0505553v1 [math.AG] 25 May 2005

2 downloads 0 Views 364KB Size Report
function was proved affirmatively by the authors [13] and Nekrasov-Okounkov [16] indepen- dently. Nekrasov also conjectured the same statements for the ...
INSTANTON COUNTING ON BLOWUP. II. K-THEORETIC PARTITION FUNCTION

arXiv:math/0505553v1 [math.AG] 25 May 2005

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA Dedicated to Vladimir Drinfeld on his fiftieth birthday Abstract. We study Nekrasov’s deformed partition function Z(ε1 , ε2 , ~a; q, β) of 5-dimensional supersymmetric Yang-Mills theory compactified on a circle. Mathematically it is the generating function of the characters of the coordinate rings of the moduli spaces of instantons on R4 . We show that it satisfies a system of functional equations, called blowup equations, whose solution is unique. As applications, we prove (a) F (ε1 , ε2 , ~a; q, β) = ε1 ε2 log Z(ε1 , ε2 , ~a; q, β) is regular at ε1 = ε2 = 0 (a part of Nekrasov’s conjecture), and (b) the genus 1 parts, which are first several Taylor coefficients of F (ε1 , ε2 , ~a; q, β), are written explicitly in terms of τ = d2 F (0, 0, ~a; q, β)/da2 in rank 2 case.

Introduction In Part I of this paper [13], we studied Nekrasov’s partition function [15] for N = 2 supersymmetric gauge theory in 4-dimension (see also [14]). It is defined as the generating function of the integral of the equivariant cohomology class 1 of the framed moduli space M(r, n) of torsion free sheaves on P2 with rank r, c2 = n: Z ∞ X inst n 1. Z (ε1 , ε2,~a; q) = q n=0

M (r,n)

Here (r + 2)-dimensional torus Te acts naturally on M(r, n), and ε1 , ε2 , ~a = (a1 , . . . , ar ) are generators of HT∗e (pt) = S ∗ (Lie Te). (More precisely, this is the instanton part of the partition function. We multiply it with the perturbative part. See §4.2.) It can be considered as series of equivariant Donaldson invariants for R4 , and there are close relation to the ordinary Donaldson invariants, such as blowup formulas, wall-crossing formulas [14, 5]. In this part II, we study a similar partition function, in which we replace the integration in the equivariant cohomology by one in equivariant K-theory: X X Z inst (ε1 , ε2 ,~a; q, β) = (qβ 2r e−rβ(ε1 +ε2 )/2 )n (−1)i ch H i (M(r, n), O). n

i

We consider eβaα , eβε1 , eβε2 as characters of Te here. The formal parameter β is introduced so that the K-theoretic partition function converges to the homological one when β → 0. It is called the partition function of the 5-dimensional supersymmetric gauge theory compactified on a circle in the physics literature, where the radius of the circle is β. 2000 Mathematics Subject Classification. Primary 14D21; Secondary 57R57, 81T13, 81T60. The first author is supported by the Grant-in-aid for Scientific Research (No.13640019, 15540023), JSPS. 1

2

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

Nekrasov [15] conjectured that F inst (ε1 , ε2 ,~a; q) = ε1 ε2 log Z inst (ε1 , ε2,~a; q) is regular at def. ε1 , ε2 = 0, and F0inst (~a; q) = F inst (0, 0,~a; q) is the instanton part of the Seiberg-Witten prepotential for N = 2 supersymmetric gauge theory [17] with gauge group SU(r). The SeibergWitten prepotential is defined by certain period integrals of hyperelliptic curves, the so-called Seiberg-Witten curves. (See [14] for detail.) This was a mathematically well formulated conjecture, which is similar to the mirror symmetry. The conjecture for the homological partition function was proved affirmatively by the authors [13] and Nekrasov-Okounkov [16] independently. Nekrasov also conjectured the same statements for the above K-theoretic version, and the technique in [16] can be applied to that case also. We study the K-theoretic partition function through the approach taken in [13], which we briefly describe now: We consider similar correlation functions given by generating functions of c(r, k, n) characters of cohomology groups of Donaldson divisors µ(C) on the moduli spaces M 2 b on the blowup plane P . As a simple application of the Atiyah-Bott-Lefschetz formula, the correlation functions can be expressed by Z inst . (See (2.2).) On the other hand, through a geometric study of moduli spaces on the blowup, we prove vanishing of certain cohomology groups. (See Theorem 2.4.) Combining these two results, we get a system of functional equations, called blowup equations, satisfied by Z inst . It determines the coefficients of qn in Z inst recursively. It also implies the regularity of F inst (ε1 , ε2,~a; q, β) and we get a differential equation for F0inst (~a; q, β), as limits of the blowup equations. (See (4.17).) We call it a contact term equation, according to the name of their homological version. The contact term equation also determines F0inst (~a; q, β) recursively. The homological version of the contact term equation was much studied in the physics literature, and the Seiberg-Witten prepotential satisfies the equations (see [6, 10] and the reference therein). By the uniqueness of its solution, the instanton part of the Seiberg-Witten prepotential is equal to F0inst (~a; q). This was our proof of Nekrasov’s conjecture in [13]. It is natural to hope that the same proof can be given for the K-theoretic version. But we do not find our K-theoretic contact term equation in the physics literature, and do not know how to prove this assertion at this moment except for r = 2 case. Although we do not give the proof of Nekrasov’s conjecture, we think that it is worthwhile to pursue our approach by various reasons: (1) The blowup equations determine not only F0inst (~a; q, β), but also several higher coefficients of the expansion of F inst (ε1 , ε2 ,~a; q, β) at ε1 = ε2 = 0. (See §5.) (2) The geometric study of moduli spaces on the blowup is probably useful for the study of the K-theoretic version of Donaldson invariants. Higher coefficients are identified with higher genus Gromov-Witten invariants for certain noncompact Calabi-Yau 3-folds (see [15], [14, §7]), and appear in the wall-crossing formula for Donaldson invariants [5]. Thus they are equally important as F0inst (~a; q, β). For the ordinary Donaldson invariants, the vanishing of first several blowup coefficients was well-known, and was proved by the dimension counting argument. Our proof of the blowup equation for the homological partition function was given by the same idea. The proof for the K-theoretic partition function is very different, and we use the Kawamata-Viewheg vanishing theorem, a result from complex algebraic geometry. But we hope that a similar result holds for the K-theoretic version of Donaldson invariants, whose existence is still conjectural. Acknowledgement. The authors are grateful to the referee for helpful suggestions and comments.

