IRREDUCIBILITY OF MODULI SPACES OF VECTOR BUNDLES ON K3 SURFACES

KŌTA YOSHIOKA

0. Introduction

Let X be a projective K3 surface defined over ℂ and H an ample divisor on X. For a coherent sheaf E on X, v(E) := ch(E)\sqrt{td_{X}} ∈ H^{∗}(X, ℤ) is the Mukai vector of E, where td_{X} is the Todd class of X. We denote the moduli space of stable sheaves E of v(E) = v by M_{H}(v). If v is primitive and H is general (i.e. H does not lie on walls [Y3]), then M_{H}(v) is a smooth projective scheme. In [Mu1], Mukai showed that M_{H}(v) has a symplectic structure. In order to get more precise information, Mukai [Mu2] introduced a quite useful notion called Mukai lattice (H^{∗}(X, ℤ), ⟨ , ⟩), where ⟨ , ⟩ is an integral primitive bilinear form on H^{∗}(X, ℤ). By the language of this lattice, we can write down Riemann-Roch theorem in a simple form. In particular we get that dim M_{H}(v) = ⟨v^{2}⟩ + 2.

If v is a primitive isotropic vector, then M_{H}(v) is a surface with a symplectic structure. Mukai proved that M_{H}(v) is a K3 surface and described the period in terms of Mukai lattice. If v is a primitive Mukai vector of ⟨v^{2}⟩ > 0, then M_{H}(v) is a higher dimensional symplectic manifold. If rk v = 1, then M_{H}(v) is the Hilbert scheme of points on X. Beauville [B] proved that it is an example of higher dimensional irreducible symplectic manifold. For an irreducible symplectic manifold, Beauville [B] defined the period and proved local Torelli theorem. As an example, he also computed the period of the Hilbert scheme of points on X. For higher rank cases, Mukai [Mu3] (rank 2 case), O’Grady [O] (any rank but primitive c_{1} case) and the author [Y4] (asymptotic case) proved that M_{H}(v) is an irreducible symplectic manifold and described the period of M_{H}(v) in terms of Mukai lattice. For classification of M_{H}(v), it is important to determine the period. Indeed, it is a birational invariant ([Mu3]), and affirmative solution of Torelli conjecture will imply that an irreducible symplectic manifold is determined by its period, up to birational equivalence.

In this paper, by using [Y4] extensively, we prove the following theorem, which is expected by many people (for example, see [D], [Mu3], [O]).

Theorem 0.1. Let v = l(r + ξ) + aω, ξ ∈ NS(X) be a primitive Mukai vector such that r + ξ is primitive and l, r > 0, where we identify H^{0}(X, ℤ) with ℤ and ω is the fundamental class of X.

(1) M_{H}(v) is not empty for a general ample divisor H if and only if ⟨v^{2}⟩ ≥ −2.

(2) Assume that ⟨v^{2}⟩ ≥ −2. Then for a general ample divisor H,

(2-1) M_{H}(v) is obtained by compositions of deformations and birational transformations from Hilb_{X}^{⟨v^{2}⟩/2+1}.

In particular M_{H}(v) is an irreducible symplectic manifold. (2-2) Let B_{M_{H}(v)} be Beauville’s bilinear form on H^{2}(M_{H}(v), ℤ). Then

θ_{v} : (v^{⊥}, ⟨ , ⟩) → (H^{2}(M_{H}(v), ℤ), B_{M_{H}(v)})

is an isometry which preserves Hodge structures for ⟨v^{2}⟩ ≥ 2, where θ_{v} : v^{⊥} → H^{2}(M_{H}(v), ℤ) is the canonical homomorphism defined by using a quasi-universal family.

Here we only use deformations of M_{H}(v) induced by deformation of complex structures of X.

We shall explain how to prove (2-1) of this theorem. In [Y3], we discussed chamber structure of polarizations. Let v = l(r + ξ) + aω, ξ ∈ NS(X) be a Mukai vector of l, r > 0. We choose an ample divisor H on X which does not lie on walls with respect to v. Then for every μ-semi-stable sheaf E of v(E) = v, if F ⊂ E satisfies (c_{1}(F), H)/ rk F = (c_{1}(E), H)/ rk E, then c_{1}(F)/ rk F = c_{1}(E)/ rk E. Thus v(F) = l′(r + ξ) + a′ω for some l′, a′. In particular, if v is primitive, then M_{H}(v) is compact. Let M(v) be the stack of coherent sheaves E of v(E) = v. We shall fix a general ample divisor H with respect to v. M(v)^{μss} (resp. M(v)^{μs}) denotes the open substack of M(v) consisting of μ-semi-stable sheaves (resp. μ-stable sheaves).

In [Y4], we proved Theorem 0.1 under the assumption ⟨v^{2}⟩ > 2l^{2} or l = 1. Hence we may assume that ⟨v^{2}⟩ ≤ 2l^{2} and l > 1. Our proof in [Y4] is systematic applications of birational maps induced by reflections of Mukai lattices. Then Theorem 0.1 (2-2) automatically follows from these birational correspondences in many cases. In [Y4, Lem. 4.5], we proved the following key lemma under the assumption lr < r_{1} < (l + 1)r.

Lemma 0.2. Let (X, H) be a polarized smooth projective surface of NS(X) = ℤH. Let (r_{1}, d_{1}) and (r, d) be pairs of integers such that r_{1}, r > 0 and dr_{1} − rd_{1} = 1. We assume that lr < r_{1}. Let E_{1} be a μ-stable vector bundle of rk(E_{1}) = r_{1} and deg(E_{1}) = d_{1}, where deg(E_{1}) = (c_{1}(E_{1}), H)/(H^{2}).

1991 Mathematics Subject Classification. 14D20.

(1) Let E be a μ-stable sheaf of rk(E) = lr and deg(E) = ld. Then every non-zero homomorphism φ : E_{1} → E is surjective in codimension 1 and ker φ is a μ-stable sheaf.

(2) Let E′ be a μ-stable vector bundle of rk(E′) = r_{1} − lr and deg(E′) = d_{1} − ld. Let ϕ : E′ → E_{1} be a non-zero homomorphism. Then ϕ is injective and E := coker ϕ is a μ-semi-stable sheaf.

Recently we realized that the most important condition in Lemma 0.2 and [Y4, Lem. 2.1, Lem. 4.1] is the following fact:

• We consider the triangle in ℝ^{2} with vertices (0, 0), (r_{1} − lr, d_{1} − ld) and (r_{1}, d_{1}). Then there is no

integral point in the interior of this triangle. This condition gives a strong restriction on homomorphisms φ, ϕ and the Harder-Narasimhan polygons of ker φ and coker ϕ. Hence we can expect to get rid of the additional condition r_{1} < (l + 1)r. This will be done in Preliminaries.

In order to use this lemma, we need to compare M(v)^{μss} and M(v)^{μs}. More precisely, we need some propositions (cf. [Y4, Lem.4.4]) on the dimension counting of various constructible substack of M(v)^{μss}. This is the most important part in this paper. In [D-L], Drezet and Le Potier computed the dimension of the substack of non-semi-stable sheaves. In their computation, the existence of exceptional vector bundle is very important. In our case, we concentrate our consideration on M(v)^{μss}. By our assumption on H, exceptional vector bundle E of v(E) = r + ξ + bω, b ∈ ℤ is important. Hence we divide our proof into two cases:

A. There is no (−2) vector of the form r + ξ + bω, i.e. ((ξ^{2}) + 2)/2r ∉ ℤ.

B. There is a (−2) vector of the form r + ξ + bω, i.e. ((ξ^{2}) + 2)/2r ∈ ℤ.

In section 2, we treat case A. In particular, we prove the following

dim(M(v)^{μss} \ M(v)^{μs}) ≤ ⟨v^{2}⟩.  (0.1)

Then we can apply Lemma 0.2. For suitable choice of 1. a primitive Mukai vector v := l(r + dH) + aω on (X, H) and

2. an exceptional vector bundle E_{1} of v(E_{1}) = r_{1} + d_{1}H + a_{1}ω, we can construct a birational map M_{H}(v) ⋯ → M_{H}(v′) sending a general μ-stable vector bundle E ∈ M_{H}(v) to F := coker(E^{∨} → Hom(E_{1}, E)^{∨} ⊗ E_{1}^{∨}) ∈ M_{H}(v′) where H and v′ satisfy that (1) (H^{2})/2 = (r_{1}⟨v^{2}⟩/2l + r)r_{1}/l − r^{2} > 0 and (2) Theorem 0.1 holds for M_{H}(v′). We remark that we need to choose a sufficiently large r_{1} for the condition (1). In the same way as in [Y4, 4.3], we get Theorem 0.1 for case A.

In section 3, we treat case B. If ⟨v^{2}⟩ < 2l^{2}, then it is known that there is no μ-stable sheaf. Hence we can not apply Lemma 0.2 in this form. When the (−2) vector is v(O_{X}) = 1 + ω, T. Nakashima found the following fact:

We set v = l − aω. Then the inequality 0 ≤ ⟨v^{2}⟩ < 2l^{2} implies that 0 ≤ a < l. We assume that a ≥ 2. Let E be a μ-stable vector bundle of v(E) = a − lω. Then H^{0}(X, E) = H^{2}(X, E) = 0 and dim H^{1}(X, E) = l − a. We consider the universal extension

0 → E → E′ → H^{1}(X, E) ⊗ O_{X} → 0.  (0.2)

It is easy to see that E′ is a stable vector bundle and we get an immersion M_{H}(a − lω)^{μs,loc} ↪ M_{H}(v), where M_{H}(a − lω)^{μs,loc} is the open subscheme of M_{H}(a − lω) consisting of μ-stable vector bundles.

This result can be easily extended to general cases. Hence what we should do is to prove the irreducibility of M_{H}(v) and the classification of M_{H}(v) consisting of non-locally free sheaves. By similar dimension counting as in case A, we shall classify non-locally free components and prove the irreducibility of M_{H}(v).

