A remark on the 2-dimensional moduli spaces of vector bundles on K3 surfaces

Takeshi ABE

1 Introduction

In [4], Mukai proved that a suface component of a moduli space of vector bundles on a K3 suface is a K3 surface if we take a general polarization. In this note, we shall consider what happens to the moduli space if we take a special polarization. Our main theorem(Theorem3.3) says that the surface component of the moduli space of semistable sheaves on a K3 surface has a surjective biratonal morphism from a K3 surface. We also investigate the singularity of the moduli space in some cases. The tool we shall use is the moduli of twisted semistable sheaves introduced by Matsuki and Wentworth[2]. By taking suitable ℚ-divisor L, we obtain a natural morphism from the moduli space of L-twisted stable sheaves to the moduli space of semistable sheaves. Considering the curves on the moduli space of L-twisted stable sheaves which contract to a point on the moduli space of semistable sheaves, we can prove the main theorem.

We shall explain the content of this paper. In section2, we fix our notations. In section3, we state the main theorem of this paper. In section4, we recall Matsuki-Wentworth’s twisted stability, which plays an important role in the proof of the main theorem. Section5 is devoted to the proof of the main theorem.

Acknowledgements. The author is grateful to Professors Masaki Maruyama, Atsushi Moriwaki and Akira Ishii for their heartful encouragement and useful advices. He also expresses thanks to Professor Hiraku Nakajima for useful comments.

2 Notation

X always denotes a projective K3 surface over ℂ and H is a fixed ample line bundle on X. As in [4], we put

H̃(X, ℤ) = H^{0}(X, ℤ) ⊕ H^{2}(X, ℤ) ⊕ H^{4}(X, ℤ)

For a coherent sheaf F on X,

v(E) = ch(F)\sqrt{td(X)} ∈ H̃(X, ℤ)
= rank(F) + c_{1}(F) + \{rank(F) + \frac{1}{2}(c_{1}(F)^{2} − 2c_{2}(F))\}

We call v(F) the Mukai vector of F. The symmetric bilinear form ( , ) on H̃(X, ℤ) is defined by the following formula:

(α, β) = α^{2} ∪ β^{2} − α^{0} ∪ β^{4} − α^{4} ∪ β^{0} ∈ H^{4}(X, ℤ) ≃ ℤ

for α = (α^{0}, α^{2}, α^{4}), β = (β^{0}, β^{2}, β^{4}). By Riemann-Roch theorem, we have (v(F_{1}), v(F_{2})) = dim Ext^{1}(F_{1}, F_{2}) − dim Hom(F_{1}, F_{2}) − dim Hom(F_{2}, F_{1}).

For a torsion-free sheaf F on X, F is said to be H-stable [resp. Hsemistable] if F is stable [resp. semistable] with respect to H in Gieseker-Maruyama’s sense. F is said to be H-slope-stable [resp. H-slope-semistable] if F is μ-stable [resp. μ-semistable] with respect to H.

P_{F} (n) := χ(F ⊗ H^{⊗n})/rankF

M_{H}(v):= the moduli space of H-stable sheaves with Mukai vector v. M̄_{H}(v):= the moduli space of H-semistable sheaves with Mukai vector v. M_{H}^{⊗L}(v):= the moduli space of L-twisted H-stable sheaves with Mukai

vector v.

M̄_{H}^{⊗L}(v):= the moduli space of L-twisted H-semistable sheaves with Mukai vector v. (See section4 for the definition of L-twisted stability.)

3 Statement of Main Theorem

In this section we state the main theorem of this paper after reviewing Mukai’s beautiful theorem.

On the moduli space of vector bundles on a K3 surface, the following Mukai’s beautiful theorem is well-known.

Theorem 3.1 (Mukai[4]). Let H̃(X, ℤ) ∋ v be isotropic. Assume that (♠) Every H-semistable sheaf F with v(F) = v is H-stable. Then M̄_{H}(v) is a K3 surface if it is not empty.

In general,without the assumption (♠), M̄_{H}(v) has singularities. The following gives us such an example.

Example 3.2. Let π : X → ℙ^{1} be an elliptic fibration. Let X ⊃ s be a section of π and f a fiber of π. Assume that Pic(X) = ℤs ⊕ ℤf. Put H := s + 5f and v := (2, H, 2) ∈ H̃(X, ℤ). Then H-semistable sheaves E with v(E) = v are one of the following types:

(i) E fits into a nonsplit short exact sequece 0 → O(3f) → E → I_{x} ⊗ O(s + 2f) → 0 for x ∈ X \ s.