INSTANTON COUNTING ON BLOWUP. II

3

1. K-theoretic partition function In this section, we define the K-theoretic version of Nekrasov’s partition function. We follow [13, 14] for which the reader can find more detail and the references. 1.1. Definition of the partition function. Let M(r, n) denote the framed moduli spaces of torsion free sheaves (E, Φ) on P2 with rank r and c2 = n. Let M0reg (r, n) be the open subvariety consisting of locally free sheaves. Let M0 (r, n) be the Uhlenbeck (partial) compactification of M reg (r, n), i.e., n G ′ M0 (r, n) = M0reg (r, n′ ) × S n−n C2 . n′ =0

We can endow this space with the structure of an affine algebraic variety so that there is a projective morphism π : M(r, n) → M0 (r, n). The corresponding map between closed points can be identified with ′

(E, Φ) 7−→ ((E ∨∨ , Φ), Supp(E ∨∨ /E)) ∈ M0reg (r, n′ ) × S n−n C2 .

where E ∨∨ is the double dual of E and Supp(E ∨∨ /E) is the support of E ∨∨ /E counted with multiplicities. Let T be the maximal torus of GLr (C) consisting of diagonal matrices and let Te = C∗ × C∗ × T . We define an action of Te on M(r, n) as follows: For (t1 , t2 ) ∈ C∗ × C∗ , let Ft1 ,t2 be an automorphism of P2 defined by Ft1 ,t2 ([z0 : z1 : z2 ]) = [z0 : t1 z1 : t2 z2 ].

For diag(e1 , . . . , er ) ∈ T let Ge1 ,...,er denote the isomorphism of Oℓ⊕r given by ∞ Oℓ⊕r ∋ (s1 , . . . , sr ) 7−→ (e1 s1 , . . . , er sr ). ∞

Then for (E, Φ) ∈ M(r, n), we define

(1.1)

 (t1 , t2 , e1 , . . . , er ) · (E, Φ) = (Ft−1 )∗ E, Φ′ , 1 ,t2

where Φ′ is the composite of homomorphisms (Ft−1 )∗ E|ℓ∞ 1 ,t2

(Ft−1,t )∗ Φ

Ge

,...,er

1 −−−1−2−−→ (Ft−1 )∗ Oℓ⊕r −→ Oℓ⊕r −−− −−→ Oℓ⊕r . 1 ,t2 ∞ ∞ ∞

Here the middle arrow is the homomorphism given by the action. In a similar way, we have a Te-action on M0 (r, n). The map π : M(r, n) → M0 (r, n) is equivariant. Notation 1.2. We denote by eα (α = 1, . . . , r) the one dimensional Te-module given by Te ∋ (t1 , t2 , e1 , . . . , er ) 7→ eα .

Similarly, t1 , t2 denote one-dimensional Te-modules. Thus the representation ring R(Te) is ± ± ± −1 isomorphic to Z[t± 1 , t2 , e1 , . . . , er ], where eα is the dual of eα .

We denote the coordinates of Lie(Te) by ε1 , ε2, a1 , . . . , ar corresponding to t1 , t2 , e1 , . . . , er . In our previous paper [13], these are generators of the equivariant cohomology group HT∗e (pt) of a single point. We relate two sets of variables as t1 = eβε1 , t2 = eβε2 , eα = eβaα , where β is a parameter. We will define the K-theory partition function so that it converges to the homological partition function when β → 0.

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

4

We define the instanton part of the partition function by X def. Z inst (ε1 , ε2 ,~a; q, β) = (qβ 2r e−rβ(ε1 +ε2 )/2 )n Zn (ε1 , ε2 ,~a; β) n

(1.3)

def.

Zn (ε1 , ε2 ,~a; β) =

X i

(−1)i ch H i(M(r, n), O).

