd element). If Δ^{2}(a ∈ H) = ∑_{i} a_{i}^{(1)} ⊗ a_{i}^{(2)} ⊗ a_{i}^{(3)} ∈ H ⊗ H ⊗ H, then put by definition Ad a = ∑_{i} a_{i}^{(2)} ⊗ (a_{i}^{(1)} × Sa_{i}^{(3)}).

Consider the series of commutative diagrams.

1. This diagram is commutative by lemma 1:

H  ⟶^{α_{L}−α_{R}}  H ⊗ Ω^{1}
↓ exp_{∗}  ↓ exp_{∗}
S(Ω^{1}[1]) ⟶^{ad} S(Ω^{1}[1]) ⊗ Ω^{1}

(16)

2. This diagram is commutative for the same reason:

H  ⟶^{Ad}  H ⊗ H
↓ exp_{∗}  ↓ exp_{∗}
S(Ω^{1}[1]) ⟶^{e^{ad}} S(Ω^{1}[1]) ⊗ S(Ω^{1}[1])

(17)

3. Figures in the squares on this diagram show the reasons for their commutativity:

H  ⟶^{Δ}  H ⊗ H  ⟶^{Ad ⊗id}  H ⊗ H ⊗ H  ⟶^{id⊗×}  H ⊗ H
↓ α_{L} (15)  ↓  (17)  ↓  (13)  ↓
Ω^{1} ⊗ H ⟶^{id⊗exp_{∗}} Ω^{1} ⊗ S(Ω^{1}[1]) ⟶^{e^{α(Ω^{1})}} Ω^{1} ⊗ S(Ω^{1}[1]) ⊗ S(Ω^{1}[1]) ⟶^{id⊗∧} Ω^{1} ⊗ S(Ω^{1}[1])

(18)

Here the unmarked vertical arrows represent maps which are equal to α̃ on the first factor and are equal to exp_{∗} on other ones.

An easy calculation gives us an identity

(id ⊗ ×) ∘ (Ad ⊗id) ∘ Δ = σ ∘ Δ,  (19)

where σ denotes a permutation of factors (with signs!) in the tensor product. Therefore moving from the upper left corner of diagram (18) to the lower right one by the upper row of arrows and the lower one and using (15), we have

ı e^{α(Ω^{1})} ∘ exp_{∗} ∘ α_{L} = exp_{∗} ∘ α_{R}  (20)

Combining (20) with (16) we obtain the statement of the theorem. □

2 Riemann–Roch theorem

Notations In this section M is a compact manifold of dimension n. By ω denote the canonical bundle. By ω_{△} denote the sheaf p_{1}^{∗}ω ⊗O_{△} = p_{2}^{∗}ω ⊗O_{△} on M ×M. By a complex of sheaves we mean a bounded complex of coherent sheaves.

1 Serre duality

Let x be a point of a manifold M and O_{x} be the structure sheaf of this point. Let z_{1}, z_{2}, …, z_{n} be local coordinates in a neighborhood U of x. Define the Koszul complex of O(U)-modules as follows:

\begin{matrix}K_{•} = O(U) ⊗ Λ^{∗} < δz_{1}, …, δz_{n} >; \\ d(δz_{i}) = z_{i}. \end{matrix}  (21)

Here < − > denotes the vector space generated by −, action of d is defined by Leibniz rule and deg δz_{i} = −1. The Koszul complex is a free resolution of the sheaf O_{x}.

Define morphisms e_{x} ∈ Ext^{0}(O, O_{x}) and f_{x} ∈ Ext^{n}(O_{x}, ω) explicitly through the Koszul complex:

\begin{matrix}e_{x}: O → K_{0} → K_{•} \\ f_{x}: K_{•} → K_{n} → ω. \end{matrix}

Here the first arrow is the isomorphism and the last one takes δz_{1} ∧ … ∧ δz_{n} to dz_{1} ∧ … ∧ dz_{n}. The reader will easily check independence of this definition on coordinates.

For a compact manifold morphisms e_{x} and f_{x} are dual to each other, moreover, the duality is defined by the composition. This follows from Serre duality. We present this theorem in the form we need in the sequel. The proof and generalizations one may find e. g. in [H].

Theorem 2 (Serre duality) For any complex of sheaves E over M there exists a linear function Tr (trace) on the space Ext^{n}(E, E ⊗ ω):

Tr: Ext^{n}(E, E ⊗ ω) → C,

such that for any two complexes of sheaves E and F the composition map

Ext^{∗}(E, F) ⊗ Ext^{n−∗}(F, E ⊗ ω) →^{∘} Ext^{n}(E, E ⊗ ω) →^{Tr} C

defines a pairing that is degenerate and supercommutative:

Tr(a · b) = (−1)^{deg a}(b ⊗ ω[n] · a).  (22)

Moreover, for morphisms e_{x} and f_{x} defined above

Tr(f_{x}e_{x}) = 1.  (23)

By Serre duality, dim Ext^{n}(O, ω) = dim Ext^{0}(O, O) = 1. An element of Ext^{n}(O, ω) that is taken to 1 by Tr we denote by v. By (23), v = f_{x}e_{x} for any point x ∈ M .

The following lemma is needed for the sequel.

Lemma 7 Let E be a complex of sheaves and s ∈ Ext^{n}(E, E ⊗ ω). For a finite-dimensional graded vector space V = ⨁ V_{i} and a grading-preserving operator A = ⨁ A_{i} consider the morphism ⨁ s ⊗ A_{i} ∈ Ext^{n}(⨁ E ⊗ V_{i}[i], ⨁ E ⊗ V_{i}[i] ⊗ ω). Then

Tr(⨁ s ⊗ A_{i}) = str A · Tr(s),

where str denotes supertrace:

str A = ∑_{i}(−1)^{i} tr A_{i}.

Proof Let v^{1}, v^{2}, … be a union of bases of vector spaces V_{i} and v_{1}, v_{2}, … be the dual basis of the dual vector space. By v^{k} also denote the multiplication operator by the vector from ⨁ Ext^{−i}(E, E ⊗ V_{i}[i]), and by v_{k} denote the contraction operator from ⨁ Ext^{i}(E ⊗ V_{i}[i], E). Then

Tr(⨁ s ⊗ A_{i}) = Tr(Av^{i} · s · v_{i})
= Tr((−1)^{deg v_{i}}Av_{i} · v^{i} · s)
= ∑_{i}(−1)^{i} tr A_{i} · Tr(s).