1. Preliminaries

1.1. Notation. Except 1.4.1 and 1.4.2, we assume that X is a K3 surface. M(v)^{ss} and M(v)^{s} denote the open substack of M(v) consisting of semi-stable sheaves and stable sheaves respectively. Since M(v)^{μss} is bounded, it is a quotient stack of an open subscheme of some quot-scheme by some general linear group (see Appendix). Hence our dimension counting of substack of M(v)^{μss} can be regarded as that of subscheme of some quot scheme. M_{H}(v)^{μs} (resp. M_{H}(v)^{loc}) be the open subscheme of M_{H}(v) consisting of μ-stable sheaves (resp. stable vector bundles).

For a (−2) vector v_{0}, R_{v_{0}} : H^{∗}(X, ℤ) → H^{∗}(X, ℤ) is the reflection defined by v_{0}, that is, R_{v_{0}}(x) = x + ⟨v_{0}, x⟩v_{0}, x ∈ H^{∗}(X, ℤ). Under suitable conditions (cf. [Y4]), R_{v_{0}} induces a birational map M_{H}(v) ⋯ → M_{H}(±R_{v_{0}}(v)).

1.2. Estimate on properly semi-stable sheaves.

Lemma 1.1. (1) Let E be a stable sheaf and F a semi-stable sheaf such that v(F)/ rk F = v(E)/ rk E. Then Hom(E, F) ⊗ E → F is injective. In particular dim Hom(E, F) ≤ rk F/ rk E.

(2) Let E be a μ-stable sheaf and F a μ-semi-stable sheaf such that c_{1}(F)/ rk F = c_{1}(E)/ rk E. Then Hom(E, F) ⊗ E → F is injective. In particular dim Hom(E, F) ≤ rk F/ rk E.

Lemma 1.2. Let v be a Mukai vector of ⟨v^{2}⟩ > 0 (we don’t assume the primitivity of v). We set

M(v)^{pss} := {E ∈ M(v)^{ss}|E is properly semi-stable }.  (1.1)

Then dim M(v)^{pss} ≤ ⟨v^{2}⟩. In particular, if M(v)^{ss} ≠ ∅, then M(v)^{s} ≠ ∅ and dim M(v)^{ss} = ⟨v^{2}⟩ + 1.

Proof. We set v = lv′, where v′ is a primitive Mukai vector. We shall prove this lemma by induction on l. Let E_{1} be a stable sheaf of v(E_{1}) = l_{1}v′ and E_{2} a semi-stable sheaf of v(E_{2}) = l_{2}v′, where l_{1} + l_{2} = l. By induction hypothesis, dim M^{ss}(v_{i}) = ⟨v_{i}^{2}⟩ + 1, i = 1, 2. We shall estimate the dimension of the substack J(v_{1}, v_{2}) whose element E fits in an extension

0 → E_{1} → E → E_{2} → 0.  (1.2)

By Lemma 1.1, dim Ext^{2}(E_{2}, E_{1}) = dim Hom(E_{1}, E_{2}) ≤ l_{2}/l_{1}. Moreover if E_{1} is general, then Hom(E_{1}, E_{2}) = 0. Hence by Lemma 4.1 in Appendix, we get

dim J(v_{1}, v_{2}) ≤ dim M(v_{1})^{ss} + dim M(v_{2})^{ss} + ⟨v_{1}, v_{2}⟩ + max{l_{2}/l_{1} − 1, 0}
= ⟨v_{1}^{2}⟩ + ⟨v_{2}^{2}⟩ + 2 + ⟨v_{1}, v_{2}⟩ + max{l_{2}/l_{1} − 1, 0}
= (⟨v^{2}⟩ + 1) − (⟨v_{1}, v_{2}⟩ − max{l_{2}/l_{1}, 1}).

(1.3)

Since

⟨v_{1}, v_{2}⟩ = l_{1} \frac{⟨v_{2}^{2}⟩}{2l_{2}} + l_{2} \frac{⟨v_{1}^{2}⟩}{2l_{1}} ≥ l,  (1.4)

we get dim J(v_{1}, v_{2}) ≤ ⟨v^{2}⟩. Therefore we get our lemma.

1.3. Semi-stable sheaves of isotropic Mukai vector.

Lemma 1.3. Let w be a primitive isotropic Mukai vector of rk w > 0. Then dim M(lw)^{ss} = l.

Proof. Let E be a semi-stable sheaf of v(E) = lw. We shall first prove that there are stable sheaves E_{1}, E_{2}, …, E_{k} of v(E_{i}) = w such that

E ≅ ⊕_{i=1}^{k}F_{i},  (1.5)

where F_{i} are S-equivalent to E_{i}^{⊕n_{i}}. Proof of the claim: By the proof of Mukai [Mu2, Prop. 4.4] (Fourier-Mukai transform for H^{∗}(X, ℚ)), there is an element E_{1} of M_{H}(w) such that Hom(E_{1}, E) ≠ 0. By induction hypothesis, E/E_{1} ≅ F_{1} ⊕F_{2} ⊕⋯⊕F_{k}, where F_{1} is S-equivalent to E_{1}^{⊕n_{1}}, n_{1} ≥ 0 and F_{i}, i > 1 are S-equivalent to E_{i}^{⊕n_{i}}. Since Ext^{1}(E_{i}, E_{1}) = 0 for i > 1, Ext^{1}(⊕_{i>1}F_{i}, E_{1}) = 0. Therefore ⊕_{i>1}F_{i} is a direct summand of E, which implies our claim.

Let E_{1} be a stable sheaf of v(E_{1}) = w. We set

J (l, E_{1}) := {E ∈ M(lw)^{ss}|E is S-equivalent to E_{1}^{⊕l}}.  (1.6)

This is a closed substack of M(lw)^{ss} (see Appendix 4.3). We shall next prove that

dim J (l, E_{1}) ≤ −1.  (1.7)

Since E_{1} is parametrized by the surface M_{H}(w), (1.7) implies that

dim M(lw)^{ss} = l.  (1.8)

For more details, see Appendix 4.3. Proof of (1.7): We set

J (l, E_{1}, n) := {E ∈ J (l, E_{1})| dim Hom(E_{1}, E) = n}.  (1.9)

By upper semi-continuity of cohomologies, this is a locally closed substack of J (l, E_{1}). If n = l, then J (l, E_{1}, l) = {E_{1}^{⊕l}} and it is a closed substack of J (l, E_{1}). For an element E of J (l, E_{1}, n), there is an exact sequence

0 → Hom(E_{1}, E) ⊗ E_{1} → E → E′ → 0  (1.10)

where E′ ∈ J (l−n, E_{1}, n′). The moduli number of E which fits in this type of extension is equal to dim J (l− n, E_{1}, n′) + nn′ − n^{2}. Indeed, Hom(E_{1}, E) ⊗ E_{1}(≅ E_{1}^{⊕n}) belongs to J (n, E_{1}, n) and dim J (n, E_{1}, n) = −n^{2}. Hence by the proof of Lemma 4.1, we get the equality. Therefore we see that

dim J (l, E_{1}) = − min_{\begin{matrix}l=n_{1}+n_{2}+···+n_{s} \\ n_{1},n_{2},...,n_{s}≥1 \end{matrix}} (∑_{i=1}^{s} n_{i}^{2} − ∑_{i=1}^{s−1} n_{i}n_{i+1}) ≤ −1.  (1.11)

Remark 1.1. If M_{H}(w) has a universal family, then Fourier-Mukai transform is defined [Br]. Then M(lw)^{ss} is transformed to the stack of 0-dimensional sheaves on M_{H}(w). In this case, by using [Y1, Thm. 0.4], we can get our lemma.

1.4. Lemma 0.2 and its extensions.

1.4.1. Proof of Lemma 0.2. In the proof of [Y4, Lem. 4.5], we shall use the inequality d/r > d_{1}/r_{1}. Then, in the proof of (1), we see that

\frac{1}{r(r_{1} − lr)} > \frac{d}{r} − \frac{deg(I)}{rk(I)} ≥ \frac{1}{r rk(I)}.  (1.12)

Hence rk(I) > r_{1} − lr, which is a contradiction. The other arguments are the same. For the proof of (2), we see that

\frac{1}{r(r_{1} − lr)} > \frac{d}{r} − \frac{deg(ϕ(E′))}{rk(ϕ(E′))} ≥ \frac{1}{r rk(ϕ(E′))}.  (1.13)

Hence rk(ϕ(E′)) > r_{1} − lr, which is a contradiction. The other arguments are the same.

Remark 1.2. In order to define deg(E) in Lemma 0.2, we assumed that NS(X) = ℤH. However deg(E) is still defined if (H^{2})|(H, D) for all D ∈ NS(X). Hence Lemma 0.2 also holds under this assumption.

1.4.2. Extensions of Lemma 0.2 and [Y4, Lem. 4.1]. We shall extend Lemma 0.2 and [Y4, Lem. 4.1]. Let (X, H) be a polarized smooth projective surface of NS(X) = ℤH.

Lemma 1.4. Let E be a μ-semi-stable vector bundle of rk(E) = lr and deg(E) = ld which is defined by a non-trivial extension

0 → F_{1} → E → F_{2} → 0,  (1.14)

where F_{1} and F_{2} are μ-stable vector bundles of deg(F_{1})/ rk(F_{1}) = deg(F_{2})/ rk(F_{2}) = d/r. Let E_{1} be a μ-stable vector bundle in Lemma 0.2. Let φ : E_{1} → E be a non-trivial homomorphism. Then φ is surjective in codimension 1, or φ(E_{1}) is a subsheaf of E_{1}. In particular, if Hom(E_{1}, F_{1}) = 0, then ker φ is μ-stable.

Proof. We assume that φ is not surjective in codimension 1. By [Y4, Lem. 5.1], deg(φ(E_{1}))/ rk(φ(E_{1})) = d/r. Assume that φ(E_{1}) → F_{2} is not 0. Then the μ-stability of F_{2} implies that it is surjective in codimension 1. By the μ-stability of F_{1}, F_{1} ∩ φ(E_{1}) = 0. Since F_{1} is locally free, the proof of [Y4, Lem. 2.1] implies that E ≅ F_{1} ⊕ F_{2}, which is a contradiction. Hence φ(E_{1}) → F_{2} is a 0-map, which means that φ(E_{1}) ⊂ F_{1}. The last assertion follows from the proof of Lemma 0.2.