Here the character ch is a formal sum of weight spaces, which are finite dimensional as shown in [13, §4]. We also have X Zn (ε1 , ε2,~a; β) = (−1)i ch H 0(M0 (r, n), Ri π∗ O). i

We will see that the higher direct image sheaves Ri π∗ O = 0 for i > 0 (Lemma 3.1). Therefore we have Zn (ε1 , ε2 ,~a; β) = ch H 0 (M0 (r, n), O). This definition has an advantage that it involves only the Uhlenbeck compactification M0 (r, n). e

1.2. Other descriptions of the partition function. Let K T (M(r, n)) denote the Groe thendieck group of Te-equivariant coherent sheaves on M(r, n) and similarly for K T (M0 (r, n)). These are modules over the representation ring R(Te) of the torus Te. As in 1.2, we iden± ± ± tify it with the Laurent polynomial ring Z[t± 1 , t2 , e1 , . . . , er ]. Since M(r, n) is nonsingular, e K T (M(r, n)) is isomorphic to the Grothendieck group of Te-equivariant vector bundles. The e e proper equivariant morphism π induces a homomorphism π∗ P : K T (M(r, n)) → K T (M0 (r, n)) by taking the alternating sum of higher direct image sheaves i (−1)i Ri π∗ . e The fixed points M0 (r, n)T consist of the single point n[0] ∈ S n C2 ⊂ M0 (r, n). Let ι0 denote the inclusion map of the fixed point. By the localization theorem for the K-theory due to Thomason [18] (a prototype was given in [1]), it is known that the homomorphism ι0∗ is an isomorphism after the localization: e

e

∼ =

e

→ K T (M0 (r, n)) ⊗R(Te) R, ι0∗ : R ∼ = K T (M0 (r, n)T ) ⊗R(Te) R −

where R = Q(t1 , t2 , e1 , . . . , em ) is the quotient field of R(Te). We have ι−1 0∗ = ch, which is a consequence of a trivial identity ch ◦ι0∗ = id. We thus get Zn (ε1 , ε2 ,~a; β) = (ι0∗ )−1 π∗ (OM (r,n) ). e

Here we denote the element in K T (M(r, n)) corresponding to OM (r,n) by the same symbol for brevity. We hope that these two meanings can be distinguished from the content. e The fixed points M(r, n)T consist of (E, Φ) = (I1 , Φ1 ) ⊕ · · · ⊕ (Ir , Φr ) such that a) Iα is an ideal sheaf of 0-dimensional subscheme Zα contained in C2 = P2 \ ℓ∞ . b) Φα is an isomorphism from (Iα )ℓ∞ to the αth factor of Oℓ⊕r . ∞ ∗ ∗ c) Iα is fixed by the action of C × C , coming from that on P2 . e We parametrize the fixed point set M(r, n)T by an r-tuple of Young diagrams Y~ = (Y1 , . . . , Yr ) so that the ideal Iα is spanned by monomials xi y j placed at (i − 1, j − 1) outside Yα . The def. P constraint is that the total number of boxes |Y~ | = α |Yα | is equal to n. Let ι denote the inclusion map of the fixed point set. We have M ∼ e e e = ι∗ : → K T (M(r, n)) ⊗R(Te) R. R∼ = K T (M(r, n)T ) ⊗R(Te) R − ~ Y

INSTANTON COUNTING ON BLOWUP. II

We then have Zn (ε1 , ε2,~a; β) =

X ~ Y

As M(r, n) is nonsingular, (ι∗ ) points formula:

−1

5

(ι∗ )−1 (OM (r,n) ).

can be explicitly given by Atiyah-Bott Lefschetz fixed

ι∗−1 (•) =

M ~ Y

V

ι∗Y~ (•) ∗ −1 TY ~ M(r, n)

,

where TY~∗ M(r, n) is the cotangent bundle of M(r, n) at a fixed point of Y~ considered as a TeV module, −1 is the alternating sum of exterior powers, and ι∗Y~ is the pull-back homomorphism with respect to the inclusion ιY~ : {Y~ } → M(r, n). Here the pull-back homomorphism is defined e via the isomorphism of K T (M(r, n)) and the Grothendieck group of Te-equivariant locally free sheaves. V In order to express −1 TY~∗ (M(r, n)), we need some notation for Young diagrams. Let Y = (λ1 ≥ λ2 ≥ · · · ) be a Young diagram, where λi is the length of the ith column. Let Y ′ = (λ′1 ≥ λ′2 ≥ . . . ) be the transpose of Y . Thus λ′j is the length of the jth row of Y . Let l(Y ) denote the number of columns of Y , i.e., l(Y ) = λ′1 . Let aY (i, j) = λi − j,

lY (i, j) = λ′j − i.

Here we set λi = 0 when i > l(Y ). Similarly λ′j = 0 when j > l(Y ′ ). When the square s = (i, j) lies in Y , these are called arm-length, leg-length, respectively in the literature. But our formula below involves these also for squares outside Y . So these take negative values in general. By [13, Theorem 2.11] we have (1.4)

Z

inst

X (qβ 2r e−rβ(ε1 +ε2 )/2 )|Y~ | X (qβ 2r e−rβ(ε1 +ε2 )/2 )|Y~ | V (ε1 , ε2 ,~a; q, β) = = , Y ~ ∗ nY (ε1 , ε2,~a; β) −1 TY ~ M(r, n) ~ Y

~ Y

α,β

α,β

where ~ nYα,β (ε1 , ε2 ,~a; β)

 Y −β(−lYβ (s)ε1 +(aYα (s)+1)ε2 +aβ −aα ) 1−e = s∈Yα

×

Y

t∈Yβ

 1 − e−β((lYα (t)+1)ε1 −aYβ (t)ε2 +aβ −aα ) .

Remark 1.5. Contrary to the convention in [13, Remark 4.4], we put the Te-module structure on the coordinate ring (and the cohomology groups) by Fg∗−1 , where Fg : M(r, n) → M(r, n) is the isomorphism given by an element g ∈ Te.