(ii) E is S-equivalent to O(4f) ⊕ O(s + f).

Therefore M̄_{H}(v) is isomorphic to the surface obtained by contracting s on X and has A_{1}-singularity.

The following is the main theorem of this paper.

Theorem 3.3. Let v = (r, l, s) ∈ H̃(X, ℤ) be primitive and isotropic. Assume that M_{H}(v) ≠ ϕ. Then M̄_{H}(v) is an irreducible 2-dimensional scheme which is regular in codimension one and there exist a K3 surface S and a surjective birational morphism f : S → M̄_{H}(v). Moreover if x ∈ M̄_{H}(v) corresponds to an S-equivalence class represented by a properly H-semistable sheaf G_{1} ⊕ ⋯ ⊕ G_{m}, where G_{1}, ⋯, G_{m} are H-stable and P_{G_{1}}(n) = ⋯ = P_{G_{m}}(n) and G_{i} ≠ G_{j} for i ≠ j, then M̄_{H}(v) is normal at x. When m = 2, x is an A_{1}-singular point. When m = 3, x is an A_{2}-singular point.

Remark 3.4. The author does not know whether M̄_{H}(v) is always normal or not.

4 Twisted Stability

In this section we recall the definition of twisted stability(which is equivalent to the notion of a-stability introduced by Ellingsrud and Göttsche[1]) and wall of the ample cone defined by Matsuki-Wentworth[2] which play an important role in the proof of Theorem3.3. We fix c_{1} ∈ Num(X) and c_{2} ∈ ℤ in this section.

Definition 4.1. A hyperplane W in Num(X)_{ℝ} is called a wall if there exist an ample line bundle A, an A-slope-semistable sheaf E with c_{1}(E) = c_{1},c_{2}(E) = c_{2} and a saturated subsheaf F of E with 0 < rankF < rankE such that

\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ≢ 0

(\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ) · A = 0

W = \{x ∈ Num(X)_{ℝ} | (\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ) · x = 0\}

For a fixed ample line bundle A, a hyperplane V in Num(X)_{ℝ} is called a subwall around A if there exist an A-slope-semistable sheaf E with c_{1}(E) = c_{1},c_{2}(E) = c_{2} and a saturated subsheaf F of E with 0 < rankF < rankE such that

\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ≢ 0

(\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ) · A = 0

V = \{x ∈ Num(X)_{ℝ} | (\frac{c_{1}(F)}{rankF} − \frac{c_{1}(E)}{rankE} ) · x + \frac{χ(F)}{rankF} − \frac{χ(E)}{rankE} = 0\}

Walls describe when the moduli space M̄_{H}(v) changes as H moves in Num(X)_{ℝ}. Subwalls tell us when the moduli space M̄_{H}^{⊗L}(v) changes as L moves in Num(X)_{ℝ}, where M̄_{H}^{⊗L}(v) is the moduli space of L-twisted Hsemistable sheaves defined below.For details, see [2].

Definition 4.2. Let E be a torsion-free sheaf on X and L a ℚ-divisor. E is said to be L-twisted H-stable [resp. L-twisted H-semistable] if for every subsheaf F of E with 0 < rankF < rankE we have P_{F} (n) + (c_{1}(F) · L)/rankF < P_{E}(n)+(c_{1}(E)·L)/rankE [resp. P_{F} (n)+(c_{1}(F)·L)/rankF ≤ P_{E}(n)+(c_{1}(E)· L)/rankE] for n >> 0.

5 Proof of Theorem3.3

Lemma 5.1. Let v = (r, l, s) be a primitive isotropic vector of H̃(X, ℤ). Assume that there exists an H-stable sheaf E with v(E) = v. If G_{1}, ⋯, G_{m} (m ≥ 2) are H-stable sheaves and k_{1}, ⋯, k_{m} are positive integers such that v = ∑_{i=1}^{m} v(G_{i})^{⊕k_{i}} and G_{i} ≄ G_{j} for i ≠ j and P_{G_{1}}(n) = ⋯ = P_{G_{m}}(n), then we have (v(G_{i})^{2}) = −2 for 1 ≤ i ≤ m. Moreover the matrix ((v(G_{i}), v(G_{j}))) is negative semi-definite and for (a_{1}, ⋯, a_{m}) ∈ ℤ^{⊕m} we have ((a_{1}v(G_{1}) + ⋯ + a_{m}v(G_{m}))^{2}) = 0 if and only if (a_{1}, ⋯, a_{m}) is a multiple of (k_{1}, ⋯, k_{m}).