□

2 Ext^{∗}(id, S) and Chern characters

Lemma 8 An isomorphism is defined

I_{P BW} : S(Ω^{1}[1])_{△} = Ext(O_{△}, ω_{△}[dim M]).

Proof This statement is a ”twisted” version of definition-proposition 1 and the proof is the same. We have

Ext(O_{△}, ω_{△}[dim M) = S(T [−1])_{△} ⊗ p_{1}^{∗}ω[dim M] = S(Ω^{1}[1])_{△}.

□ Corollary Ext_{M×M}^{i}(O_{△}, ω_{△}) = ⨁_{j+k=i} H_{M}^{j}(Ω^{n−k}).

This isomorphism we also denote by I_{P BW}.

The functor − ⊗ ω[n] is called the Serre functor. Obviously, S = Φ_{ω_{△}[n]} (for definition of Φ see subsection 1.3). Any element of Ext^{i}(O_{△}, ω_{△}) defines a morphism of functors Ext^{i−n}(id, S). Describe this morphism explicitly.

Lemma 9 Morphism of functors Θ_{I_{P BW} (Ω)} for Ω ∈ H^{i}(Ω^{j}) acts on a complex of sheaves E as follows

Θ_{I_{P BW} (Ω)}(E) = top(e^{α(E)} ∧ Ω) ∈ Ext^{i−j}(E, SE),  (24)

where e^{α(E)} ∈ ⨁_{i} Ext^{i}(E, E ⊗ Ω^{i}) is the exponent of the Atiyah class and top is the projection onto the forms of top weight.

Proof See lemma 3. □

Definition 2 Chern character Ch_{E} of a complex of sheaves E on M is a linear function on the space Ext^{0}(id, S) defined by the formula:

Ch_{E}(a) = Tr(a_{E}).  (25)

The reader may easily check that this definition is connected with the usual definition of Chern character ch(E) ∈ ⨁_{i} H^{i}(Ω^{i}) as follows:

Ch_{E}(Θ_{P BW (l)}) = ∫ l ∧ ch(E),  (26)

where l ∈ ⨁_{i} H^{i}(Ω^{i}) and ∫ is the Poincaré pairing on the space ⨁_{i} H^{i}(Ω^{i}).

3 Riemann–Roch theorem

Consider morphisms e_{x} and f_{x} introduced in subsection 1 for all points x simultaneously.

Lemma 10 There exist unique morphisms

\begin{matrix}e ∈ Ext_{M×M}^{0}(O, O_{△}) \\ f ∈ Ext_{M×M}^{n}(O_{△}, p_{2}^{∗}ω) \end{matrix}

such that their restrictions on p_{1}^{−1}(x) for any x are equal to e_{x} and f_{x} respectively.

Proof Morphism e is simply the unit section of the sheaf O_{△}. To define f consider the Leray spectral sequence:

H_{M}^{∗}(Rp_{1∗}(Ext(O_{△}, p_{2}^{∗}ω))) ⇒ Ext_{M×M}^{∗}(O_{△}, p_{2}^{∗}ω).

This sequence degenerates in the second term because Rp_{1∗}(Ext(O_{△}, p_{2}^{∗}ω)) is a (shifted) line bundle. Above we define a canonical trivialization of this bundle and the unit section gives us the morphism we need. □

Morphisms e and f define morphisms of functors Θ_{e} ∈ Ext^{0}(Φ_{O}, id) and Θ_{f} ∈ Ext^{n}(id, Φ_{p_{2}^{∗}ω}). The reader will easily check that

Φ_{O}(−) = O ⊗ Ext^{∗}(O, −)

and

Φ_{p_{2}^{∗}ω} = SΦ_{O}[−n].

Taking compositions of Θ_{e} and Θ_{f} and calculating the trace we get Riemann–Roch theorem.

Theorem 3 (Riemann–Roch theorem) There exists an element

Td ∈ Ext^{n}(O_{△}, ω_{△}),

which is called Todd genus, such that for any complex of sheaves E

∑(−1)^{i} dim H^{i}(E) = Ch_{E}(Θ_{Td}).

In other words (see the remark after definition 2),

∑(−1)^{i} dim H^{i}(E) = ∫ ch(E) ∧ I_{P BW} (Td).

Proof Put Td = ef where e and f are defined in lemma 10. Then

Ch_{E}(Θ_{Td}) = Tr(Θ_{ef}(E)) = Tr(Θ_{e}(Φ_{O}(E))Θ_{f}(E))
= Tr(Θ_{f}(E)Θ_{e}(Φ_{O}(E))) = Tr(Θ_{fe}(Φ_{O}(E))).

Consider the product fe ∈ H^{n}(p_{1}^{∗}O ⊗ p_{2}^{∗}ω) = ∑_{i} H^{i}(O) ⊗ H^{n−i}(ω). Only summands with the first factor of degree 0 contribute in the trace. By the remark after theorem 2, the only such a summand has the form 1⊗p_{2}^{∗}v, where v ∈ H^{n}(ω) is defined in the remark. Thus Θ_{fe}(E) ∈ Ext^{n}(O⊗H^{∗}(E), ω⊗H^{∗}(E)) is a multiplication by v. By lemma 7,

Tr(Θ_{fe}(E)) = str id_{H^{∗}(E)} · Tr(v) = ∑(−1)^{i} dim H^{i}(E).

□

4 Holomorphic Lefschetz formula

Before calculating the Todd genus let us consider an example of applying the trick, which we use in the proof of Riemann–Roch theorem, in another analogous situation.

Let φ: M → M be an automorphism of M such that its fixed points are non-degenerate. Let E be a complex of sheaves on M. Suppose a morphism g: φ_{∗}E → E is given (note that φ_{∗} = (φ^{−1})^{∗}, because φ is an automorphism). Then g acts on cohomologies of E as follows:

H^{∗}(E) →^{φ_{∗}} H^{∗}(φ_{∗}E) →^{g} H^{∗}(E)

Denote this action by g_{H} ∈ End(H^{∗}(E)).

For a complex of sheaves F introduce its fiber at a point x by F_{x} = F ⊗^{L}O_{x}, where O_{x} is the structure sheaf of the point. F_{x} is a module over O_{x} that is a vector space. For a fixed point x ∈ Fix φ vector spaces E_{x} and φ_{∗}E_{x} are naturally isomorphic. This yields an action of g on the fiber E_{x}. Denote this action by g_{x} ∈ End(E_{x}).

Supertraces of two actions described are related by the following theorem (see e. g. [GH]).