This combinatorial expression was the original definition of the instanton part of the partition function due to Nekrasov [15], except we put the additional factor e−rβ(ε1 +ε2 )/2 . This factor is the half of the canonical bundle of M(r, n) (see Lemma 3.6), hence Zn is the index of the Dirac operator, rather than the Dolbeault operator. Also, the factor makes the symmetry of the partition function nicer, as we see in Lemma 4.3. Moreover, it is clear that this K-theoretic partition function converges to the homological partition function studied in [13, 14] as β → 0.

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

6

2. Blowup equation and a main result b2 be the blowup of P2 at [1 : 0 : 0]. Let 2.1. Correlation functions on blowup. Let P b2 → P2 denote the projection. Let C be the exceptional divisor, O(C) the corresponding p: P line bundle, and O(mC) its mth power. b2 with rank c k, n Let M(r, b) be the framed moduli space of torsion free sheaves (E, Φ) on P b2 ]i = n c (E)2 . This is also r, hc1 (E), [C]i = −k and h∆(E), [P b where ∆(E) = c2 (E) − r−1 2r 1 nonsingular of dimension 2b nr. (Remark that n b may not be integer in general.) By tensoring a line bundle if necessary, we assume 0 ≤ k < r hereafter. By [14, Theorem 3.3] there is a projective morphism defined by c(r, k, n π b: M b) → M0 (r, n) 7→

(E, Φ)

 ((p∗ E)∨∨ , Φ), Supp(p∗ E ∨∨ /p∗ E) + Supp(R1 p∗ E) ,

where n = n b − k(r−k) . 2r b2 × M(r, c k, n Let (E, Φ) be the universal family on P b). It has a natural Te-structure from ∗ ∗ the construction. As C is C × C -invariant, its fundamental class [C] defines a class in the c(r, k, n equivariant homology group. We define an equivariant Q-divisor µ(C) on M b) by  h i e def. c k, n c(r, k, n b))Q , µ(C) = p2∗ ∆(E) ∩ C × M b) ∈ AT2bnr−1 (M(r,

b2 × M c(r, k, n c(r, k, n where p2 : P b) → M b) is the projection to the second factor. As customary, we denote the above as the ‘slant product’ ∆(E)/[C]. In the ordinary nonequivariant situation, it is known that µ(C) comes from the determinant line bundle  det R• p2∗ E|C ⊗ p∗1 OC (−1) .

(See [8, §1] for more detail). This construction works in our equivariant setting, so µ(C) comes from an equivariant Q-line bundle. We will identify µ(C) as the latter element hereafter, but e c e c(r, k, n b)) (see [19, Lemma 1.3] k, n b)) ∼ this does not make any trouble as PicT (M(r, = AT2bnr−1 (M or [4, Theorem 1]). We want to define the K-theoretic direct image π b∗ (O(dµ(C))) of the Te-equivariant Q-line b2 × bundle µ(C). For this purpose, we compute det E for the universal family (E, Φ) on P c(r, k, n c(r, k, n b2 ) = 0, det E ∼ b). M b). Since h1 (P = L ⊠ OPb 2 (kC) for a Te-line bundle L on M L Q r ∼ ∼ e Since E|ℓ∞ ×M = α=1 Oℓ∞ ×M(r,k,b c(r,k,b c α eα . Hence n) n) eα as T -sheaves, we have L = OM P c(r,k,bn) c1 (E) = aα + kC, therefore we have  !2  r−1 X µ(C) = c2 (E) − aα + kC  /[C] 2r α = c2 (E)/[C] + λ,

P (2k α aα + k 2 (ε1 + ε2 )) ∈ H 2 (B Te, Q). Here we have used 1/[C] = 0, C/[C] = where λ = r−1 2r −1, C 2 /[C] = −(ε1 + ε2 ) (cf. [13, Proof of Lemma 5.8]). Now we define the K-theoretic direct image as follows: Definition 2.1. def.

e

π b∗ (O(dµ(C))) = π b∗ (O(dc2 (E)/[C])) ⊗ OM0 (r,n) (dλ) ∈ K T (M0 (r, n)) ⊗R(Te) R′ ,

INSTANTON COUNTING ON BLOWUP. II

1/r

1/r

1/r

7

1/r

where R′ = Q[t1 , t2 , e1 , ..., er ]. e

Note that π b∗ (O(dµ(C))) is in K T (M0 (r, n)) if k is 0 or r, or d = 0. We now define correlation functions on blowup: X def. inst Zbk,d (ε1 , ε2 ,~a; q, β) = (qβ 2r e−rβ(ε1 +ε2 )/2 )nb (ι0∗ )−1 (b π∗ (O(dµ(C)))) . n b

2.2. Correlation functions via the partition function. As in §1.2 we can use Atiyah-Bott inst Lefschetz fixed points formula to get another description of Zbk,d . The necessary computation for the weights of tangent spaces and divisor µ(C) at fixed points was already done in homological version [13, §3, §6]. So we only state the answer: inst (2.2) Zbk,d (ε1 , ε2 ,~a; q, β) =

X

~~

~

(eβ(ε1 +ε2 )(d−r/2) qβ2r )(k,k)/2 eβ(k,~a)d × Q ~k l (ε , ε ,~ a ) 1 2 α ~ ∈∆ α ~ {~k}=−k/r

Z inst (ε1 , ε2 − ε1 ,~a + ε1~k; eβε1 (d−r/2) q, β)Z inst (ε1 − ε2 , ε2 ,~a + ε2~k; eβε2 (d−r/2) q, β).