Proof. Claim 1. (v, v(G_{j})) = 0 for 1 ≤ j ≤ m.

Proof of Claim1. We have 0 = (v^{2}) = ∑(v(E), v(G_{j})). By the stability of E and the semistability of G_{i}, we have (v(E), v(G_{j})) ≥ 0. Hence the claim1 holds true.

Assume that (v(G_{1})^{2}) ≥ 0. We have 0 ≤ ((v − \frac{r}{rankG_{1}} v(G_{1}))^{2}) = (l − \frac{r}{rankG_{1}} c_{1}(G_{1}))^{2} ≤ 0. Thus l ≡ \frac{r}{rankG_{1}} c_{1}(G_{1}) by Hodge index theorem. We have v = \frac{r}{rankG_{1}} v(G_{1}). This contradicts the primitivity of v. Therefore we have (v(G_{i})^{2}) = −2. In the rest of this proof, we assume that m ≥ 3. The case when m = 2 is easy.

Claim 2. For (a_{1}, ⋯, a_{m}) ∈ ℤ^{⊕m}, we have ((a_{1}v(G_{1}) + ⋯ + a_{m}v(G_{m}))^{2}) ≤ 0. If 0 ≤ a_{i} ≤ k_{i} and (a_{1}, ⋯, a_{m}) ≠ (0, ⋯, 0), (k_{1}, ⋯, k_{m}), then ((a_{1}v(G_{1})+ ⋯ + a_{m}v(G_{m}))^{2}) < 0.

Proof of Claim2. We may assume that a_{i} ≥ 0 for all i. (v − r(a_{1}v(G_{1}) + ⋯ + a_{m}v(G_{m}))/(a_{1}rankG_{1} + ⋯ + a_{m}rankG_{m}))^{2} = (l − r · c_{1}(G_{1}^{⊕a_{1}} ⊕ ⋯ ⊕ G_{m}^{⊕a_{m}})/(a_{1}rankG_{1} + ⋯ + a_{m}rankG_{m}))^{2} ≤ 0 by Hodge index theorem. The latter statement follows from the primitivity of v. The proof of Claim2 is completed.

By Claim2, we have (v(G_{i}), v(G_{j})) = 0 or 1 for i ≠ j. Let Γ be a graph with vertices v_{1}, ⋯, v_{m} such that v_{i} and v_{j} are connected by one edge if (v(G_{i}), v(G_{j})) = 1 and disconnected if (v(G_{i}), v(G_{j})) = 0. By the primitivity of v, Γ is a connected graph, which implies the last statement.

Remark 5.2. By the characterization of Cartan matrices, we know that the matrix (−(v(G_{i}), v(G_{j}))/2) in the above lemma is a Cartan matrix of Ã_{n},D̃_{n},Ẽ_{6},Ẽ_{7} or Ẽ_{8}.

Lemma 5.3. M̄_{H}(v) − M_{H}(v) is a finite set.

Proof. It is obvious from lemma5.1.

We fix a ℚ-divisor L near H in Num(X)_{ℝ} so that there is no point except for H on the segment connecting L and H which lies on a wall or a subwall around H.

Lemma 5.4. If F is an L-twisted H-stable sheaf with v(F) = v, then F is H-semistable. In particular, there exists a natural morphism f : M_{H}^{⊗L}(v) → M̄_{H}(v).

Proof. Assume that F is an L-twisted H-stable sheaf with v(F) = v which is not H-semistable. There is a saturated subsheaf G of F such that:

\frac{c_{1}(G) · H}{rankG} = \frac{c_{1}(F) · H}{rankF}  (1)
\frac{χ(G)}{rankG} > \frac{χ(F)}{rankF}  (2)

Since F is L-twisted H-stable, we have

\frac{χ(G)}{rankG} + \frac{c_{1}(G) · L}{rankG} < \frac{χ(F)}{rankF} + \frac{c_{1}(F) · L}{rankF}  (3)

The two inequalities (2) and (3) contradict the choice of L.