Lemma 1.5. Keep the notations in Lemma 1.4. Assume that Ext^{1}(E_{1}, F_{2}) = 0. Then a non-trivial extension of E by E_{1} is μ-stable.

Proof. Let E′ be a non-trivial extension of E by E_{1}.

0 → E_{1} → E′ → E → 0.  (1.15)

We shall prove that E′ is μ-stable. We consider the following diagram which is induced by the extension (1.15).

\begin{aligned}& & & & 0 & & 0 \\ & & & & ↑ & & ↑ \\ & & & & F_{2} & & F_{2} \\ & & & & ↑ & & ↑ \\ 0 & ⟶ & E_{1} & ⟶ & E′ & ⟶ & E & ⟶ & 0 \\ & & ∥ & & ↑ & & ↑ \\ 0 & ⟶ & E_{1} & ⟶ & E′′ & ⟶ & F_{1} & ⟶ & 0 \\ & & & & ↑ & & ↑ \\ & & & & 0 & & 0 \end{aligned}  (1.16)

By Serre duality, Ext^{1}(F_{2}, E_{1}) = 0. Hence Ext^{1}(E, E_{1}) → Ext^{1}(F_{1}, E_{1}) is injective. Thus the last horizontal sequence of (1.16) does not split. By [Y4, Lem. 4.1], E′′ is μ-stable. If the middle vertical sequence splits, then E ≅ F_{1} ⊕ F_{2}, which is a contradiction. Hence E′ is a non-trivial extension of F_{2} by a locally free sheaf E′′. By the construction of E′′, the conditions in [Y4, Lem. 4.1] hold. Hence applying [Y4, Lem. 4.1] again, we see that E′ is μ-stable.

1.4.3. Extension of [Y4, Lem. 4.2]. Let (X, H) be a polarized K3 surface of Pic(X) = ℤH. Let E_{1} be an exceptional vector bundle of v(E_{1}) := r_{1} + d_{1}H + a_{1}ω and let v = l(r + dH) + aω be a primitive Mukai vector of dr_{1} − d_{1}r = 1. We set

M_{H}(v)_{i}^{μs} := {E ∈ M_{H}(v)^{μs}| dim Hom(E_{1}, E) = i − ⟨v, v(E_{1})⟩}.  (1.17)

Assume that r_{1} > lr. In the same way as in [Y4, Lem. 4.2], we see that

Lemma 1.6. (1) If ⟨v, v(E_{1})⟩ < 0, then

codim_{M_{H}(v)^{μs}} M_{H}(v)_{i}^{μs} ≥ −⟨v, v(E_{1})⟩ + 1 ≥ 2  (1.18)

for i ≥ 1.

(2) If ⟨v, v(E_{1})⟩ ≥ 0, then

codim_{M_{H}(v)^{μs,loc}}(M_{H}(v)_{i}^{μs} ∩ M_{H}(v)^{μs,loc}) ≥ ⟨v, v(E_{1})⟩ + 1 ≥ 1  (1.19)

for i > ⟨v, v(E_{1})⟩, where M_{H}(v)^{μs,loc} = M_{H}(v)^{μs} ∩ M_{H}(v)^{loc}.

Proof. We only prove the second claim. Hence we assume that i > ⟨v, v(E_{1})⟩. For E ∈ M_{H}(v)_{i}^{μs} ∩ M_{H}(v)^{μs,loc}, we choose a homomorphism ϕ : E_{1} → E. By Lemma 0.2, ϕ is surjective in codimension 1. We set G := coker(E^{∨} → E_{1}^{∨}). By Lemma 0.2, F is a stable sheaf of v(G) = w := v(E_{1})^{∨} − v^{∨}. It is easy to see that dim Hom(E_{1}^{∨}, G) = dim Ext^{1}(E_{1}, E) + 1 = ⟨v, v(E_{1})⟩ + i + 1. Hence ϕ : E_{1} → E is parametrized by an open subscheme of a projective bundle of dimension (⟨v, v(E_{1})⟩ + i) over the subscheme M_{H}(w)_{i}, where

M_{H}(w)_{i} : = {G ∈ M_{H}(w)| dim Hom(E_{1}^{∨}, G) = ⟨v, v(E_{1})⟩ + i + 1}

= {G ∈ M_{H}(w)| dim Hom(E_{1}^{∨}, G) = ⟨w, v(E_{1})^{∨}⟩ + (i − 1)}. (1.20)

Thus we see that

\begin{aligned}dim M_{H}(v)_{i}^{μs} ∩ M_{H}(v)^{μs,loc} & ≤ ⟨w^{2}⟩ + 2 + (⟨v, v(E_{1})⟩ + i) − (i − 1) \\ & = ⟨v^{2}⟩ + 2 − (⟨v, v(E_{1})⟩ + 1). \end{aligned}  (1.21)

2. Case A

2.1. Estimate. In this section, we fix a primitive Mukai vector r + ξ, ξ ∈ NS(X). We assume that

((ξ^{2}) + 2)/2r ∉ ℤ.  (2.1)

We shall prove Theorem 0.1 for a primitive Mukai vector v := l(r + ξ) + aω ∈ H^{∗}(X, ℤ). We shall first estimate the dimensions of various locally closed substacks of M(v).

Lemma 2.1. If M(v)^{μss} ≠ ∅, then ⟨v^{2}⟩ ≥ 0. If the equality holds, then M(v)^{μss} = M(v)^{ss}.

Proof. Let E be a μ-semi-stable sheaf of v(E) = v and E is S-equivalent to ⊕_{i=1}^{s}E_{i} with respect to μ-stability, where E_{i}, 1 ≤ i ≤ s are μ-stable sheaves. We set

v(E_{i}) := l_{i}(r + ξ) + a_{i}ω, 1 ≤ i ≤ s.  (2.2)

By our assumption (2.1), ⟨v(E_{i})^{2}⟩ = l_{i}(l_{i}(ξ^{2}) − 2ra_{i}) ≠ −2. Thus ⟨v(E_{i})^{2}⟩ ≥ 0 for all i. Since

\frac{⟨v^{2}⟩}{l} = ∑_{i=1}^{s} \frac{⟨v(E_{i})^{2}⟩}{l_{i}},  (2.3)

we get ⟨v^{2}⟩ ≥ 0. If ⟨v^{2}⟩ = 0, then ⟨v(E_{i})^{2}⟩ = 0 for all i. Since ⟨v(E_{i})^{2}⟩/ rk(E_{i})^{2} = (ξ^{2}) − 2a_{i}/rl_{i} and χ(E_{i})/ rk(E_{i}) = 1 + a_{i}/rl_{i}, we see that χ(E_{i})/ rk(E_{i}) = χ(E)/ rk(E) for all i. Thus E is semi-stable.

Corollary 2.2. If ⟨v^{2}⟩ = 0, then M(v)^{μss} consists of locally free sheaves.

Definition 2.1. Let w = l_{0}(r + ξ) + a_{0}ω be the primitive Mukai vector such that ⟨w^{2}⟩ = 0.

By Lemma 2.1 and Corollary 2.2, M_{H}(w) consists of μ-stable locally free sheaves.

Lemma 2.3. (1)
dim(M(v)^{μss} \ M(v)^{ss}) ≤ ⟨v^{2}⟩.  (2.4)

(2) Assume that ⟨v^{2}⟩ > 0. Then

dim(M(v)^{μss} \ M(v)^{s}) ≤ ⟨v^{2}⟩.  (2.5)

In particular, if M(v)^{μss} ≠ ∅, then M(v)^{s} ≠ ∅ and dim M(v)^{μss} = ⟨v^{2}⟩ + 1.

Proof. By Lemma 1.2, it is sufficient to prove (1). Let F be a μ-semi-stable sheaf of v(F) = v. We assume that F is not stable. Let

0 ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{s} = F  (2.6)

be the Harder-Narasimhan filtration of F. We set

v_{i} := v(F_{i}/F_{i−1}) = l_{i}(r + ξ) + a_{i}ω, 1 ≤ i ≤ s.  (2.7)

Since χ(F_{i}/F_{i−1})/ rk(F_{i}/F_{i−1}) > χ(F_{i+1}/F_{i})/ rk(F_{i+1}/F_{i}), we get that

\frac{a_{0}}{l_{0}} ≥ \frac{a_{1}}{l_{1}} > \frac{a_{2}}{l_{2}} > ⋯ > \frac{a_{s}}{l_{s}} .  (2.8)

Let F^{HN}(v_{1}, v_{2}, …, v_{s}) be the substack of M(v)^{μss} whose element E has the Harder-Narasimhan filtration of the above type. We shall prove that dim F^{HN}(v_{1}, v_{2}, …, v_{s}) ≤ ⟨v^{2}⟩. Since Hom(F_{i}/F_{i−1}, F_{j}/F_{j−1}) = 0 for i < j, Lemma 4.2 in Appendix implies that

dim F^{HN}(v_{1}, v_{2}, …, v_{s}) = ∑_{i=1}^{s} dim M(v_{i})^{ss} + ∑_{i<j}⟨v_{j}, v_{i}⟩.  (2.9)

For i < j, by using Lemma 2.1 and (2.8), we see that

\begin{aligned}⟨v_{i}, v_{j}⟩ & = l_{i}l_{j}(ξ^{2}) − (l_{i}a_{j} + l_{j}a_{i})r \\ & = l_{i}l_{j}(ξ^{2}) − 2l_{j}a_{i}r + (a_{i}l_{j} − a_{j}l_{i})r \\ & = l_{j}(l_{i}(ξ^{2}) − 2a_{i}r) + (a_{i}l_{j} − a_{j}l_{i})r \\ & ≥ (a_{i}l_{j} − a_{j}l_{i})r ≥ r ≥ 2, \end{aligned}  (2.10)

where the inequality r ≥ 2 comes from our assumption (2.1). Hence if ⟨v_{i}^{2}⟩ > 0 for all i, then, by using Lemma 1.2, we see that

dim F^{HN}(v_{1}, v_{2}, …, v_{s}) ≤ (⟨v^{2}⟩ + 1) − (∑_{i<j}⟨v_{i}, v_{j}⟩ − s + 1) ≤ ⟨v^{2}⟩.  (2.11)

Assume that ⟨v_{i}^{2}⟩ = 0, i.e. v_{i} = l′_{i}w, l_{i} ∈ ℤ. Then i = 1 and (a_{1}l_{j} − a_{j}l_{1}) is divisible by l′_{i}. Hence

\begin{aligned}⟨v_{1}, v_{j}⟩ − l′_{1} & = l′_{1}(⟨w, v_{j}⟩ − 1) \\ & ≥ l′_{1}(r − 1) > 0. \end{aligned}  (2.12)

In this case, by using Lemma 1.3, we see that

dim F^{HN}(v_{1}, v_{2}, …, v_{s}) ≤ (⟨v^{2}⟩ + 1) − (∑_{i<j}⟨v_{i}, v_{j}⟩ − (l′_{1} + s − 1) + 1) ≤ ⟨v^{2}⟩.  (2.13)

Hence we get our lemma.