Theorem 4 (Holomorphic Lefschetz formula)

str g_{H} = ∑_{x∈Fix φ} \frac{str g_{x}}{det(1 − φ′(x)^{−1})},

where φ′(x) is the derivative of φ at x.

Proof Let

O_{Γ} = (φ × id)^{∗}O_{△}

be the structure sheaf of the graph of the map φ. Obviously, Φ_{O_{Γ}} = φ_{∗}. Introduce notations:

e_{Γ} = (φ × id)^{∗}e: O → O_{Γ}
f_{Γ} = (φ × id)^{∗}f : O_{Γ} → p_{2}^{∗}ω.

We shall show that left-hand and right-hand sides of Lefschetz formula are equal to Tr(gΘ_{e_{Γ}}Θ_{f}(E)). L.H.S. Consider a diagram:

O ⊗ H ⟶^{Θ_{e}} E ⟶^{Θ_{f}} SO ⊗ H ⟶^{Θ_{e_{Γ}}} Sφ_{∗}E ⟶^{g} SE
↑ g_{H}  ↑ g
O ⊗ H ⟶^{Θ_{e_{Γ}}} φ_{∗}E

where by H is denoted spaces H^{∗}(φ_{∗}E) = H^{∗}(E). We have

Tr(gΘ_{e_{Γ}}Θ_{f}) = Tr(Θ_{f}gΘ_{e_{Γ}}) = Tr(Θ_{f}Θ_{e}g_{H})
= Tr(Θ_{fe}g_{H}) = Tr(vg_{H}) = str g_{H}.

(27)

R.H.S. Consider the morphism e_{Γ}f ∈ Ext^{n}(O_{△}, O_{Γ} ⊗ p_{2}^{∗}ω). By the hypothesis of the theorem,

supp Ext(O_{△}, O_{Γ} ⊗ p_{2}^{∗}ω) = △ ∩ Γ = {(x, x)|x ∈ Fix φ}

is a set of isolated points. Consider the morphism e_{Γ}f in a neighborhood of any such point x. By a straightforward calculation (adduced below), we get in a neighborhood of x

e_{Γ}f = \frac{1}{det(1 − φ′(x)^{−1})}(p_{2}^{∗}f_{x} ⊗ O_{Γ})(p_{2}^{∗}e_{x} ⊗ O_{△}),  (28)

where e_{x} and f_{x} are defined before theorem 2. This local statement leads to the global result:

e_{Γ}f = ∑_{x∈Fix φ} \frac{1}{det(1 − φ′(x)^{−1})}(p_{2}^{∗}f_{x} ⊗ O_{Γ})(p_{2}^{∗}e_{x} ⊗ O_{△}),

Now we have

Tr(gΘ_{e_{Γ}}Θ_{f}) = Tr(gΘ_{e_{Γ}f}) = ∑_{x∈Fix φ} \frac{1}{det(1−φ′(x)^{−1})} Tr(gf_{x}e_{x})

= ∑_{x∈Fix φ} \frac{1}{det(1−φ′(x)^{−1})} Tr(e_{x}gf_{x}) = ∑_{x∈Fix φ} \frac{1}{det(1−φ′(x)^{−1})} Tr(e_{x}f_{x}g_{x})

= ∑_{x∈Fix φ} \frac{1}{det(1−φ′(x)^{−1})} Tr(vg_{x}) = ∑_{x∈Fix φ} \frac{str g_{x}}{det(1−φ′(x)^{−1})}

(29)

By composing (27) and (29) we conclude the proof. □

Calculation Prove (28).

Let z_{1}, …, z_{n} be local coordinates on M in a neighborhood of the fixed point x. Denote X_{i} = p_{1}^{∗}z_{i} and Y_{i} = p_{2}^{∗}z_{i}.

Write down Koszul resolutions of the sheaves O_{△}, O_{Γ} and the structure sheaf of the point O_{(x,x)} on M × M:

K_{•}(O_{△}) = O ⊗ Λ^{∗} < δ(Y_{1} − X_{1}), …, δ(Y_{n} − X_{n}) >
K_{•}(O_{Γ}) = (φ × id)^{∗}K_{•}(O_{△}) = O ⊗ Λ^{∗} < δ(Y_{1} − φ(X), …, δ(Y_{n} − φ(X)) >
K_{•}(O_{(x,x)}) = O ⊗ Λ^{∗} < δ(X_{1}), …, δ(X_{n}), δ(Y_{1}), …, δ(Y_{n}) > .

Here notations are the same as in (21). We treat δ as a linear operation: δ(X_{i} − Y_{i}) = δ(X_{i}) − δ(Y_{i}) and so on.

One may suppose that φ is linear in chosen coordinates. The reader will have no difficulty in showing that three morphisms appeared in (28) acts on the Koszul complexes as follows:

e_{Γ}f : K_{−n}(O_{△}) ∋ δ(Y_{1} − X_{1}) ∧ … ∧ δ(Y_{n} − X_{n}) ↦
dY_{1} ∧ … ∧ dY_{n} ∈ K_{0}(O_{Γ}) ⊗ p_{2}^{∗}ω
p_{2}^{∗}e_{x} ⊗ O_{△} : K_{−k}(O_{△}) ∋ δ(Y_{i_{1}} − X_{i_{1}}) ∧ … ∧ δ(Y_{i_{k}} − X_{i_{k}}) ↦
(δ(Y_{i_{1}}) − δ(X_{i_{1}})) ∧ … ∧ (δ(Y_{i_{k}}) − δ(X_{i_{k}})) ∈ K_{−k}(O_{(x,x)}
p_{2}^{∗}f_{x} ⊗ O_{Γ} : K_{−n}(O_{(x,x)}) ∋ a ↦

\frac{a ∧ δ(Y_{1} − φ_{1}(X)) ∧ … ∧ δ(Y_{n} − φ_{n}(X))}{δY_{1} ∧ … ∧ δY_{n} ∧ δ(Y_{1} − φ_{1}(X)) ∧ … ∧ δ(Y_{n} − φ_{n}(X))} dY_{1} ∧ … ∧ dY_{n} ∈ K_{0}(O_{Γ}) ⊗ p_{2}^{∗}ω

It follows that

(p_{2}^{∗}f_{x} ⊗ O_{Γ})(p_{2}^{∗}e_{x} ⊗ O_{△}) δ(Y_{1} − X_{1}) ∧ … ∧ δ(Y_{n} − X_{n})

= \frac{δ(Y_{1} − X_{1}) ∧ … ∧ δ(Y_{n} − X_{n}) ∧ δ(Y_{1} − φ_{1}(X)) ∧ … ∧ δ(Y_{n} − φ_{n}(X))}{δY_{1} ∧ … ∧ δY_{n} ∧ δ(Y_{1} − φ_{1}(X)) ∧ … ∧ δ(Y_{n} − φ_{n}(X))} dY_{1} ∧ … ∧ dY_{n}

= det(1 − φ′(x)^{−1}) dY_{1} ∧ … ∧ dY_{n} = det(1 − φ′(x)^{−1}) e_{Γ}f δ(Y_{1} − X_{1}) ∧ … ∧ δ(Y_{n} − X_{n}).