Lemma 5.5. If F is an L-twisted H-semistable sheaf with v(F) = v, then F is L-twisted H-stable. In particular, M̄_{H}^{⊗L}(v) = M_{H}^{⊗L}(v) is a K3 surface.

Proof. The proof of the former statement is quite similar to that of lemma5.4. We leave it to the reader. The latter follows from Corollary4.6 in [4]. (There the proof is given only for H-stable sheaves,but it is valid for L-twisted Hstable ones.)

Lemma 5.6. f : M_{H}^{⊗L}(v) → M̄_{H}(v) is surjective.

Proof. We employ the argument used in Proposition4.4 in [4]. Let E is a quasi-universal sheaf on X ×M_{H}^{⊗L}(v) with similitude σ. Given a properly Hsemistable sheaf G_{1}⊕⋯⊕G_{m}, where G_{1}, ⋯, G_{m} are H-stable and P_{G_{1}}(n) = ⋯ = P_{G_{m}}(n), we shall show that there exists an E ∈ M_{H}^{⊗L}(v) which is S-equivalent to G_{1} ⊕ ⋯ ⊕ G_{m}. By Grothendieck-Riemann-Roch formula, Φ(F) := ch(∑(−1)^{i}Ext_{π_{M}}^{i} (E, π_{X}^{∗}F)) does not depend on the choice of F with v(F) = v, where π_{M} : X × M_{H}^{⊗L}(v) → M_{H}^{⊗L}(v) and π_{X} : X × M_{H}^{⊗L}(v) → X. For F ∈ M_{H}^{⊗L}(v), Φ(F) = σ·(fundamental class). If Hom(F, G_{1}⊕⋯⊕G_{m}) = Hom(G_{1} ⊕⋯⊕G_{m}, F) = 0 for any F ∈ M_{H}^{⊗L}(v), then Φ(G_{1} ⊕⋯⊕G_{m}) = 0. Therefore we have Hom(F, G_{i}) ≠ 0 or Hom(G_{i}, F) ≠ 0 for some F ∈ M_{H}^{⊗L}(v) and 1 ≤ i ≤ m. Say we have Hom(G_{1}, F) ≠ 0 and let α be a nonzero element of Hom(G_{1}, F). Then α is injective and v(F_{1}) = v(G_{2} ⊕ ⋯ ⊕ G_{m}), where F_{1} := cokerα. By lemma5.1, (v(G_{2} ⊕ ⋯ ⊕ G_{m})^{2}) < 0. Therefore we have Hom(F_{1}, G_{i}) ≠ 0 or Hom(G_{i}, F_{1}) ≠ 0 for some 2 ≤ i ≤ m. Continuing this argument, we know that F is S-equivalent to G_{1} ⊕ ⋯ ⊕ G_{m}.

Lemma 5.7. If x ∈ M̄_{H}(v) corresponds to an S-equivalence class represented by a properly H-semistable sheaf G_{1} ⊕ ⋯ ⊕ G_{m}, where G_{1}, ⋯, G_{m} are Hstable and P_{G_{1}}(n) = ⋯ = P_{G_{m}}(n) and G_{i} ≠ G_{j} for i ≠ j, then M̄_{H}(v) is normal at x.

Proof. Let U be an open neiborhood of x. Since U is constructed as a categorical quotient of a subscheme R of a quot scheme, it suffices to check the normality of R. A point q of R corresponds to a surjective morphism O^{⊕N} ↠ E ⊗ H^{α}, where α is a fixed large number, N = dim H^{0}(X, E ⊗ H^{α}) and E is H-semistable.

Using the assumption that G_{i} ≠ G_{j} for i ≠ j, E is one of the following:

(i) E is simple.

(ii) E is isomorphic to F_{1} ⊕ ⋯ ⊕ F_{m}, where F_{i} is simple for any i and m ≥ 2.

Let Z be the closed subset of R which consists of the points corresponding to a surjective morphism O^{⊕N} ↠ E ⊗ H^{α}, where E is of type (ii) in the above.

Claim 3. R \ Z is smooth, connected and of dimension N^{2} + 1.