We need explanations of several notations. The vector P ~a is considered as an element of the Cartan subalgebra h of slr by imposing the condition α aα = 0. Then ∆ is the set of roots of P slr . The vector ~k runs over the coweight lattice P = {~k = (k1 , . . . , kr ) ∈ Qr | kα = 0, ∃k ∈ def. Z ∀α kα ≡ −k/r mod Z} with the constraint {~k} = kα (mod Z) = −k/r. Finally we set Y  (1 − eβ(iε1 +jε2 −h~a,~αi) ) if h~k, α ~ i < 0,     i,j≥0    ~ i+j≤−h Yk,~αi−1  ~k (2.3) lα~ (ε1 , ε2 ,~a) = 1 − eβ(−(i+1)ε1 −(j+1)ε2 −h~a,~αi) if h~k, α ~ i > 1,    i,j≥0   i+j≤h~k,~ αi−2    1 otherwise for a root α ~ ∈ ∆.

2.3. Main results. We can now state our main results: Theorem 2.4. (1)(d = 0 case) (2.5)

k(r−k) inst Zbk,0 (ε1 , ε2,~a; q, β) = (qβ 2r e−rβ(ε1 +ε2 )/2 ) 2r Z inst (ε1 , ε2,~a; q, β).

(2)(0 < d < r case) (2.6) (3)(d = r case)

( Z inst (ε1 , ε2 ,~a; q, β) inst Zbk,d (ε1 , ε2 ,~a; q, β) = 0

for k = 0, for 0 < k < r.

inst (2.7) Zbk,r (ε1 , ε2 ,~a; q, β) = (−1)k(r−k) (t1 t2 )k(r−k)/2 (qβ2r e−rβ(ε1 +ε2 )/2 )

k(r−k) 2r

Z inst (ε1 , ε2 ,~a; q, β).

For the homological partition function, similar formulas were obtained ([13, 6.12], [14, §5.1]). The proof was the same as that of first several coefficients of the blowup formula for Donaldson invariants. It essentially follows from the dimension counting argument. We hope that the above formulas can be also considered as blowup formulas in low degree for the K-theoretic equivariant Donaldson invariants, whose existence is still conjectural.

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

8

Combining Theorem 2.4 with (2.2), we get functional equations satisfied by Z inst . These equations, which we call blowup equations, are powerful, and we have several consequences as we will see in later sections. We give the first application now: Corollary 2.8. Let Zn ≡ Zn (ε1 , ε2,~a; β) be the coefficient of qn in Z inst (ε1 , ε2 ,~a; q, β) as in (1.3). Then Zn is determined from Z0 = 1 inductively by the blowup equations from Theorem 2.4 with k = 0, d = 0, 1, 2. The proof is contained in that of Theorem 4.4. 3. Vanishing theorem We will prove Theorem 2.4 in this section. The crucial result here is the vanishing theorem of higher direct image sheaves (see Proposition 3.4). A reader who has interests only in applications of equations in Theorem 2.4 can safely skip this section. 3.1. It is known that M0 (r, n) is a normal variety by [3]. Since π : M(r, n) → M0 (r, n) is birational, π∗ (OM (r,n) ) = OM0 (r,n) . Since M(r, n) is a holomorphic symplectic manifold, KM (r,n) ∼ = OM (r,n) as sheaves (see Lemma 3.6 for a different proof). By the Grauert-Riemenschneider vanishing theorem, the higher direct image sheaves vanish, i.e., Ri π∗ (OM (r,n) ) = 0 for i > 0. Hence we have the following. e

Lemma 3.1. We have the following equality in the equivariant K-group K T (M0 (r, n)): π∗ (OM (r,n) ) = OM0 (r,n) . Theorem 2.4 is equivalent to the following: e

Proposition 3.2. We have the following equalities in K T (M0 (r, n)) ⊗R(Te) R: (1) π b∗ (OM c(r,k,b n) ) = OM0 (r,n) . (2) If 0 < d < r, then

(3)

( OM0 (r,n) , π b∗ (OM c(r,k,b n) (dµ(C))) = 0,

k=0 0 < k < r.

k(r−k) (t1 t2 )k(r−k)/2 OM0 (r,n) . π b∗ (OM c(r,k,b n) (rµ(C))) = (−1)

The equation for the case where 0 ≤ d < r is a consequence of the following two propositions. Proposition 3.3. In the category of the equivariant coherent sheaves CohTe (M0 (r, n)), we have the following equalities: (1) π b∗ (OM c(r,k,b n) ) = π∗ (OM (r,n) ). (2) If d ≥ 1, then

Proposition 3.4. for i > 0 and d < r.

( OM0 (r,n) , π b∗ (OM c(r,k,b n) (dµ(C))) = 0,

k=0 0 < k < r.