5 Todd genus

Return to Riemann–Roch theorem. Recall that theorem 3 states existence of a particular class Td ∈ Ext^{n}(O_{△}, O_{△} ⊗ p_{2}^{∗}ω), which is constructed in the proof. It remains to find an element from ⨁_{i} H^{i}(Ω^{i}) that corresponds to Td under PBW isomorphism (see corollary of lemma 8). Before do it prove two lemmas.

Let us multiply Td by O_{△}, that is introduce the morphism

id ⊗ Td ∈ Ext^{n}(O_{△} ⊗^{L} O_{△}, O_{△} ⊗^{L} O_{△} ⊗ p_{2}^{∗}ω).

Consider the direct image of this morphism p_{2∗}(id ⊗ Td) ∈ Ext^{n}(S(Ω^{1}[1]), S(Ω^{1}[1]) ⊗ ω), where we

identify S(Ω^{1}[1]) with p_{2∗}(O_{△} ⊗^{L} O_{△}) by the right exponential map (see definition-proposition 2).

Lemma 11 Let i: ω[n] → S(Ω^{1}[1]) be the inclusion of top weight forms. Then

p_{2∗}(id ⊗ Td) ∘ i = 1 ⊗ id.

Proof By the proof of theorem 3, Td = ef. Local calculation shows that

Im(p_{2∗}(id ⊗ Td) ∘ i − 1 ⊗ id) ⊂ ⨁_{i=1}^{n} Ω^{i}[i].

On the other hand

p_{2∗}(id ⊗ ef) = p_{2∗}(id ⊗ e)p_{2∗}(id ⊗ f).

But p_{2∗}(id ⊗ e) ∈ Ext^{0}(O, S(Ω^{1}[1])) is simply an inclusion of O into S(Ω^{1}[1]). It follows that image of p_{2∗}(id ⊗ Td) lies in O, so Im(p_{2∗}(id ⊗ Td) ∘ i − 1 ⊗ id) = 0. □

Consider the object End(S(Ω^{1}[1])). By ı: S(Ω^{1}[1]) → End(S(Ω^{1}[1])) and ı: Λ^{i}T [−i] → End(S(Ω^{1}[1])) denote maps taking a form to the operator of multiplication by the form and a polyvector field to the operator of contraction by this field respectively. The standard fact from Clifford algebras theory claims that the composition map

S(Ω^{1}[1]) ⊗ Λ^{∗}T ⟶^{ı⊗ı} End(S(Ω^{1}[1])) ⊗ End(S(Ω^{1}[1])) ⟶^{∘} End(S(Ω^{1}[1]))  (30)

is an isomorphism. By D_{i} denote image of S(Ω^{1}[1]) ⊗ Λ^{i}T [−i] in End(S(Ω^{1}[1])):

End(S(Ω^{1}[1])) = ⨁_{i} D_{i}.  (31)

Equip S(Ω^{1}[1]) with a non-degenerate supersymmetric pairing taking values in ω[n] by the formula

< −, − >= top(− ∧ −),

where top means the projection on the forms of top weight.

The pairing defines the conjugation operation + on End(S(Ω^{1}[1])) by the formula

< La, b >=< a, L^{+}b >,  (32)

where a and b are local sections. Note that

\begin{matrix}L^{++} = L \\ (L_{1}L_{2})^{+} = (−1)^{deg L_{1} deg L_{2}}L_{2}^{+}L_{1}^{+}. \end{matrix}  (33)

Using isomorphism (30) we have

(ıΩ^{i} · ıΛ^{j}T)^{+} = (−1)^{ij}(−1)^{j}ıΛ^{j}T · ıΩ^{i}.  (34)

For the family of operators L_{L} depending on the tangent bundle, which was introduced in theorem 1, conjugate operators have a very particular form.

Lemma 12 Put

td = det(f(α(T)) ∈ ⨁_{i} H^{i}(Ω^{i}),

where f(z) = z/(1 − e^{z}) and α(T) ∈ H^{1}(Ω^{1} ⊗ End(T)) is the Atiyah class of the tangent bundle. Then

L_{L}^{+} = −ıtd · L_{L} · ıtd^{−1}.

Proof Consider End(S(Ω^{1}[1])) as a Lie superalgebra. Then D_{1} (see (31)) is a subalgebra. End(S(Ω^{1}[1])) is analogous to differential operators algebra, D_{1} is analogous to vector fields Lie algebra. S(Ω^{1}[1]) is analogous to functions ring and D_{1} acts on it by derivations.

Consider an analog of volume form S_{S(Ω^{1}[1])}^{n}D_{1}, that is the top antisymmetric power of D_{1} over S(Ω^{1}[1]). S_{S(Ω^{1}[1])}^{n}D_{1} is a one-dimensional S(Ω^{1}[1])-module. Consider the element Id ∈ H^{0}(T ⊗ Ω^{1}) ⊂ H^{1}(D_{1} ⊗ Ω^{1}). Then Id^{n} ∈ H^{n}(S_{S(Ω^{1}[1])}^{n}D_{1} ⊗ Ω^{n}) gives a generator of S_{S(Ω^{1}[1])}^{n}D_{1}.

D_{1} naturally acts on S_{S(Ω^{1}[1])}^{n}D_{1}. A straightforward calculation gives the commutative diagram

S(Ω^{1}[1]) ⟶^{Id^{n}} S_{S(Ω^{1}[1])}^{n}D_{1} ⊗ ω
↓ L^{+}  ↓ −L
S(Ω^{1}[1]) ⟶^{Id^{n}} S_{S(Ω^{1}[1])}^{n}D_{1} ⊗ ω

(35)

where L ∈ D_{1} Recall that in theorem 1 a family of operators L_{L} on S(Ω^{1}[1]) were introduced. They describe the

direct image of action of the Atiyah class of the diagonal on O_{△} ⊗^{L} O_{△} by acting on the second factor. Introduce also L_{R} = L_{L} − ad. This family describes the direct image of action of the Atiyah class on

O_{△} ⊗^{L} O_{△} by acting on the first factor. Indeed, these families commute with each other.