Proposition 2.4. Assume that ⟨v^{2}⟩ > 0. Then

dim(M(v)^{s} \ M(v)^{μs}) ≤ ⟨v^{2}⟩.  (2.14)

In particular, M(v)^{μs} ≠ ∅, if M(v)^{s} ≠ ∅.

Proof. Let E be a stable sheaf and E_{1} be a μ-stable subsheaf of E such that E/E_{1} is torsion free. We set

\begin{matrix}v_{1} := v(E_{1}) = l_{1}(r + ξ) + a_{1}ω, \\ v_{2} := v(E/E_{1}) = l_{2}(r + ξ) + a_{2}ω. \end{matrix}  (2.15)

Since χ(E_{1})/ rk E_{1} < χ(E)/ rk E, we get ⟨v(E_{1})^{2}⟩ > 0 and

\frac{a_{1}}{l_{1}} < \frac{a_{2}}{l_{2}} .  (2.16)

Let J(v_{1}, v_{2}) be the substack of M(v)^{s} consisting of E which has a subsheaf F_{1} ⊂ E. By using Lemma 4.1 in Appendix, we shall estimate dim J(v_{1}, v_{2}). By Lemma 1.1, dim Hom(E_{1}, E/E_{1}) ≤ l_{2}/l_{1}, and if E_{1} is general, then Hom(E_{1}, E/E_{1}) = 0. We shall bound the dimension of the substack

N (v_{1}, v_{2}) := {(E_{1}, E_{2}) ∈ M(v_{1})^{μss} × M(v_{2})^{μss}| dim Hom(E_{1}, E_{2}) ≠ 0}.  (2.17)

For a fixed E_{2} ∈ M(v_{2})^{ss},

#{E_{1}^{∨∨}|E_{1} ∈ M(v_{1})^{μs}, Hom(E_{1}, E_{2}) ≠ 0} < ∞.  (2.18)

Hence, by using [Y1, Thm. 0.4], we see that

dim{E_{1} ∈ M(v_{1})^{μs}| Hom(E_{1}, E_{2}) ≠ 0} ≤ dim M(v_{1})^{μs} − 2 − (rk v_{1} − 1).  (2.19)

Thus dim N (v_{1}, v_{2}) ≤ dim M(v_{1})^{μss} + dim M(v_{2})^{μss} − 3. Moreover, taking (1.5) into account, if l_{1} ≠ l_{0} and v_{2} = l′_{2}w, l′_{2} ∈ ℤ, then we get N (v_{1}, v_{2}) = ∅.

If ⟨v_{2}^{2}⟩ > 0, then Lemma 2.3 implies that dim M(v_{2})^{μss} = ⟨v_{2}^{2}⟩ + 1. Hence Lemma 4.1 implies that

dim M(v)^{s} − dim J(v_{1}, v_{2}) = min (⟨v_{1}, v_{2}⟩ − \frac{l_{2}}{l_{1}} + 2, ⟨v_{1}, v_{2}⟩ − 1)

= l_{1} \frac{⟨v_{2}^{2}⟩}{2l_{2}} + l_{2} \frac{⟨v_{1}^{2}⟩}{2l_{1}} − max (\frac{l_{2}}{l_{1}} − 2, 1) > 0.

(2.20)

We next treat the case where ⟨v_{2}^{2}⟩ = 0. Then v_{2} = l′_{2}w, l′_{2} ∈ ℤ. By Lemma 2.3 (1) and Lemma 1.3, dim M(v_{2})^{μss} = ⟨v_{2}^{2}⟩ + l′_{2}. If l_{1} = l_{0}, then l_{2}/l_{1} = l′_{2}. So we see that

dim M(v)^{s} − dim J(v_{1}, v_{2}) = min (l′_{2}(⟨v_{1}, w⟩ − 1 − 1) + 3, l′_{2}(⟨v_{1}, w⟩ − 1))

= min{l′_{2}((a_{0}l_{1} − a_{1}l_{0})r − 2) + 3, l′_{2}((a_{0}l_{1} − a_{1}l_{0})r − 1)} > 0. (2.21)

Remark 2.1. By the proof of Lemma 2.3 and Proposition 2.4, we see that

codim_{M(v)^{μss}}(M(v)^{μss} \ M(v)^{μs}) ≥ 2 for r ≥ 3.  (2.22)

Moreover if r = 2, then the general member E of M(v)^{s} \ M(v)^{μs} fits in the following exact sequence

0 → E_{1} → E → E_{2} → 0  (2.23)

where E_{1} is a μ-stable vector bundle and E_{2} is a μ-stable vector bundle of v(E_{2}) = w.

Indeed, if codim_{M(v)^{s}}(M(v)^{s} \ M(v)^{μs}) = 1, then (1) ⟨v_{2}^{2}⟩ > 0, l_{1} = l_{2} = ⟨v_{1}^{2}⟩/2 = ⟨v_{2}^{2}⟩/2 = 1, or (2) r = 2, l′_{2} = 1 and a_{0}l_{1} − a_{1}l_{0} = 1. By (2.16), case (1) does not occur.

2.2. Proof of Theorem 0.1 for case A.

2.2.1. The case of r > 2. In the same way as in the proof of [Y4, Thm. 0.1], we shall prove Theorem 0.1 if r > 2. For more details, see [Y4, sect. 4]. Let v = l(r + ξ) + aω be a primitive Mukai vector on a K3 surface X such that r + ξ, ξ ∈ NS(X) is primitive. Then we can find a primitive Mukai vector v′ = l(r +d′H′)+a′ω on a polarized K3 surface (X′, H′) of Pic(X′) = ℤH′ and an exceptional vector bundle G of v(G) := v_{1} = r_{1} + d_{1}H′ + a_{1}ω′ such that (1) r + d′H′ is primitive, (2) ⟨v^{2}⟩ = ⟨(v′)^{2}⟩, (3) a ≡ a′ mod l,

(4) d′r_{1} − d_{1}r = 1 and (5) ⟨v(G), v′⟩ = −1, where ω′ is the fundamental class of X′. Then we get

−R_{v_{1}}(v′)^{∨} = (r_{1} − lr) − (d_{1} − ld′)H′ + (a_{1} − a′)ω′.  (2.24)

We may assume that r_{1} − lr > 0. Since r_{1} − lr and d_{1} − ld′ are relatively prime, [Y4, Thm. 0.1] implies that M_{H′}(−R_{v_{1}}(v′)^{∨}) ≠ ∅ and Theorem 0.1 holds for this space. By using Lemma 0.2 (2), we see that M(v′)^{μss} ≠ ∅. Then Lemma 2.3 and Proposition 2.4 imply that M_{H′}(v′)^{μs} ≠ ∅. Applying Lemma 0.2 again, we get a birational map

M_{H′}(v′) ⋯ → M_{H′}(−R_{v_{1}}(v′)^{∨}),  (2.25)

which means that Theorem 0.1 (1) and (2-1) hold for M_{H′}(v′). Then [Y4, Prop. 1.1] implies that Theorem 0.1 (1) and (2-1) also hold for M_{H}(v), where H is a general ample divisor on X. Moreover by Remark 2.1, (2-2) also holds if r > 2.

2.2.2. The case of r = 2. We next treat the case where r = 2. It is sufficient to extend the birational map (2.25) to a general member E ∈ M_{H′}(v′) which fits in (2.23). Let G be the exceptional vector bundle in 2.2.1. By using Lemma 1.4 and 1.5, we can prove the following: Assume that ⟨v′, v(G)⟩ = −1. Then, for a general member E which fits in the exact sequence (2.23),

(1) Ext^{1}(G, E) = 0,

(2) ϕ : Hom(G, E) ⊗ G → E is surjective in codimension 1 and ker ϕ is stable.

Proof. Since ⟨(v′)^{2}⟩ > 0, we can write v′ = xw − yω, where x, y ∈ ℚ and x, y > 0. Since ⟨v′, v(G)⟩ = x⟨w, v(G)⟩ + y rk(G) = −1, ⟨w, v(G)⟩ < 0. Hence ⟨v(E_{1}), v(G)⟩ ≥ 0 and ⟨v(E_{2}), v(G)⟩ < 0. Applying Lemma 1.6, we may assume that Hom(G, E_{1}) = Ext^{1}(G, E_{2}) = 0. By Lemma 1.5, we can use the same argument as in the proof of [Y4, Lem. 4.2]. Hence (1) holds. Applying Lemma 1.4, we get (2).

Therefore the dual of ϕ : E^{∨} → G^{∨} is injective and the cokernel is stable. Thus the reflection induces a birational map M_{H}(v) \ Z → M_{H}(v(G)^{∨} − v^{∨}) such that codim_{M_{H}(v)} Z ≥ 2. Therefore Theorem 0.1 also holds for this case.

2.2.3. Some remarks. Under the assumptions for case A, the following holds.

(1) M_{H}(v) ≠ ∅ if and only if ⟨v^{2}⟩ ≥ 0.

(2) M_{H}(v)^{μs} ≠ ∅ if and only if ⟨v^{2}⟩ ≥ 0. In particular, M_{H}(v) contains μ-stable vector bundles if M_{H}(v) ≠ ∅.