Ri π b∗ (OM c(r,k,b n) (dµ(C))) = 0

INSTANTON COUNTING ON BLOWUP. II

9

Proof of Proposition 3.3: We start with the proof of (1). We note that π b is a Grassmannian bundle over M0reg (r, n) with a fiber Gr(r, k). In fact, the inverse image of (E, Φ) ∈ M0reg (r, n) under π b consists of sheaves E ′ , which fit in an exact sequence 0 → E ′ (−C) → p∗ E → OC⊕r−k → 0.

reg π∗ (OM Thus OM0 (r,n) → π b∗ (OM c(r,k,b c(r,k,b n) )) → n) ) is an isomorphism on M0 (r, n). Hence Spec(b M0 (r, n) is finite and birational. Since M0 (r, n) is normal, the assertion holds. We next prove (2). Assume that k = 0. Then the Poincar´e dual of µ(C) is the closed subset c 0, n of M(r, b) (see [2]): c(r, 0, n {(E, Φ) ∈ M b)|E|C ∼ 6= OC⊕r }. Hence we have a Te-equivariant homomorphism O c → Oc (µ(C)), which induces M (r,0,b n)

M (r,0,b n)

Te-equivariant inclusions OM c(r,0,b c(r,0,b n) ⊂ OM n) (dµ(C)) for d ≥ 1. By taking the direct images, we have a Te-equivariant inclusion (3.5)

OM0 (r,n) = π b∗ (OM b∗ (OM c(r,0,b c(r,0,b n) ) ⊂ π n) (dµ(C))).

reg We note that OM0reg (r,n) → π b∗ (OM c(r,0,b n) (dµ(C)))|M0 (r,n) is an isomorphism. Since M0 (r, n) is reg normal and dim(M0 (r, n)\M0 (r, n)) ≤ dim M0 (r, n)−2, the torsion freeness of π b∗ (OM c(r,0,b n) (dµ(C))) implies that π b∗ (OM (dµ(C))) = O as a coherent sheaf. Then (3.5) implies that c(r,0,b M0 (r,n) n) b∗ (O c )∼ π b∗ (O c (dµ(C))) ∼ = OM (r,n) =π

M (r,0,b n)

M (r,0,b n)

0

as Te-equivariant sheaves. Thus (2) holds for k = 0. We next assume that 0 < k < r. As we will see in Proposition 3.7, −µ(C) is π b-big. Since π b is a Gr(r, k)-bundle over M0reg (r, n) with reg dim Gr(r, k) = k(r − k) > 0, π b∗ (OM b∗ (OM c(r,0,b c(r,0,b n) (dµ(C)))|M0 (r,n) = 0. Since π n) (dµ(C))) is torsion free, (2) holds also in these cases.  Proof of Proposition 3.4: We note that we do not need the Te-structure to prove the claim. So we forget the Te-action. In order to apply the Kawamata-Viehweg vanishing theorem to the c(r, k, n Q-line bundle µ(C), we first compute the canonical line bundle of M b). For a later use, e we compute it as a T -equivariant sheaf. Lemma 3.6. We have the following equalities:

e

KM (r,n) = OM (r,n) (rµ(KP2 + 2ℓ∞ )) = (t1 t2 )−rn OM (r,n) ∈ PicT (M(r, n)) ⊗ Q,

Te c −rb n b)) ⊗ Q. KM OM b 2 + 2ℓ∞ )) = (t1 t2 ) c(r,k,b c(r,k,b c(r,k,b n) = OM n) (rµ(KP n) (rµ(C)) ∈ Pic (M (r, k, n

Proof. We only compute KM(r,k,b c n) . The computation of KM (r,n) is similar and simpler. Let b2 × M(r, b2 × M c k, n c(r, k, n c(r, k, n (E, Φ) be the universal family on P b) and p2 : P b) → M b) be the 1 ∼ projection. Note that TM c(r,k,b n) = Extp2 (E, E(−ℓ∞ )). Using the (equivariant) GrothendieckRiemann-Roch theorem, we see that c1 (KM c(r,k,b n) ) = c1 (Rp2∗ (E ∨ ⊗ E(−ℓ∞ )))   = p2∗ ch(E ∨ ⊗ E)e−ℓ∞ ToddPb 2 1     2ℓ∞ + KPb 2 2 ∨ +··· = p2∗ (r − r∆(E) + ch4 (E ⊗ E) + · · · ) 1 − 2 1 = r∆(E)/(KPb 2 + 2ℓ∞ ) = rµ(KPb 2 + 2ℓ∞ ),

10

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

e

c(r, k, n b))Q . where [...]1 means the codimension 1 component in the equivariant Chow group AT2bnr−1 (M e e c(r, k, n c(r, k, n b)) (see [19, Lemma 1.3] or [4, Theorem 1]), KM b)) ∼ Since PicT (M = AT2bnr−1 (M c(r,k,b n) = L r Te c ∼ b)) ⊗ Q. Since E|ℓ∞ ×M OM b 2 + 2ℓ∞ )) ∈ Pic (M (r, k, n c c(r,k,b c(r,k,b α=1 Oℓ∞ ×M(r,k,b n) eα as n) = n) (rµ(KP 1 c(r, k, n Te-sheaves, we have p2∗ (∆(E )) = 0 in A (M b))Q . Therefore O c (µ(ℓ∞ )) ∼ = c OM c(r,k,n)

|ℓ∞ ×M (r,k,b n)

Te

M (r,k,n)

as a Te-line bundle. Since KP2 ∼ = p∗ (KP2 )(C), we get = (t1 t2 )−1 OP2 (−3ℓ∞ ) and KPb 2 ∼ Te c −rb n b)) ⊗ Q, KM OM c(r,k,b c(r,k,b n) = (t1 t2 ) n) (rµ(C)) ∈ Pic (M (r, k, n

b2] = n where we used ∆(E)/[P b.