In according with theorem 1 families of operators L_{L} and L_{R} lie into D_{1}. On may easily check that the element L_{R}^{n} ∈ H^{n}(S_{S(Ω^{1}[1])}^{n}D_{1} ⊗ ω) gives a generator of S_{S(Ω^{1}[1])}^{n}D_{1} as S(Ω^{1}[1])-module. Moreover,

L_{L} · L_{R}^{n} = 0,

because {L_{L}, L_{R}} = 0. Thus by diagram (35),

L_{L}^{+}((Id^{n})^{−1}L_{R}^{n}) = 0.

A calculation (by using formulas α(T) = − α(Ω^{1}) and (14)) shows

(Id^{n})^{−1}L_{R}^{n} = td.

It follows that

L_{L}^{+}td = 0.

Finally we have for local sections a and b

< L_{L}a, b >= top(L_{L}a · b) = top(L_{L}a · td^{−1} b td)
= top(L_{L}(a · td^{−1} b) td) − (−1)^{deg a} top(a · L_{L}(td^{−1} b) td)
=< L_{L}(a td^{−1} b), td > −(−1)^{deg a} < a, td L_{L}(td^{−1} b) >
=< a td^{−1} b, L_{L}^{+}td > −(−1)^{deg a} < a, td L_{L}(td^{−1} b) >
= −(−1)^{deg a} < a, td L_{L}(td^{−1} b) > .

This concludes the proof. □

Theorem 5 (Riemann–Roch theorem, part 2) The Todd genus Td introduced in theorem 3 has the form

Td = I_{P BW} (det f(α(T))),

where f(z) = z/(1 − e^{−z}) and α(T) ∈ H^{1}(Ω^{1} ⊗ End(T)) is the Atiyah class of the tangent bundle.

Proof Let x ∈ ⨁_{i} H^{i}(Λ^{n−i}T ⊗ ω) be the required element that is

Td = I_{P BW} (x).

By definition of PBW isomorphism and L_{L}, we have

p_{2∗}(id ⊗ Td) = e^{L_{L}} ⊢ x,  (36)

where e^{L_{L}} ∈ H^{0}(End(S(Ω^{1}[1])) ⊗ S(Ω^{1}[1])) and ⊢: Λ^{i}T ⊗ Ω^{i} → O is the contraction map.

Strategy of the proof is to reconstruct Td by its action on p_{2∗}(O_{△} ⊗^{L} O_{△}) that is by

p_{2∗}(id ⊗ Td) ∈ H^{n}(End(S(Ω^{1}[1])) ⊗ ω).

Introduce two isomorphisms f_{r,l}: End(S(Ω^{1}[1])) → S(Ω^{1}[1])⊗Λ^{∗}T. We call this isomorphisms the left and the right forms. The left form f_{l} is given by composition map (30) and f_{r} differs by the permutation of factors:

f_{l}^{−1}: S(Ω^{1}[1]) ⊗ Λ^{∗}T →^{σ} Λ^{∗}T ⊗ S(Ω^{1}[1]) ⟶^{ı⊗ı} End(S(Ω^{1}[1])) ⊗ End(S(Ω^{1}[1])) →^{∘} End(S(Ω^{1}[1])),

where σ is the factors permutation (with signs!). By |_{0} denote the constant term map

|_{0}: S(Ω^{1}[1]) ⊗ Λ^{∗}T → Λ^{∗}T

defined by the projection of S(Ω^{1}[1]) to O.

By lemma 11,

f_{r}(p_{2∗}(id ⊗ Td))|_{0} = Id^{n} ∈ H^{0}(Λ^{n}T ⊗ ω),

where Id ∈ H^{0}(T ⊗Ω^{1}) is the canonical element. On the other hand, by definition of PBW isomorphism (see remark after definition-proposition 2),

f_{l}(p_{2∗}(id ⊗ I_{P BW} (−)))|_{0} = −;

in particular,

f_{l}(p_{2∗}(id ⊗ Td))|_{0} = x.

By (34) we have

f_{l}(−^{+}) = If_{r}(−),

where I multiplies Λ^{i}T by (−1)^{i}. By (36) and lemma 12 we have

(p_{2∗}(id ⊗ Td))^{+} = (e^{L_{L}} ⊢ x)^{+} = e^{L_{L}^{+}} ⊢ x
= e^{−tdL_{L}td^{−1}} ⊢ x = td (e^{L_{L}^{+}} ⊢ Ix) td^{−1}
= td (p_{2∗}(id ⊗ I_{P BW} (Ix))) td^{−1}.

So

f_{r}(p_{2∗}(id ⊗ Td)) = If_{l}(td (p_{2∗}(id ⊗ I_{P BW} (Ix))) td^{−1}).

Taking constant terms of both sides we get

Id^{n} = I(td ⊢ Ix),

where ⊢: S(Ω^{1}[1]) ⊗ Λ^{∗}T → Λ^{∗}T is the contraction map. This equality implies the statement of the theorem. □

3 Hochshild homology and relative Riemann–Roch theorem

Notations Let M be a smooth complex manifold. By Δ : M → M ×M denote the diagonal embedding. By O_{△} and ω_{△} denote the sheaves Δ_{∗}O and Δ_{∗}ω respectively, where ω is the canonical bundle. We identify ω_{△} with O_{△} ⊗p_{2}^{∗}ω and ω_{△} ⊗ω_{M×M}^{−1} with O_{△} ⊗p_{2}^{∗}ω^{−1} in natural way, where p_{2} is the projection on the second factor. By S(Ω^{1}[1]) denote differential forms, more exactly S(Ω^{1}[1]) = ⊕_{i}Ω^{i}[i].

By S denote the functor − ⊗ ω[dim M]. Below functors f^{∗}, ⊗, f_{∗} and f_{!} mean Lf^{∗}, ⊗^{L}, Rf_{∗} and Rf_{!}. By D^{♭}(M) denote the derived category of bounded complexes of coherent sheaves on M.

1 Relative Serre duality

For a proper morphism of smooth manifolds f : Y → X define functor f_{!} from D^{♭}(Y) to D^{♭}(X) by the formula

f_{!} = S_{X}^{−1} ∘ f_{∗} ∘ S_{Y} .