3. Case B

3.1. The case of ⟨v^{2}⟩ < 2l^{2}. In this section, we assume that there is a (−2) vector v_{0} of the form v_{0} = r + ξ + bω, b ∈ ℤ. Let E_{0} be the element of M_{H}(v_{0}). We shall prove Theorem 0.1 for a primitive Mukai vector v := lv_{0} − aω. By [Y4, Thm. 0.1], we may assume that ⟨v^{2}⟩ ≤ 2l^{2}. We first treat the case where ⟨v^{2}⟩ < 2l^{2}. Clearly ⟨v^{2}⟩ ≥ −2, if M_{H}(v) ≠ ∅. If ⟨v^{2}⟩ = −2, then 2l(a rk v_{0} − l) = −2. Hence we get l = 1, which was treated in [Y4]. So we assume that ⟨v^{2}⟩ ≥ 0, that is

l ≤ a rk v_{0} < 2l.  (3.1)

Based on the next key lemma, we shall prove Theorem 0.1 in 3.1.2 and 3.1.3.

Lemma 3.1. codim_{M_{H}(v)}(M_{H}(v) \ M_{H}(v)^{loc}) ≥ 2 unless

v = \begin{cases}(rk v_{0})v_{0} − ω, \\ lv_{0} − (l + 1)ω, rk v_{0} = 1. \end{cases}  (3.2)

3.1.1. Proof of Lemma 3.1. We set

M(v, n)^{s} := {E ∈ M(v)^{s}| dim(E^{∨∨}/E) = n}.  (3.3)

For E ∈ M(v, n)^{s}, let

0 ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{s} = E^{∨∨}  (3.4)

be the Harder-Narasimhan filtration of E^{∨∨}. We set

v_{i} := v(F_{i}/F_{i−1}) = l_{i}v_{0} − a_{i}ω, 1 ≤ i ≤ s.  (3.5)

We shall estimate the codimension of the substack

M(v, n; v_{1}, v_{2}, …, v_{s}) := {E ∈ M(v, n)^{s}|E^{∨∨} ∈ F^{HN}(v_{1}, v_{2}, …, v_{s})},  (3.6)

where F^{HN}(v_{1}, v_{2}, …, v_{s}) is defined as in the proof of Lemma 2.3. We divide our consideration into two cases

(I) s ≥ 2, that is, E^{∨∨} is not semi-stable. (II) s = 1, that is, E^{∨∨} is semi-stable. Case (I). (I-a) If ⟨v_{1}^{2}⟩ ≥ 0, then we can use almost the same arguments as in the proof of Lemma 2.3. The difference comes from the inequality r ≥ 2 which was used in (2.10) and (2.12). Thus

dim F^{HN}(v_{1}, v_{2}, …, v_{s}) ≤ ⟨v(E^{∨∨})^{2}⟩ + 1.  (3.7)

The equality holds only if r = 1 and ⟨v_{1}^{2}⟩ = 0. Since F_{s} is locally free, F_{1} is also locally free. On the other hand, if r = 1 and ⟨v_{1}^{2}⟩ = 0, then M(v_{1})^{ss} consists of non-locally free sheaves (cf. Lemma 1.3). Therefore the equality does not hold. Hence, by using [Y1, Thm. 0.4], we see that

\begin{aligned}dim M(v, n; v_{1}, v_{2}, …, v_{s}) & < ⟨(v − nω)^{2}⟩ + 1 + n(rk v + 1) \\ & ≤ ⟨v^{2}⟩ + 1 − n(rk v − 1). \end{aligned}  (3.8)

Thus we get a desired estimate

codim_{M(v)^{s}} M(v, n; v_{1}, v_{2}, …, v_{s}) ≥ 2.  (3.9)

(I-b) We assume that ⟨v_{1}^{2}⟩ < 0. Then F_{1} = E_{0}^{⊕l_{1}}, which implies that dim M(v_{1})^{ss} = −l_{1}^{2} = ⟨v_{1}^{2}⟩ + l_{1}^{2}. For convenience sake, we set

v′_{2} := ∑_{i=2}^{s} v_{i} = l′_{2}v_{0} − a′_{2}ω.  (3.10)

(I-b-1) We first assume that

⟨(v′_{2})^{2}⟩ = 2l′_{2}(a′_{2} rk v_{0} − l′_{2}) > 0.  (3.11)

Then, since ⟨v_{i}^{2}⟩ ≥ 0 for all i ≥ 2, dim F^{HN}(v_{2}, …, v_{s}) ≤ ⟨(v′_{2})^{2}⟩ + 1. Thus we get

⟨v(E^{∨∨})^{2}⟩ + 1 − dim F^{HN}(v_{1}, v_{2}, …, v_{s}) ≥ ⟨v_{1}, v′_{2}⟩ − l_{1}^{2}.  (3.12)

We shall prove that

⟨v_{1}, v′_{2}⟩ − l_{1}^{2} + n(l rk v_{0} − 1) ≥ 2.  (3.13)

Proof. Since E is stable, F_{1} ∩ E satisfies that

\frac{χ(F_{1} ∩ E)}{rk F_{1}} < \frac{χ(E)}{rk E}.  (3.14)

Since χ(F_{1} ∩ E) ≥ χ(F_{1}) − n and v_{1} = l_{1}v_{0} (i.e. a_{1} = 0), we see that

nl′_{2} − a′_{2}l_{1} > 0.  (3.15)

By our assumption (3.1), (a′_{2} + n) rk v_{0} < 2(l_{1} + l′_{2}). By using (3.11), we see that

n ≤ \frac{2l_{1} + l′_{2} − 1}{rk v_{0}}.  (3.16)

By using (3.11) and (3.15), we see that

nl_{1} rk v_{0} ≥ l_{1}^{2} + 1.  (3.17)

Assume that rk v_{0} ≥ 2. Then (3.16) implies that n < l_{1} + l′_{2}/2. By using (3.11) and (3.17), we see that

⟨v_{1}, v′_{2}⟩ − l_{1}^{2} + n(l rk v_{0} − 1) =l_{1}(−2l′_{2} + a′_{2} rk v_{0}) + n((l_{1} + l′_{2}) rk v_{0} − 1) − l_{1}^{2}
= − l_{1}l′_{2} + l_{1}(a′_{2} rk v_{0} − l′_{2}) + nl_{1} rk v_{0} + l_{1}l′_{2} + l′_{2}(n rk v_{0} − l_{1}) − n − l_{1}^{2}
≥l_{1}(a′_{2} rk v_{0} − l′_{2}) + l′_{2}(n rk v_{0} − l_{1}) + 1 − n
≥l_{1} + l′_{2} + 1 − n > 1.

(3.18)

Assume that rk v_{0} = 1. Then similar computations work if a′_{2} rk v_{0} − l′_{2} > 1. So we assume that rk v_{0} = a′_{2} rk v_{0} − l′_{2} = 1. Then (3.17) implies that n − l_{1} > 0. Hence we see that

l_{1}(−2l′_{2} + a′_{2} rk v_{0}) + n((l_{1} + l′_{2}) rk v_{0} − 1) − l_{1}^{2} = (l_{1} + l′_{2} − 1)(n − l_{1}) ≥ (l − 1) ≥ 1.  (3.19)

If the equality holds, then l = 2 and n − l_{1} = 1. In this case, we get that l_{1} = l′_{2} = 1 and n = 2. By (3.15), we get a contradiction. Thus the left hand side of (3.19) is greater than or equal to 2. Therefore (3.13) holds.

By (3.13) and [Y1, Thm. 0.4], we get a desired estimate

codim_{M(v)^{s}} M(v, n; v_{1}, v_{2}, …, v_{s}) ≥ 2.  (3.20)

(I-b-2) We next assume that

⟨(v′_{2})^{2}⟩ = 2l′_{2}(a′_{2} rk v_{0} − l′_{2}) = 0.  (3.21)

Then s = 2 and ⟨v_{0}, v′_{2}⟩ = −l′_{2}. Since Hom(E_{0}, F_{2}/F_{1}) = 0, dim Hom(F_{2}/F_{1}, E_{0}) ≥ l′_{2}. By Lemma 1.1, coevaluation map

ψ : F_{2}/F_{1} → E_{0} ⊗ Hom(F_{2}/F_{1}, E_{0})^{∨}  (3.22)

is surjective in codimension 1 (cf. Lemma 1.1). Therefore we get

(i) dim Hom(F_{2}/F_{1}, E_{0}) = l′_{2}, (ii) dim Ext^{1}(F_{2}/F_{1}, E_{0}) = 0 and (iii) ψ is isomorphic in codimension 1.

By the definition of Harder-Narasimhan filtration, ψ is not isomorphic. Thus F_{2}/F_{1} is not locally free. By (ii), E^{∨∨} ≅ F_{1} ⊕ F_{2}/F_{1}, which contradicts the locally freeness of E^{∨∨}. Thus this case does not occur. By (I-a), (I-b-1) and (I-b-2), we get a desired bound for the case (I).

Case (II). We divide our consideration into three cases (II-a) ⟨v^{2}⟩ > 0, (II-b) ⟨v^{2}⟩ = 0 and (II-c) ⟨v^{2}⟩ < 0. (II-a) If ⟨v_{1}^{2}⟩ > 0, then

dim M(v, n; v_{1}) = dim M(v_{1}) + n(rk v + 1)
= ⟨v_{1}^{2}⟩ + 1 + n(rk v + 1)
= ⟨v^{2}⟩ + 1 − n(rk v − 1).

(3.23)

Hence if rk v ≥ 3, then codim_{M(v)^{s}} M(v, n; v_{1}) ≥ 2. If rk v = 2, then the condition (3.1) implies that a = 3. Hence we get ⟨v_{1}^{2}⟩ = ⟨v^{2}⟩ − 2n rk v ≤ 0. Therefore this case does not occur.

(II-b) If ⟨v_{1}^{2}⟩ = 0, then the argument in (I-b-2) implies that M(v_{1})^{ss} consists of non-locally free sheaves, which is a contradiction.

(II-c) We assume that ⟨v_{1}^{2}⟩ < 0, that is, E^{∨∨} = E_{0}^{⊕l}. Then (3.1) implies that n rk v_{0} − l ≥ 0.