Proof of Claim3. The obstruction of smoothness at q = [O^{⊕N} ↠ E ⊗H^{α}] of R (we denote it by ob(q)) is in Ext^{1}(K, E ⊗ H^{α}), where K := ker(O^{⊕N} ↠ E ⊗ H^{α}). In [3], it is proved that ob(q) is mapped to 0 by the morphism Ext^{1}(K, E⊗H^{α}) → H^{2}(X, O_{X}) which is the composite of Ext^{1}(K, E⊗H^{α}) → Ext^{1}(K, K) and Ext^{1}(K, K) →^{tr} H^{2}(X, O_{X}). Therefore the deformation theory at q has dimension dim Ext^{1}(K, E ⊗ H^{α}) − 1. Since E of the type (i) is simple and dim Ext^{1}(K, E ⊗ H^{α}) − 1 = dim Hom(E, E) − 1, R \ Z is smooth. Moreover for q = [O^{⊕N} ↠ E⊗H^{α}] in R\Z, dim_{q} R = dim Hom(K, E⊗H^{α}) = N^{2} + 1. Let U be the open set of R \ Z which consists of the points corresponding to a surjective morphismO^{⊕N} ↠ E ⊗ H^{α}, where E is H-stable. Since H-stable sheaves are L-twisted H-stable and M_{H}^{⊗L}(v) is connected,U is connected. We can easily see that dim(R \(Z ∪U)) < N^{2} +1. Hence R \Z is connected. The proof of Claim3 is completed.

Since R\Z is dense in R, R is irreducible. In order to prove the normality of R, we use Serre criterion. Let q = [O^{⊕N} ↠ E ⊗ H^{α}] be a point in Z and put K := ker(O^{⊕N} ↠ E ⊗ H^{α}). We have

dim Ô_{q,R} ≥ dim T_{q,R} − (dim Ext^{1}(K, E ⊗ H^{α}) − 1)
= −(v(K), v(E ⊗ H^{α})) + 1
= N^{2} + 1

On the otherhand dim Ô_{q,R} = N^{2} + 1. Therefore the above inequality is an equality, hence Ô_{q,R} is a locally complete intersection, in particular satisfies (S_{2}). It is easily seen that codim(Z, R) ≥ 2.

Now we come to the proof of the main theorem.

Proof of Theorem3.3. Most assertions were proved in the previous lemmas. Therefore we have only to show the following:

(i) f^{−1}(x) is a (−2)-curve for x = G_{1} ⊕ G_{2} ∈ M̄_{H}(v), where G_{1},G_{2} are H-stable and P_{G_{1}}(n) = P_{G_{2}}(n).

(ii) f^{−1}(y) is a union of two (−2)-curves which intersect each other at one point transversely for y = G_{1} ⊕ G_{2} ⊕ G_{3} ∈ M̄_{H}(v), where G_{1},G_{2},G_{3} are H-stable and P_{G_{1}}(n) = P_{G_{2}}(n) = P_{G_{3}}(n).

The proof of (i) and (ii) are quite similar and we shall show only (ii). Let y = G_{1} ⊕ G_{2} ⊕ G_{3} be as in (iii) above. We may assume that (c_{1}(G_{1}) · L)/rankG_{1} < (c_{1}(G_{2}) · L)/rankG_{2} < (c_{1}(G_{3}) · L)/rankG_{3}. Note that we have either (l · L)/r < (c_{1}(G_{2}) · L)/rankG_{2} or (l · L)/r > (c_{1}(G_{2}) · L)/rankG_{2}. The arguments are similar and we consider the former case.

Claim 4. (v(G_{i}), v(G_{j})) = 1 for i ≠ j

Proof of Claim4. By lemma 5.1 and Claim1 we have (v(G_{i})^{2}) = −2 and (v, v(G_{i})) = 0. Hence we obtain (v(G_{i}), v(G_{j})) = 1 for i ≠ j.

Since dim Ext^{1}(G_{1}, G_{2}) = 1 by Claim4, there exists uniquely up to ℂ^{×} a nonsplit exact sequences 0 → G_{1} → E_{1} → G_{2} → 0. Nonsplit exact sequence 0 → E_{1} → E_{2} → G_{3} → 0 is parametrized by ℙ(Ext^{1}(G_{3}, E_{1})^{∨}) ≅ ℙ^{1} up to ℂ^{×}. We obtain the following commutative diagram:

0  0
↓  ↓
G_{1}  G_{1}
↓  ↓
0 ⟶ E_{1} ⟶ E_{2} ⟶ G_{3} ⟶ 0
↓  ↓  ∥
0 ⟶ G_{2} ⟶ E_{3} ⟶ G_{3} ⟶ 0
↓  ↓
0  0

Similarly, starting from a nonsplit exact sequence 0 → G_{1} → E′_{1} → G_{3} → 0, we obtain the following diagram:

0  0
↓  ↓
G_{1}  G_{1}
↓  ↓
0 ⟶ E′_{1} ⟶ E′_{2} ⟶ G_{2} ⟶ 0
↓  ↓  ∥
0 ⟶ G_{3} ⟶ E′_{3} ⟶ G_{2} ⟶ 0
↓  ↓
0  0

Claim 5. E_{2} and E′_{2} in the above diagrams are L-twisted H-semistable. Every L-twisted H-semistable sheaf F which is S-equivalent to G_{1} ⊕G_{2} ⊕G_{3} is isomorphic to either E_{2} or E′_{2} in the above.

Proof of Claim5. The latter statement is obvious. We shall show the former statement. We check L-twisted H-semistability of E′_{2}. Suppose that E′_{2} is not L-twisted H-semistable. We can find a saturated subsheaf A of E′_{2} such that:

\frac{c_{1}(A) · H}{rankA} = \frac{c_{1}(E′_{2}) · H}{rankE′_{2}}  (4)
\frac{c_{1}(A) · L}{rankA} + \frac{χ(A)}{rankA} > \frac{c_{1}(E′_{2}) · L}{rankE′_{2}} + \frac{χ(E′_{2})}{rankE′_{2}}  (5)

Since L is not on a subwall, we have χ(A)/rankA = χ(E′_{2})/rankE′_{2}. This implies that P_{A}(n) = P_{E′_{2}}(n). Noting that the two exact sequences 0 → E′_{1} → E′_{2} → G_{2} → 0 and 0 → G_{1} → E′_{1} → G_{3} → 0 are nonsplit, A is S-equivalent to G_{1} or G_{1} ⊕ G_{2} or G_{1} ⊕ G_{3}. In any case, we have

P_{A}(n) + \frac{c_{1}(A) · L}{rankA} < P_{E′_{2}}(n) + \frac{c_{1}(E′_{2}) · L}{rankE′_{2}}

since we are assuming (c_{1}(E′_{2}) · L)/rankE′_{2} < (c_{1}(G_{2}) · L)/rankG_{2}. This contradicts (4) and (5). The proof of Claim5 is completed.

By Claim5, we obtain two morphisms g : ℙ(Ext^{1}(G_{3}, E_{1})^{∨}) → M_{H}^{⊗L} and g′ : ℙ(Ext^{1}(G_{2}, E′_{1})^{∨}) → M_{H}^{⊗L}. By construction g and g′ are injective. Put C = Img and C′ = Img′. Let F be isomorphic to an E′_{2} in the above diagram. The point [F] in M_{H}^{⊗L}(v) is on C if and only if Hom(F, G_{3}) ≠ 0, which is equivalent to the splitness of the exact sequence 0 → G_{3} → E′_{3} → G_{2} → 0. It is easily seen that 0 → G_{3} → E′_{3} → G_{2} → 0 splits at just one point of ℙ(Ext^{1}(G_{2}, E′_{1})^{∨}). Therefore we know C ≠ C′ and C ∩ C′ is one point. Since C ∪ C′ is contracted to a point of the normal surface M̄_{H}(v), the fact that g and g′ are embeddings and the transversality at C ∩ C′ follow from calculating the intersection matrix of C and C′ or more economically looking at the classification of rational double points. The proof of Theorem3.3 is completed.

References

[1] G.Ellingsrud and L.Göttsche, Variation of moduli spaces and Donaldson invariants under change of polarization, J. Reine Angew. Math. 467 (1995), 1-49.

[2] K.Matsuki and R.Wentworth, Mumford-Thaddeus principle on the moduli spaces of vector bundles on a surface, Inter.J.Math.,vol8(1997), 97-148

[3] S.Mukai, Symplectic structure of the moduli space of sheaves on an abelian or K3 surface, Invent. Math. 77 (1984), no. 1, 101-116.

[4] S.Mukai, Muduli of vector bundles on K3 surfaces I, Vector bundles on Algebraic Varietes, Oxford,1987, 341-413

e-mail address: abeken@kusm.kyoto-u.ac.jp