By relative Serre duality, functor f_{!} is left adjoint to f^{∗}. We cite here this statement without proof, referring to [H].

Theorem 6 (Relative Serre duality) For a proper morphism of smooth manifolds f : Y → X and objects E ∈ D^{♭}(Y) and F ∈ D^{♭}(X) a bifunctorial isomorphism

D_{f} : Ext(E, f^{∗}F) → Ext(f_{!}E, F)  (37)

is defined. For a pair of morphisms Z →^{f} Y →^{g} X,

D_{g∘f} = D_{g} ∘ D_{f}.  (38)

For a smooth closed embedding f : Y → X, isomorphism D_{f} is defined by the canonical isomorphism

Ext(f_{∗}O_{Y} , O_{X}) ≅ S_{X}^{−1}f_{∗}S_{Y} O_{Y} .  (39)

Remark In theorem 2 the trace function

Tr : Ext(E, SE) → C

were introduced for an object E ∈ D^{♭}(X) on a compact manifold X. For the structure sheaf the trace is equal to the image of 1 ∈ Ext(O_{X}, p_{X}^{∗}O_{pt}) under the duality isomorphism

D_{p_{X}} : Ext(O_{X}, p_{X}^{∗}O_{pt}) → Ext(p_{X!}O_{X}, O_{pt}),

where p_{X} : X → pt is the morphism of the manifold to a point. For a general object E consider a morphism from E to SE as an element x ∈ Ext(End(E), SO). Then the trace of this morphism equals

Tr(x · id),

where x · id ∈ Ext(O, SO) and id ∈ Ext(O, End(E)) is the unit section. In other words, the trace is equal to the value of the linear function D_{p_{X}} (S^{−1}x) ∈ Ext(End(E), O_{pt}) on the unit section.

2 Inverse image of Ext(id, S)’s

For a proper morphism f : Y → X introduce morphisms of functors

\begin{matrix}E_{f} : f^{∗}f_{∗} → id \\ F_{f} : id → f^{∗}f_{!} \end{matrix}  (40)

by formulas

\begin{matrix}E_{f}(E) = C_{f}^{−1}(id_{f_{∗}E}) \\ F_{f}(E) = D_{f}^{−1}(id_{f_{!}E}) \end{matrix}  (41)

where C_{f} denotes the isomorphism Ext(f^{∗}−, −) → Ext(−, f_{∗}−) and D is from theorem 6.

Definition 3 Let f : Y → X be a proper morphism of smooth manifolds and a ∈ Ext_{X}(id, S) be a morphism from the identity functor to S in D^{♭}(X). Then by the inverse image of a we call an element

f^{∗}a ∈ Ext_{Y} (id, S),

such that S^{−1}f^{∗}a(E) is the composition

S^{−1}E ⟶^{F_{f} (S^{−1}E)} f^{∗}f_{!}S^{−1}E ⟶ f^{∗}f_{∗}E ⟶^{E_{f} (E)} E,  (42)

where the middle arrow is the inverse image of the morphism S^{−1}a(f_{∗}E) ∈ Ext(S^{−1}f_{∗}E, f_{∗}E) = Ext(f_{!}S^{−1}E, f_{∗}E).

An important property of the inverse image is the compatibility with the Chern character. Recall (definition 2), that the Chern character of an object E ∈ D^{♭}(X) on a compact manifold X is a linear function on Ext_{X}(id, S) defined by the formula

Ch_{E}(a) = Tr a(E).  (43)

Proposition 1 Let f : Y → X be a morphism of compact smooth manifolds, E ∈ D^{♭}(Y) and a ∈ Ext_{X}(id, S). Then

Ch_{f_{∗}E}(a) = Ch_{E}(f^{∗}a).

Proof Introduce a map

π : f_{!} Ext(E, S^{−1}E) → Ext(f_{∗}E, f_{!}S^{−1}E)

corresponding under D to the map

Ext(E, S^{−1}E) → Ext(f^{∗}f_{∗}E, f^{∗}f_{!}S^{−1}E)

that is the left multiplication by E_{f}(E) and the right multiplication by F_{f}(S^{−1}E) (compare with (42)). Calculate the trace of the composition of maps

S^{−1}O ⟶^{f_{∗}id} f_{!} Ext(E, S^{−1}E) ⟶^{π} Ext(f_{∗}E, f_{!}S^{−1}E) ⟶^{S^{−1}a(f_{∗}E)} O

in two ways. Here the first arrow

f_{∗}id ∈ Ext(S^{−1}O, f_{!} End(E)) = Ext(O, f_{∗} End(E))

is the direct image of the unit section and the last arrow

S^{−1}a(f_{∗}E) ∈ Ext(Ext(f_{∗}E, f_{!}S^{−1}E), O) = Ext(S^{−1}f_{∗}E, f_{∗}E)

is due to the action of the morphism of functors a on f_{∗}E.

It is easy to see that f_{∗}id · π = S^{−1}id_{f_{∗}E}, where id_{f_{∗}E} ∈ Ext(O, End(E)) is the unit section. Consequently (see the remark after theorem 6 and (43)),

Tr(Sf_{∗}id · Sπ · a(f_{∗}E)) = Tr(id · a(f_{∗}E)) = Ch_{f_{∗}E}(a).

On the other hand, by definition,

π · S^{−1}a(f_{∗}E) ∈ Ext(f_{!} Ext(E, S^{−1}E), O)

corresponds under D to

S^{−1}f^{∗}a(E) ∈ Ext(Ext(E, S^{−1}E), f^{∗}O) = Ext(S^{−1}E, E).

Thus, by naturalness of D, Tr(Sf_{∗}id · Sπ · a(f_{∗}E)) is equal to the value of the linear function

D_{p_{X}} (Sπ · a(f_{∗}E)) ∈ Ext(p_{X!}(f_{!} Ext(E, S^{−1}E)), O_{pt})

on the section f_{∗}id (here p_{X} : X → pt and p_{Y} : Y → pt). By (38) for p_{Y} = p_{X} ∘ f, it is equal to the value of the function

D_{p_{Y}} (f^{∗}a(E)) ∈ Ext(p_{Y!} Ext(E, S^{−1}E)), O_{pt}) = Ext(p_{Y ∗} End(E), O_{pt})

on the unit section, that is equal to Ch_{E}(f^{∗}a) (see the remark after theorem 6 and (43)). □

3 Hochshild homology

For a manifold M the space HH_{M} = Tor_{M×M}(O_{△}, O_{△}) is called the Hochshild homology of M. For any morphism f : Y → X the inverse image

f_{HH}^{∗} : HH_{X} → HH_{Y}

is obviously defined.