(II-c-1) We first assume that n rk v_{0} − l ≥ 1. Then we get

codim_{M(v)^{s}} M(v, n) = (2nl rk v_{0} − 2l^{2} + 1) − (n(l rk v_{0} + 1) − l^{2})

= n(l rk v_{0} − 1) − (l^{2} − 1)

≥ \frac{l + 1}{rk v_{0}}(l rk v_{0} − 1) − (l^{2} − 1)

= \frac{(l + 1)(rk v_{0} − 1)}{rk v_{0}} ≥ 0,

(3.24)

and the equality holds if and only if rk v_{0} = 1 and n rk v_{0} − l = 1. By the computation of (3.24), it is easy to show that codim_{M(v)^{s}} M(v, n) ≥ 2, if rk v_{0} ≥ 2, or n rk v_{0} − l ≥ 2.

(II-c-2) If n rk v_{0} − l = 0, then the primitivity of v implies that n = 1. Therefore we get a desired bound for the case (II).

By (I) and (II), we complete the proof of Lemma 3.1.

3.1.2. Components containing locally free sheaves. We shall prove Theorem 0.1 unless v = (rk v_{0})v_{0} − ω, or rk v_{0} = 1 and v = lv_{0} − (l + 1)ω. For a locally free sheaf E ∈ M(v)^{s}, we consider the dual of E. Since ⟨v_{0}, v⟩ = a rk v_{0} − 2l < 0 and E is stable, l′ := dim Ext^{2}(E_{0}, E) ≥ 2l − a rk v_{0} > 0. By Serre duality, we get an exact sequence

0 → (E_{0}^{∨})^{⊕l′} → E^{∨} → F → 0,  (3.25)

where F ∈ M(v^{∨} − l′v_{0}^{∨})^{μss}. Then we see that

Hom(E_{0}^{∨}, F) = Ext^{2}(E_{0}^{∨}, F) = 0.  (3.26)

Since l′ < l, we see that

\begin{aligned}⟨v(F)^{2}⟩ & = 2l(a rk v_{0} − l) − 2l′(l′ − 2l + a rk v_{0}) \\ & = 2(l − l′)(a rk v_{0} − l + l′) \\ & ≥ 2(l − l′)l > 2(l − l′)^{2}. \end{aligned}  (3.27)

Hence, by [Y4, Lem. 4.4], we get dim M(v^{∨} − l′v_{0}^{∨})^{μss} = ⟨(v − l′v_{0})^{2}⟩ + 1. Taking into account (3.26), we see that the moduli number of E^{∨} which fits in the exact sequence (3.25) is given by

dim M(v^{∨} − l′v_{0}^{∨})^{μss} + dim Gr(⟨v^{∨} − l′v_{0}^{∨}, v_{0}^{∨}⟩, l′) = ⟨v^{2}⟩ + 1 − l′(l′ + ⟨v_{0}, v⟩),  (3.28)

(cf. [Y4, Lem. 2.6]). We set

M_{H}(v)^{∗} := \{E ∈ M_{H}(v)^{loc} | \begin{matrix}Ext^{1}(E_{0}, E) = 0, \\ coker(Hom(E, E_{0}) ⊗ E_{0}^{∨} → E^{∨}) ∈ M_{H}(R_{v_{0}}(v^{∨})) \end{matrix}\} .  (3.29)

Then, by using [Y4, Lem. 4.4], Lemma 3.1 and (3.28), we see that codim_{M_{H}(v)}(M_{H}(v) \ M_{H}(v)^{∗}) ≥ 2. Since Theorem 0.1 holds for M_{H}(R_{v_{0}}(v^{∨})), M_{H}(R_{v_{0}}(v^{∨})) is irreducible. Therefore the morphism M_{H}(v)^{∗} → M_{H}(R_{v_{0}}(v^{∨})) is birational, if M_{H}(v) ≠ ∅. Conversely, for a μ-stable vector bundle F ∈ M_{H}(R_{v_{0}}(v^{∨})), we consider the universal extension

0 → Ext^{1}(F, E_{0}^{∨})^{∨} ⊗ E_{0}^{∨} → E′ → F → 0.  (3.30)

We claim that (E′)^{∨} is a stable sheaf of v((E′)^{∨}) = v. This means that M_{H}(v) ≠ ∅ and Theorem 0.1 holds for this case. Proof of the claim: Assume that (E′)^{∨} is not stable. Since (E′)^{∨} is a μ-semi-stable vector bundle, there is a subbundle G such that

(1) (E′)^{∨}/G is torsion free,

(2) v(G) = l_{1}v_{0} − a_{1}ω and

(3) a_{1}/l_{1} < a/l.

Since F^{∨} is μ-stable and (E′)^{∨}/G is torsion free, F^{∨}∩G = 0, or F^{∨}. If F^{∨}∩G = 0, then G → Ext^{1}(F, E_{0})⊗E_{0} is injective. Since the slopes of G and E_{0} are the same and G is locally free, we see that G ≅ E_{0}^{⊕l_{1}}, which means that (3.30) is not the universal extension. Therefore F^{∨} ∩ G must be equal to F^{∨}. This means that G contains F^{∨} and v(G/F^{∨}) = (l_{1} + l − a rk v_{0})v_{0} − (a_{1} − a)ω. Since G/F^{∨} is a subsheaf of Ext^{1}(F, E_{0}) ⊗ E_{0}, a_{1} − a ≥ 0. On the other hand, (3) and l_{1} ≤ l implies that a_{1} < a, which is a contradiction. Hence (E′)^{∨} is stable.

3.1.3. Non-locally free components. We shall prove Theorem 0.1 for

v = \begin{cases}(rk v_{0})v_{0} − ω, \\ lv_{0} − (l + 1)ω, rk v_{0} = 1. \end{cases}  (3.31)

Proposition 3.2. Assume that v = (rk v_{0})v_{0} − ω. Then M_{H}(v) ≅ X.

Proof. By the argument in (I-b-2), E ∈ M_{H}(v) satisfies that E^{∨∨} = E_{0}^{⊕ rk v_{0}}. Since v = v(E_{0}^{⊕ rk v_{0}}) − ω, E is the kernel of a quotient E_{0}^{⊕ rk v_{0}} → ℂ_{x}, x ∈ X. We shall construct a family of stable sheaves {E_{x}}_{x∈X} of v(E_{x}) = v and prove our proposition. For convenience sake, we set X_{i} := X, i = 1, 2. Let Δ ⊂ X_{2} × X_{1} be the diagonal. We denote the projections X_{2} × X_{1} → X_{i}, i = 1, 2 by p_{i}. We shall consider the evaluation map

ϕ : E_{0}^{∨} ⊠ E_{0} → (E_{0}^{∨} ⊠ E_{0})_{|Δ} → O_{Δ}.  (3.32)

Then it is easy to see that ϕ_{x} := ϕ_{|{x}×X_{1}}, x ∈ X_{2} is surjective and the induced homomorphism

Hom(E_{0}, (E_{0}^{∨} ⊠ E_{0})_{|{x}×X}) → Hom(E_{0}, ℂ_{x})  (3.33)

is an isomorphism. Hence ker ϕ_{x} is stable. Since O_{Δ} is flat over X_{2}, E := ker ϕ is flat over X_{2} and E_{|{x}×X} = ker ϕ_{x}, x ∈ X_{2} is stable. Thus we get a morphism X_{2} → M_{H}(v), which is an isomorphism.

Corollary 3.3. R_{v_{0}} is the Fourier-Mukai transform defined by E.

Proof. Let F : D(X_{1}) → D(X_{2}) be the functor defined by

F(x) := R Hom_{p_{2}}(E, p_{1}^{∗}(x)), x ∈ D(X_{1})  (3.34)

and F̂ : D(X_{2}) → D(X_{1}) the functor defined by

F̂(y) := Rp_{1∗}(E ⊗ p_{2}^{∗}(y)), y ∈ D(X_{2}).  (3.35)

Then F̂[2] gives the inverse of F. Let E be a coherent sheaf on X_{1} such that

\begin{matrix}Hom(E_{0}, E) = 0, \\ Ext^{2}(E_{0}, E) = 0. \end{matrix}  (3.36)

We shall prove that E satisfies WIT_{1} for F, i.e.

Hom_{p_{2}}(E, p_{1}^{∗}(E)) = Ext_{p_{2}}^{2}(E, p_{1}^{∗}(E)) = 0  (3.37)

and F^{1} := Ext_{p_{2}}^{1}(E, p_{1}^{∗}(E)) fits in the universal extension of E by E_{0}: Since O_{Δ} is flat over X_{1} and K_{X} is trivial, we get

Hom^{i}(O_{Δ}, p_{1}^{∗}(E)) = \begin{cases}0, i = 0, 1, \\ O_{Δ} ⊗ p_{1}^{∗}(E), i = 2. \end{cases}  (3.38)

By using local-global spectral sequence, (3.36) and (3.38), we see that (3.37) holds and we get an exact sequence

0 → Ext^{1}(E_{0}, E) ⊗ E_{0} → Ext_{p_{2}}^{1}(E, p_{1}^{∗}(E)) → E → 0.  (3.39)

We claim that this sequence gives the universal extension of E by E_{0}. Proof of the claim: If it is not the universal extension, since Ext^{1}(E_{0}, E) ≅ Ext^{1}(E, E_{0})^{∨}, E_{0} must be a direct summand of F^{1}(E). Since E satisfies WIT_{1} for F, F^{1}(E) satisfies WIT_{1} for F̂, which implies that E_{0} also satisfies WIT_{1} for F̂. In particular, R^{2}p_{1∗}(E ⊗ p_{2}^{∗}(E_{0})) = 0. On the other hand, a direct computation shows that R^{2}p_{1∗}(E ⊗ p_{2}^{∗}(E_{0})) ≅ E_{0}, which is a contradiction. Therefore (3.39) is the universal extension. Thus F^{1}(E) is the reflection of E by v_{0}. As in [Y5], we define F^{H} : H^{∗}(X_{1}, ℤ) → H^{∗}(X_{2}, ℤ) by

F^{H}(x) := p_{2∗} (ch(E)^{∨}p_{1}^{∗}\sqrt{td_{X_{1}}}p_{2}^{∗}\sqrt{td_{X_{2}}}p_{1}^{∗}(x)) , x ∈ H^{∗}(X_{1}, ℤ).  (3.40)

Since ch(E)^{∨} = p_{1}^{∗}(ch(E_{0}))^{∨}p_{2}^{∗}(ch(E_{0})) − ch(O_{Δ})^{∨}, we also see that F^{H}(x) = −(x + ⟨x, v(E_{0})⟩v(E_{0})).