Te c b)) ⊗ Q. By the By this lemma, OM c(r,k,b c(r,k,b n) ((d − r)µ(C)) ∈ Pic (M (r, k, n n) (dµ(C)) = KM c(r, k, n following proposition, −µ(C) on M b) is π b-nef and π b-big. Then the relative version of the Kawamata-Viehweg vanishing theorem [7, Theorem 1-2-3] implies that

∼ i b∗ (K c Ri π b∗ (OM c(r,k,b M (r,k,b n) ((d − r)µ(C) + D)) = 0 n) (dµ(C))) = R π

c(r, k, n for d < r, where D = 0 in A2bnr−1 (M b)) ⊗ Q. Thus Proposition 3.4 holds.



c(r, k, n Proposition 3.7. The Q-divisor −µ(C) on M b) is π b-nef and π b-big.

Before proving this proposition, we shall treat the remaining case (3) d = r and finish the Te proof of Proposition 3.2. Since π b∗ (OM c(r,k,b n) ) = OM0 (r,n) in K (M0 (r, n)), the Grothendiecke

k(r−k) KM0 (r,n) = (−1)k(r−k) π∗ (KM (r,n) ) in K T (M0 (r, n)). Serre duality implies that π b∗ (KM c(r,k,b n) ) = (−1) By using Lemma 3.6, we get (3). The proof of Proposition 3.7 occupies the rest of this section. Let us briefly sketch the idea of the proof. Let M(r, −H, n′ ) be the moduli space of H-stable sheaves E on P2 with c1 (E) = c1 , ∆(E) = n′ = n + (r − 1)(2r − 1)/r. The first step of the proof is to construct an embedding M(r, n) → M(r, −H, n′ ) by using idea in [11]. Next we show that there is an induced commutative diagram

(3.8)

M(r, n) ֒→ M(r, −H, n′ ) π ↓ ↓π ′ M0 (r, n) → M0 (r, −H, n′ ),

where M0 (r, −H, n′ ) is the Uhlenbeck compactification of the open subset consisting of locally free sheaves in M(r, −H, n′ ), and π ′ is the morphism defined exactly as π. Similarly we can make a commutative diagram on blowup: (3.9)

c(r, k, n c(r, kC − H, n M b) ֒→ M b′ ) π b ↓ ↓πb ′ M0 (r, n) → M0 (r, −H, n′ ),

b2 with n c(r, kC −H, n where M b′ ) is the moduli space of (H −εC)-stable sheaves on P b′ −b n = n′ −n. From the diagram it is enough to show that −µ(C) is π b′ -nef and π b′ -big. But this assertion follows from the following known results: 1) The divisor µ(H − εC) is nef and big as it gives c(r, kC − H, n c0 (r, kC − H, n a birational morphism M b′ ) → M b′ ) [9]. 2) µ(H) is the pull-back of a Q-Cartier divisor H on M0 (r, −H, n′ ) by the construction of π b′ in [14, Appendix F].

INSTANTON COUNTING ON BLOWUP. II

11

3.2. An embedding M(r, n) → M(r, −H, n′ ). We construct an embedding M(r, n) → M(r, −H, n′ ) using parabolic sheaves [12]. Our construction is inspired by [11]. We collect basic facts on parabolic sheaves in [12]. Definition 3.10. A parabolic sheaf (E∗ , α∗ ) with respect to ℓ∞ is a pair of a filtration E∗ : E(−ℓ∞ ) ⊂ El ⊂ · · · ⊂ E2 ⊂ E1 = E on P2 and a sequence of rational numbers 0 ≤ α1 < α2 < · · · < αl < 1. In this paper, we consider parabolic sheaves (E∗ , α1 , α2 ) such that (3.11)

E∗ : E(−ℓ∞ ) ⊂ E ′ ⊂ E

and E/E ′ = Oℓ∞ (r − 1) in the Grothendieck group K(ℓ∞ ). Definition 3.12. A parabolic sheaf (E∗ , α1 , α2 ) is µ-semi-stable, if E is torsion free and (3.13)

deg G + α1 rankℓ∞ (G/G ∩ E ′ ) + α2 rankℓ∞ (G ∩ E ′ /G(−ℓ∞ )) rank G deg E + α1 rankℓ∞ (E/E ′ ) + α2 rankℓ∞ (E ′ /E(−ℓ∞ )) ≤ rank E

for a saturated subsheaf G ⊂ E with 0 < rank G < rank E, where deg denotes the degree with respect to H and rankℓ∞ the rank on ℓ∞ . If the inequality is strict for all G, then we say (E∗ , α1 , α2 ) is µ-stable. We set α = α2 − α1 , then 0 < α < 1 and (3.13) is equivalent to the following condition: (3.14)

deg E − α rankℓ∞ (E/E ′ ) deg G − α rankℓ∞ (G/G ∩ E ′ ) ≤ rank G rank E

for a saturated subsheaf G ⊂ E with 0 < rank G < rank E. Therefore we define our parabolic sheaf as a filtration E∗ : E(−ℓ∞ ) ⊂ E ′ ⊂ E with a parameter α, and we define the stability of (E∗ , α) as the condition (3.14). From now on, we fix the parameter α with 0 < α < r/(r − 1)2 . Thus our parabolic sheaf is the filtration E∗ : E(−ℓ∞ ) ⊂ E ′ ⊂ E. Definition 3.15. M denotes the moduli space of µ-stable parabolic sheaves (E∗ , α) on P2 such that (rank E, c1 (E), ∆(E)) = (r, 0, n) and E/E ′ = Oℓ∞ (r − 1) in K(ℓ∞ ). An easy computation shows that (c1 (E ′ ), ∆(E ′ )) = (−H, n′ ). Lemma 3.16. Let E∗ : E(−ℓ∞ ) ⊂ E ′ ⊂ E be a parabolic sheaf with (rank E, c1 (E), ∆(E)) = (r, 0, n) and E/E ′ = Oℓ∞ (r − 1). (1) E∗ is µ-stable if and only if E is µ-semi-stable and rankℓ∞ (G/G ∩ E ′ ) = 1 for all saturated subsheaf 0 6= G ⊂ E with deg(G) = 0. (2) E∗ is µ-stable if and only if E ′ is µ-stable. Hence we have a morphism φ : M → M(r, −H, n′ ).