Theorem 6 supplies the duality isomorphism

D_{Δ} : HH_{M} = Ext(O_{M}, Δ^{∗}Δ_{∗}O_{M}) → Ext(Δ_{!}O_{M}, Δ_{∗}O_{M}) = Ext(O_{△}, ω_{△}[dim M]).  (44)

In section 2.2 we introduced the map Θ : Ext(O_{△}, ω_{△}[dim M]) → Ext_{M}(id, S).

Theorem 7 For a proper morphism f : Y → X of smooth manifolds and a ∈ HH_{X} the following holds

Θ_{D_{Δ}(f_{HH}^{∗}a)} = f^{∗}Θ_{D_{Δ}(a)}.

Proof To prove the theorem take advantage of the following lemma.

Lemma 13 For a proper morphism f : Y → X of smooth manifolds consider the commutative diagram

\begin{matrix}Y & ⟶^{Δ} & Y × Y \\ ↓ f & & ↓ F \\ X & ⟶^{Δ} & X × X, \end{matrix}  (45)

where F = f × f.

1.

\begin{matrix}Φ(F^{∗}Δ_{∗}O_{X}) = f^{∗}f_{∗} \\ Φ(F^{∗}Δ_{!}O_{X}) = f^{∗}S^{−1}f_{∗}, \end{matrix}

where Φ denotes the functor from D^{♭}(Y) to itself corresponding to the object on Y × Y (see section 1.3).

2. Let ε ∈ Ext(F^{∗}Δ_{∗}O_{X}, Δ_{∗}O_{Y}) and ϕ ∈ Ext(Δ_{!}O_{Y} , F^{∗}Δ_{!}O_{X}) be the images of 1 ∈ Ext(O_{Y} , O_{Y}) = Ext(f^{∗}O_{X}, O_{Y}) = Ext(O_{Y} , f^{∗}O_{X}) under the composition of maps

Ext(f^{∗}O_{X}, O_{Y}) → Ext(O_{X}, f_{∗}O_{Y}) ⟶^{Δ_{∗}}

Ext(Δ_{∗}O_{X}, Δ_{∗}f_{∗}O_{Y}) → Ext(Δ_{∗}O_{X}, F_{∗}Δ_{∗}O_{Y}) → Ext(F^{∗}Δ_{∗}O_{X}, Δ_{∗}O_{Y})

and

Ext(O_{Y} , f^{∗}O_{X}) → Ext(f_{!}O_{Y} , O_{X}) ⟶^{Δ_{!}}

Ext(Δ_{!}f_{!}O_{Y} , Δ_{!}O_{X}) → Ext(F_{!}Δ_{!}O_{Y} , Δ_{!}O_{X}) → Ext(Δ_{!}O_{Y} , F^{∗}Δ_{!}O_{X}),

where unmarked arrows designate natural isomorphisms.

Then

Θ_{ε} = E_{f} ∈ Ext(id, f^{∗}f_{∗})

and

Θ_{ϕ} = F_{f}S^{−1} ∈ Ext(S^{−1}, f^{∗}f_{!}S^{−1}).

Proof The first statement is obvious.

To prove the second one we have to show that C_{f}(Θ_{ε}(E)) = id_{f_{∗}E} and D_{f}(Θ_{ϕ}(E)) = id_{f_{!}E} for any E ∈ D^{♭}(Y), where C_{f} : Ext(f^{∗}−, −) → Ext(−, f_{∗}−) is the natural isomorphism and D_{f} is the duality isomorphism (theorem 6). By naturalness of C and D, this equalities are equivalent to following ones

C_{f}(id_{f^{∗}f_{∗}E}) · f_{∗}Θ_{ε}(E) = id_{f_{∗}E}  (46)

and

f_{!}Θ_{ϕ}(E) · D_{f}(id_{f^{∗}f_{!}E}) = id_{f_{!}E}.  (47)

Prove them.

It can easily be checked that f_{∗}f^{∗}f_{∗}E = f_{∗}E ⊗ f_{∗}O, the map C_{f}(id_{f^{∗}f_{∗}E}) ∈ Ext(f_{∗}E, f_{∗}E ⊗ f_{∗}O) takes a local section x to x ⊗ 1, and f_{∗}Θ_{ε}(E) ∈ Ext(f_{∗}E ⊗ f_{∗}O, f_{∗}E) takes a local section x ⊗ y to x · y ( by · we mean the action of f_{∗}O on f_{∗}E). Thus, the composition of this two morphisms is the identity and (46) is proved.

For E ∈ D^{♭}(Y) denote Ext(E, O) by E^{∨}. As it follows from theorem 6 (by using the functorial property of D), the following holds:

f_{∗}(E^{∨}) = (f_{!}E)^{∨}.

The reader will have no difficulty in showing that equality (47) is obtained of (46) by substituting in it E^{∨} except E and applying Ext(−, O) to it. Thus (46) leads (47). This concludes the proof. □

Return to the theorem. It is easy to see that to get the inverse image of a ∈ HH_{X} = Ext(O_{X}, Δ^{∗}Δ_{∗}O_{X}) one should take f^{∗}a ∈ Ext(O_{Y} , f^{∗}Δ^{∗}Δ_{∗}O_{X}) = Ext(O_{Y} , Δ^{∗}F^{∗}Δ_{∗}O_{X}) (see diagram (45)) and apply an element of

Ext(F^{∗}Δ_{∗}O_{X}, Δ_{∗}O_{Y}) = Ext(Δ_{∗}O_{X}, F_{∗}Δ_{∗}O_{Y}) = Ext(Δ_{∗}O_{X}, Δ_{∗}f_{∗}O_{Y})

which is the direct image of 1 ∈ Ext(O_{X}, f_{∗}O_{Y}). By definition, this element is map ε from lemma 13. Thus, the inverse image of a ∈ HH_{X} = Ext(O_{X}, Δ^{∗}Δ_{∗}O_{X}) is equal to the composition

O_{Y} ⟶^{f^{∗}a} f^{∗}Δ^{∗}Δ_{∗}O_{X} = Δ^{∗}F^{∗}Δ_{∗}O_{X} ⟶^{Δ^{∗}ε} Δ^{∗}Δ_{∗}O_{Y} .  (48)

Consider D_{Δ}(a) ∈ Ext(Δ_{!}O_{X}, Δ_{∗}O_{X}). By naturalness of D, a is equal to the composition

O_{X} ⟶^{D_{Δ}^{−1}(id_{Δ_{!}O_{X}})} Δ^{∗}Δ_{!}O_{X} ⟶^{Δ^{∗}D_{Δ}(a)} Δ^{∗}Δ_{∗}O_{X}.