Finally we shall treat M_{H}(lv_{0} − (l + 1)ω), rk v_{0} = 1.

Proposition 3.4. If rk v_{0} = 1, then M_{H}(lv_{0} −(l+1)ω) ≅ Hilb_{X}^{l+1} and θ_{v} is an isometry of Hodge structures.

Proof. We may assume that v_{0} = 1 + ω. Let E be an element of M_{H}(l − ω). We shall first prove that E^{∨∨} ≅ O_{X}^{⊕l}. To see this, it is sufficient to prove that

n := dim(E^{∨∨}/E) = l + 1.  (3.41)

Proof of (3.41): Since χ(E) = l −1 and E is stable, Serre duality implies that dim H^{0}(X, E^{∨}) ≥ l −1. Hence we get an exact sequence

0 → O_{X}^{⊕(l−1)} → E^{∨} → I_{Z} → 0  (3.42)

where I_{Z} ∈ Hilb_{X}^{l+1−n}. If n = 0, then E is locally free. By taking the dual of (3.42), we get a section of E, which contradicts the stability of E. We assume that 0 < n < l + 1. Then dim Ext^{1}(I_{Z}, O_{X}) = dim H^{1}(X, I_{Z}) = l − n. Hence we get a decomposition E^{∨} = O_{X}^{⊕(n−1)} ⊕ F, where F fits in an exact sequence

0 → O_{X}^{⊕(l−n)} → F → I_{Z} → 0.  (3.43)

Then E^{∨∨} has a subsheaf O_{X}^{⊕n}. The stability of E implies that

\frac{χ(O_{X}^{⊕n} ∩ E)}{n} < \frac{χ(E)}{l}.  (3.44)

Since χ(O_{X}^{⊕n} ∩ E) ≥ χ(O_{X}^{⊕n}) − n = n, this is impossible. Therefore (3.42) holds, which implies that E^{∨∨} = O_{X}^{⊕l}.

Conversely for a general quotient ϕ : O_{X}^{⊕l} → ⊕_{i=1}^{l+1}ℂ_{x_{i}}, x_{1}, x_{2}, …, x_{l+1} ∈ X, it is easy to see that ker ϕ is stable. Thus M_{H}(l − ω) ≠ ∅.

We shall prove that it is isomorphic to Hilb_{X}^{l+1}. For this purpose, we shall consider a functor G : D(X_{1}) → D(X_{2})_{op} which is the composition of reflection by v(O_{X}) with the taking dual functor:

G(x) := R Hom_{p_{2}}(p_{1}^{∗}(x), I_{Δ}), x ∈ D(X_{1}),  (3.45)

where we use the same notation as in Proposition 3.2 and D(X_{2})_{op} is the opposite category of D(X_{2}). Then G gives an equivalence of categories. For E ∈ M_{H}(l − ω), we shall prove that

(a) Ext_{p_{2}}^{i}(p_{1}^{∗}(E), I_{Δ}) = 0, i = 0, 2 and

(b) G^{1}(E) := Ext_{p_{2}}^{1}(p_{1}^{∗}(E), I_{Δ}) is an ideal sheaf of colength l + 1.

Then the map M_{H}(l −ω) → Hilb_{X}^{l+1} sending E to G^{1}(E) gives an isomorphism of moduli spaces. The second assertion follow from [Y5, Prop. 2.5] or a direct computation by using the equality

G(E) = R Hom_{p_{M_{H} (l−ω)}}(E, O_{M_{H}(l−ω)×X_{1}}) ⊠ O_{X_{2}} − RHom(E, O_{M_{H}(l−ω)×X_{2}})  (3.46)

as an element of Grothendieck group of M_{H}(l−ω)×X_{2}, where E is a quasi-universal family on M_{H}(l−ω)×X.

Proof of (a), (b): By Serre duality and the stability of E, Ext^{2}(E, I_{x}) = 0 for all x ∈ X. Also we see that Hom(E, I_{x}) = 0 for x ∉ Supp(E^{∨∨}/E). Hence by the base change theorem and its proof, we see that (a) holds and G^{1}(E) is torsion free. It is easy to see that v(G(E)) = −R_{v(O_{X})}(v(E)^{∨}) = −1 + lω. Hence v(G^{1}(E)) = 1 − lω. Therefore G^{1}(E) is an ideal sheaf of colength l + 1.

3.2. The case of ⟨v^{2}⟩ = 2l^{2}. We next treat the case where ⟨v^{2}⟩ = 2l^{2}. Let E be a general member of M_{H}(v) \ M_{H}(v)^{μs}. Then by the proof of [Y4, Lem. 4.4], E fits in an exact sequence

0 → E_{1} → E → E_{0} → 0  (3.47)

where E_{1} is a μ-stable vector bundle.

Indeed, if ⟨v^{2}⟩ = 2l^{2}, then the primitivity of v implies that (i) v = lv_{0} − 2ω, l = rk v_{0} or (ii) v = lv_{0} − ω, 2l = rk v_{0}. In particular rk v_{0} > 1. In the notation of the proof of [Y4, Lem. 4.4], we see that k = 1. Hence E fits in the above exact sequence. Then in the same way as in 2.2.2, we get Theorem 0.1 in this case.

3.3. Some remarks. By the proof of Theorem 0.1 for case B and [Y4, Lem. 4.4], we also get the following.

(1) M_{H}(v)^{μs} ≠ ∅ if and only if ⟨v^{2}⟩ ≥ 2l^{2}.

(2) M_{H}(v)^{loc} = ∅ if and only if v = (rk v_{0})v_{0} − ω, or rk v_{0} = 1 and v = lv_{0} − (l + 1)ω.

4. Appendix

In this appendix, we shall explain our method for dimension counting of substacks of M(v)^{μss}. Since most results are appeared in another forms (cf. [D-R], [H-N]), we only give an outline.

4.1. Notation. For an ample divisor H′ on X, let Q(mH′, v) be the open subscheme of the quot-scheme Quot_{O_{X}(−mH′)^{⊕N} /X/ℂ} consisting of points

λ : O_{X}(−mH′)^{⊕N} → E  (4.1)

such that 1. v(E) = v,

2. λ induces an isomorphism H^{0}(X, O_{X}^{⊕N}) ≅ H^{0}(X, E(mH′)),

3. H^{i}(X, E(mH′)) = 0, i > 0. Let O_{Q(mH′,v)×X}(−mH′)^{⊕N} → Q_{v} be the universal quotient. We set V_{v} := O_{X}(−mH′)^{⊕N}. For our purpose, the choice of mH is not so important. Hence we simply denote Q(mH′, v) by Q(v).

Let q_{v} : Q(v) → M(v) be the natural map. We denote the pull-backs q_{v}^{−1}(M(v)^{μss}), q_{v}^{−1}(M(v)^{ss}), … by Q(v)^{μss}, Q(v)^{ss}, … respectively. If we choose a suitable Q(v), then q_{v} : Q(v)^{μss} → M(v)^{μss} is surjective and M(v)^{μss} is a quotient stack of Q(v)^{μss} by a natural action of G_{v} := GL(N):

M(v)^{μss} ≅ [Q(v)^{μss}/G_{v}] .  (4.2)

¿From now on, we assume that q_{v} : Q(v)^{μss} → M(v)^{μss} is surjective.

4.2. Stack of filtrations.

Definition 4.1. F(v_{1}, v_{2}) is the stack of filtrations F_{1} ⊂ E, E ∈ M(v) such that 1. F_{1} is a μ-semi-stable sheaf of v(F_{1}) = v_{1}.

2. E/F_{1} is a μ-semi-stable sheaf of v(E/F_{1}) = v_{2}. Let p_{v} : F(v_{1}, v_{2}) → M(v)^{μss} be the projection sending (F_{1} ⊂ E) to E and p_{v_{1},v_{2}} : F(v_{1}, v_{2}) → M(v_{1})^{μss} × M(v_{2})^{μss} the morphism sending (F_{1} ⊂ E) to (F_{1}, E/F_{1}).

We consider an open subscheme F (v_{1}, v_{2}) of Quot_{Q_{v}/Q(v)^{μss}×X/Q(v)^{μss}} consisting of quotients (Q_{v})_{x} → E_{2}, x ∈ Q(v)^{μss} such that E_{2} is a μ-semi-stable sheaf of v(E_{2}) = v_{2}. Then

F(v_{1}, v_{2}) = [F (v_{1}, v_{2})/G_{v}].  (4.3)

We shall give another expression of F(v_{1}, v_{2}) which is useful to compute the dimensions of substacks of F(v_{1}, v_{2}) and its projections to M(v).