Substituting in (48), we get

f_{HH}^{∗}a = f^{∗}D_{Δ}^{−1}(id_{Δ_{!}O_{X}}) · f^{∗}Δ^{∗}D_{Δ}(a) · Δ^{∗}ε
= f^{∗}D_{Δ}^{−1}(id_{Δ_{!}O_{X}}) · Δ^{∗}F^{∗}D_{Δ}(a) · Δ^{∗}ε.

By naturalness of D,

D_{Δ}(f_{HH}^{∗}a) = Δ_{!}f^{∗}D_{Δ}^{−1}(id_{Δ_{!}O_{X}}) · F^{∗}D_{Δ}(a) · ε.

It is easy to see, that Δ_{!}f^{∗}D_{Δ}^{−1}(id_{Δ_{!}O_{X}}) is equal to map ϕ from lemma 13. Thus, D_{Δ}(f_{HH}^{∗}a) is equal to the composition

Δ_{!}O_{Y} ⟶^{ϕ} F^{∗}Δ_{!}O_{X} ⟶^{F^{∗}D_{Δ}(a)} F^{∗}Δ_{∗}O_{X} ⟶^{ε} Δ_{∗}O_{Y} .  (49)

Applying lemma 13 to (49) and comparing with (42) we get the statement of the theorem. □

4 Relative Riemann–Roch theorem

By the corollary of definition–proposition 2 for a smooth manifold M the isomorphism

exp_{∗} : HH_{M} → H_{M}^{∗}(Ω^{∗}).

is defined. It is well known, that for any morphism of smooth manifolds f : Y → X the following diagram is commutative

HH_{Y} ⟶^{exp_{∗}} H_{Y}^{∗} (Ω^{∗})
↑ f_{HH}^{∗}  ↑ f_{Ω^{∗}}^{∗}
HH_{X} ⟶^{exp_{∗}} H_{X}^{∗}(Ω^{∗}),

(50)

where f_{Ω^{∗}}^{∗} denotes the “usual” inverse image of differential forms (in fact, (50) may be considered as a definition of the “usual” inverse image).

Besides in lemma 8 we introduced an isomorphism

I_{P BW} : H_{M}^{∗}(Ω^{∗}) → Ext_{M×M}(O_{△}, ω_{△}[dim M]).

For a compact manifold M and E ∈ D^{♭}(M),

Ch_{E}(Θ_{I_{P BW} (l)}) = ∫ l ∧ ch(E),  (51)

where l ∈ H^{∗}(Ω^{∗}), ∫ is the Poincaré pairing on H^{∗}(Ω^{∗}) and ch is the usual Chern character (see remark after definition 2).

Finally, in the previous subsection we define the duality isomorphism D_{Δ} : HH_{M} → Ext_{M×M}(O_{△}, ω_{△}[dim M]).

Theorem 8 For a smooth manifold M the following diagram is commutative

HH_{M}  ⟶^{D_{Δ}} Ext(O_{△}, ω_{△}[dim M])
↓ exp_{∗}  ↑ I_{P BW}
H_{M}^{∗}(Ω^{∗}) ⟵^{td_{M}}  H_{M}^{∗}(Ω^{∗}),

where td_{M} means the multiplication by the Todd genus td ∈ ⨁_{i} H^{i}(Ω^{i}) (see e. g. theorem 5).

Proof Below we identify Δ^{∗}Δ_{∗}O_{M} with ⊕_{i}Ω^{i}[i] and Δ_{∗}Δ^{∗}Δ_{∗}O_{M} = O_{△} ⊗ O_{△} with Δ_{∗} ⊕_{i} Ω^{i}[i] by exp_{∗}.

Consider the canonical morphism

D_{Δ}^{−1}(id_{Δ_{!}O}) ∈ Ext(O, Δ^{∗}Δ_{!}O) = Ext(O, S^{−1}Δ^{∗}Δ_{∗}O).

Local analysis and (39) shows that S^{−1}D_{Δ}^{−1}(id_{Δ_{!}O}) is the embedding of shifted form of the highest weight Ω^{n}[n] into ⊕_{i}Ω^{i}[i] as a direct summand.

By naturalness of D, the duality isomorphism D_{Δ}^{−1} for v ∈ Ext(Δ_{!}O, Δ_{∗}O) is given by the composition

O ⟶^{D_{Δ}^{−1}(id_{Δ_{!}O})} Δ^{∗}Δ_{!}O ⟶^{Δ^{∗}v} Δ^{∗}Δ_{∗}O.

It means, that D_{Δ}^{−1}(v) is equal to the result of action of v on

1 ∈ Ext_{M}(O, ⊕_{i}Ω^{i}[i] ⊗ ω^{−1}[−n]) = Ext_{M×M}(O, O_{△} ⊗ O_{△} ⊗ p_{2}^{∗}ω^{−1})

(we mean the “twisted” version of action of Ext(O_{△}, O_{△}) on O_{△} ⊗ O_{△}). By the proof of theorem 5, result of such an action for v = I_{P BW} (α) equals α · td. This proves the theorem. □

Combining lemma 1, theorem 7, theorem 8, (50) and (51) we get the following statement.

Theorem 9 (Relative Riemann–Roch theorem) For a morphism of smooth compact manifolds f : Y → X, E ∈ D^{♭}(Y) and l ∈ H_{X}^{∗}(Ω^{∗}) the following holds

∫ td_{X} ∧ l ∧ ch(f_{∗}E) = ∫ td_{Y} ∧ f^{∗}l ∧ ch(E).

References

[A] Atiyah M. F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. vol.85 (1957) No.1, pp. 181–207.

[B] Bondal A., Non-commutative deformations and Poisson brackets on projective spaces, preprint MPI/93-67.

[NB] Bourbaki, N. Algèbre, Chapitre 10, Algèbre homologique (Eléments de mathématique), Paris, Masson, 1980.

[GH] Griffiths Ph., Harris J., Principles of algebraic geometry, New York, Wiley, 1978.

[H] Hartshorne R., Residues and duality, Heidelberg, Springer, 1966.

[K] Kapranov M., Rozansky–Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113.

[Ko] Kontsevich M., Deformation quantization of Poisson manifolds, I, q-alg/9709040.

e-mail: nikita@mccme.ru