We shall choose Q(mH′, v)^{μss}, Q(mH′, v_{1})^{μss} and Q(mH′, v_{2})^{μss} for the same mH′. Then V_{v} = V_{v_{1}} ⊕V_{v_{2}}. For simplicity, we set Q_{i} := Q_{v_{i}}, V_{i} := V_{v_{i}}, … and K_{i} are the universal subsheaves of O_{Q(v_{i})^{μss}} ⊗ V_{i}, i = 1, 2. We define a scheme ϖ : Y → Q(v_{1})^{μss} × Q(v_{2})^{μss} by

Y := {ψ : (K_{2})_{x_{2}} → (Q_{1})_{x_{1}}|(x_{1}, x_{2}) ∈ Q(v_{1})^{μss} × Q(v_{2})^{μss}}.  (4.4)

Then Y parameterizes subsheaves K ⊂ V such that K ∩V_{1} = (K_{1})_{x_{1}} and K/K ∩V_{1} = (K_{2})_{x_{2}}: For a quotient ψ : (K_{2})_{x_{2}} → (Q_{1})_{x_{1}}, the subsheaf K of V_{v} is defined by

K := {(a_{1}, a_{2}) ∈ V_{1} ⊕ V_{2}|a_{2} ∈ (K_{2})_{x_{2}}, ψ(a_{2}) = a_{1}mod(K_{1})_{x_{1}}}.  (4.5)

Considering the quotient V_{v} → V_{v}/K, Y also parameterizes the following exact and commutative diagram:

\begin{aligned}& & 0 & & 0 & & 0 \\ & & ↑ & & ↑ & & ↑ \\ 0 & ⟶ & (Q_{1})_{x_{1}} & ⟶ & E & ⟶ & (Q_{2})_{x_{2}} & ⟶ & 0 \\ & & ↑ & & ↑ & & ↑ \\ 0 & ⟶ & V_{1} & ⟶ & V_{v} & ⟶ & V_{2} & ⟶ & 0 \end{aligned}  (4.6)

Let ξ : Y × G_{v} → F (v_{1}, v_{2}) be the morphism sending (y : V_{v} → E, g) ∈ Y × G_{2} to

(y ∘ g : V_{v} → E, E → Q_{x_{2}}) ∈ F (v_{1}, v_{2}),  (4.7)

where ϖ(y) = (x_{1}, x_{2}). Let P be the parabolic subgroup of G_{v} fixing V_{1}. Then Y has a natural action of P and ξ induces a morphism Y ×_{P} G_{v} → F (v_{1}, v_{2}), which is G_{v}-equivariant. It is easy to see that this morphism is an isomorphism (cf. [Y2, appendix]). Therefore

\begin{aligned}F(v_{1}, v_{2}) & ≅ [Y ×_{P} G_{v}/G_{v}] \\ & ≅ [Y/P] . \end{aligned}  (4.8)

By using (4.8), we shall prove the following.

Lemma 4.1. We set

N^{n}(v_{1}, v_{2}) : = {(E_{1}, E_{2}) ∈ M(v_{1})^{μss} × M(v_{2})^{μss}| dim Hom(E_{1}, E_{2}) = n},
F^{n}(v_{1}, v_{2}) : = p_{v_{1},v_{2}}^{−1}(N^{n}(v_{1}, v_{2}))
= {(F_{1} ⊂ E) ∈ F(v_{1}, v_{2})| dim Hom(F_{1}, E/F_{1}) = n}.

(4.9)

Then,

dim F^{n}(v_{1}, v_{2}) = dim N^{n}(v_{1}, v_{2}) + ⟨v_{1}, v_{2}⟩ + n.  (4.10)

Proof. We set

Q^{n}(v_{1}, v_{2}) := {(x_{1}, x_{2}) ∈ Q(v_{1})^{μss} × Q(v_{2})^{μss}| dim Hom((Q_{1})_{x_{1}}, (Q_{2})_{x_{2}}) = n}.  (4.11)

For (x_{1}, x_{2}) ∈ Q^{n}(v_{1}, v_{2}), there is an exact sequence

0 → Hom((Q_{2})_{x_{2}}, (Q_{1})_{x_{1}}) → Hom(V_{2}, (Q_{1})_{x_{1}}) → Hom((K_{2})_{x_{2}}, (Q_{1})_{x_{1}}) → Ext^{1}((Q_{2})_{x_{2}}, (Q_{1})_{x_{1}}) → 0. (4.12)

Since dim Hom(V_{2}, (Q_{1})_{x_{1}}) = rk V_{1} rk V_{2} and dim Ext^{2}((Q_{2})_{x_{2}}, (Q_{1})_{x_{1}}) = n, Hom(K′_{2}, Q′_{1}) is a locally free sheaf of rank ⟨v_{1}, v_{2}⟩ + rk V_{1} rk V_{2} + n on Q^{n}(v_{1}, v_{2}), where K′_{2} and Q′_{1} are pull-backs of K_{2} and Q_{1} to Q^{n}(v_{1}, v_{2}) respectively. We set Y^{n} := 𝕍(Hom(K′_{2}, Q′_{1})^{∨}) → Q^{n}(v_{1}, v_{2}). Then F^{n}(v_{1}, v_{2}) ≅ [Y^{n}/P]. Hence we get that

dim F^{n}(v_{1}, v_{2}) = dim Y^{n} − dim P
= dim Q^{n}(v_{1}, v_{2}) + ⟨v_{1}, v_{2}⟩ + n − (dim G_{1} + dim G_{2})
= dim N^{n}(v_{1}, v_{2}) + ⟨v_{1}, v_{2}⟩ + n.

(4.13)

By similar method as in the proof of Lemma 4.1, we can also prove the following.

Lemma 4.2. Let F^{0}(v_{1}, v_{2}, …, v_{s}) be the stack of filtrations

0 ⊂ F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{s} = E, E ∈ M(v)  (4.14)

such that 1. F_{i}/F_{i−1}, 1 ≤ i ≤ s are semi-stable of v(F_{i}/F_{i−1}) = v_{i}.

2. Hom(F_{i}/F_{i−1}, F_{j}/F_{j−1}) = 0, i < j. Then

dim F^{0}(v_{1}, v_{2}, …, v_{s}) = ∑_{i=1}^{s} M(v_{i})^{ss} + ∑_{i<j}⟨v_{i}, v_{j}⟩.  (4.15)

4.3. Supplement for the proof of Lemma 1.3. We shall explain how to derive (1.8) from (1.7). Let q_{l} : Q(lw)^{ss} → \overline{M}_{H}(lw) := Q(lw)^{ss}/G_{lw} be the quotient map. For a sequence of positive integers l_{1} ≤ l_{2} ≤ ⋯ ≤ l_{s} of ∑_{i=1}^{s} l_{i} = l, we set

\begin{aligned}\overline{M}_{H}(lw; l_{1}, l_{2}, …, l_{s}) : & = \{⊕_{i=1}^{s}E_{i}^{⊕l_{i}} ∈ \overline{M}_{H}(lw)|E_{1}, E_{2}, …, E_{s} ∈ M_{H}(w), E_{i} ≠ E_{j} for i ≠ j\}, \\ Q(lw; l_{1}, l_{2}, …, l_{s})^{ss} : & = q_{l}^{−1}(\overline{M}_{H}(lw; l_{1}, l_{2}, …, l_{s})). \end{aligned} (4.16)

For simplicity, we set G := G_{lw} and G_{i} := G_{l_{i}w}, i = 1, 2, …, s. For quotients ϕ_{i} : V_{i} → Q_{x_{i}} ∈ Q(l_{i}w; l_{i})^{ss}, i = 1, 2, …, s and an element g ∈ G, we define a quotient

(⊕_{i=1}^{s}ϕ_{i}) ∘ g : V → ⊕_{i=1}^{s}Q_{x_{i}}.  (4.17)

It will define a morphism ∏_{i=1}^{s} Q(l_{i}w; l_{i})^{ss} × G → Q(lw). Let ∏_{i=1}^{s} Q(l_{i}w; l_{i})^{ss} ×_{∏_{i} G_{i}} G be the quotient of ∏_{i=1}^{s} Q(l_{i}w; l_{i})^{ss} × G by a natural action of ∏_{i} G_{i}. Then the above morphism induces a morphism

π_{l_{1},l_{2},...,l_{s}} : ∏_{i=1}^{s} Q(l_{i}w; l_{i})^{ss} ×_{∏_{i} G_{i}} G → Q(lw).  (4.18)

By the construction of π_{l_{1},l_{2},...,l_{s}}, im π_{l_{1},l_{2},...,l_{s}} contains Q(lw; l_{1}, l_{2}, …, l_{s})^{ss}. Hence

dim [Q(lw; l_{1}, l_{2}, …, l_{s})^{ss}/G] ≤ ∑_{i=1}^{s} dim [Q(l_{i}w; l_{i})^{ss}/G_{i}] .  (4.19)

We shall prove that

dim [Q(l_{i}w; l_{i})^{ss}/G_{i}] ≤ 1.  (4.20)

Then dim [Q(lw; l_{1}, l_{2}, …, l_{s})^{ss}/G] ≤ s ≤ l. Clearly dim [Q(lw; 1, 1, …, 1)^{ss}/G] = l. Hence dim[Q(lw)^{ss}/G] = l.

Proof of (4.20): Since q_{l_{i}} : Q(l_{i}w; l_{i})^{ss} → \overline{M}_{H}(l_{i}w; l_{i}) is surjective and dim \overline{M}_{H}(l_{i}w; l_{i}) = 2, it is sufficient to prove that

dim [q_{l_{i}}^{−1}(E^{⊕l_{i}})/G_{i}] ≤ −1, E ∈ M_{H}(w).  (4.21)

By definition, J (l_{i}, E) = [q_{l_{i}}^{−1}(E^{⊕l_{i}})/G_{i}]. Hence we obtain this claim from (1.7).

Acknowledgement. This paper heavily depends on Mukai’s wonderful works [Mu1,2,3]. I would like to thank his papers, and also him for valuable discussion on his works. I would also like to thank Max Planck Institut für Mathematik for support and hospitality.

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[Y1] Yoshioka, K., The Betti numbers of the moduli space of stable sheaves of rank 2 on P^{2}, J. reine angew. Math. 453 (1994), 193–220

[Y2] Yoshioka, K., The Betti numbers of the moduli space of stable sheaves of rank 2 on a ruled surface, Math. Ann. 302 (1995), 519–540

[Y3] Yoshioka, K., Chamber structure of polarizations and the moduli of stable sheaves on a ruled surface, Internat. J. Math. 7 (1996), 411–431

[Y4] Yoshioka, K., An application of exceptional bundles to the moduli of stable sheaves on a K3 surface, alg-geom/9705027 Some examples of Mukai’s reflections on K3 surfaces, (extended version) J. reine angew. Math. to appear

[Y5] Yoshioka, K., Moduli spaces of stable sheaves on abelian surfaces, preprint

Max Planck Institut für Mathematik, Vitatsgasse 7, D-53111 Bonn, Germany & Department of mathematics, Faculty of Science, Kobe University, Kobe, 657, Japan

   E-mail address: yoshioka@mpim-bonn.mpg.de, yoshioka@math.kobe-u.ac.jp