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

12

Proof. We first prove (1). Assume that E∗ is µ-stable. If E is not µ-semi-stable, then there is a saturated subsheaf G ⊂ E with deg G > 0. Then deg E − α rankℓ∞ (E/E ′ ) deg G − α rankℓ∞ (G/G ∩ E ′ ) − rank  rank G  E −1 rankℓ∞ (G/E ′ ∩ G) 1 ≤ +α − r−1 rank G r   −1 1 1 ≤ ≤ +α 1− ((r − 1)2 α − r) < 0, r−1 r r(r − 1) which is a contradiction. Therefore E is µ-semi-stable. If E is properly µ-semi-stable, then there is a saturated subsheaf G ⊂ E with deg G = 0. In this case, we must have rankℓ∞ (G/G ∩ E ′ )/rank G > 1/r. Since G/G∩E ′ ⊂ E/E ′ and rankℓ∞ (E/E ′ ) = 1, rankℓ∞ (G/G∩ E ′ ) = 1. We show the inverse direction. We assume that E is µ-stable. Then for a saturated subsheaf G ⊂ E, we have deg G < 0. Then we see that deg E − α rankℓ∞ (E/E ′ ) deg G − α rankℓ∞ (G/G ∩ E ′ ) − rank  rank G  E ′ 1 1 rankℓ∞ (G/E ∩ G) 1 α ≥ ≥ +α − − > 0. r−1 rank G r r−1 r Thus E∗ is µ-stable. If E is properly µ-semi-stable, we assume that rankℓ∞ (G/G ∩ E ′ ) = 1 for all saturated subsheaf G ⊂ E with deg G = 0. Then we also see that E∗ is µ-stable. We next prove (2). If E ′ is not µ-stable, then there is a saturated subsheaf G′ ⊂ E ′ with 0 < rank G′ < rank E ′ and deg(E ′ ) 1 deg G′ ≥ =− . ′ ′ rank G rank E r Therefore deg G′ ≥ 0. Since E ′ /G′ is torsion free, G′|ℓ∞ → E|ℓ′ ∞ is injective, and hence G′ ∩ E ′ (−ℓ∞ ) = G′ (−ℓ∞ ). We set G := (G′ ∩ E(−ℓ∞ ))(ℓ∞ ). Since (G′ ∩ E(−ℓ∞ )) ∩ E ′ (−ℓ∞ ) = G′ ∩ E ′ (−ℓ∞ ) = G′ (−ℓ∞ ), we get G ∩ E ′ = G′ . By (1), E is µ-semi-stable, therefore deg G ≤ 0. As 0 ≤ deg G′ ≤ deg G ≤ 0, we get deg G = deg G′ = 0. Hence rankℓ∞ (G/G′ ) = 0, which implies that E(−ℓ∞ ) ⊂ E ′ ⊂ E is not µ-semi-stable. Conversely suppose E ′ is µ-stable. Then for a subsheaf G ⊂ E, we have deg G ∩ E ′ < 0. As deg G = deg(G ∩ E ′ ) + rankℓ∞ (G/G ∩ E ′ ), we have deg G < 0 or deg G = 0 and rank(G/G ∩ E ′ ) = 1. By (1), E∗ is µ-stable.  Let M′ be the open subscheme of M such that M′ = {E∗ ∈ M | E/E ′ and E ′ /E(−ℓ∞ ) are semi-stable locally free sheaves on ℓ∞ } ={E∗ ∈ M | E/E ′ ∼ = Oℓ∞ (−1)⊕(r−1) }. = Oℓ∞ (r − 1), E ′ /E(−ℓ∞ ) ∼

If E(−ℓ∞ ) ⊂ E ′ ⊂ E belongs to M′ , then E|ℓ∞ is a locally free Oℓ∞ -module. Hence E is locally free in a neighborhood of ℓ∞ , and hence E ′ = ker(E → Oℓ∞ (r − 1)) is also locally free in a neighborhood of ℓ∞ . Lemma 3.17. Let E∗ : E(−ℓ∞ ) ⊂ E ′ ⊂ E be a point of M. (1) If E∗ belongs to M′, then E|ℓ′ ∞ ∼ = Oℓ∞ (r − 2) ⊕ Oℓ∞ (−1)⊕(r−1) . (2) If E|ℓ′ ∞ ∼ = Oℓ∞ (r − 2) ⊕ Oℓ∞ (−1)⊕(r−1) , then (3.18)

E = ker(E ′ → Hom(E ′ , Oℓ∞ (−1))∨ ⊗ Oℓ∞ (−1)) ⊗ OP2 (ℓ∞ ).

In particular E∗ ∈ M is uniquely determined by E ′ . Moreover E∗ belongs to M′ .

INSTANTON COUNTING ON BLOWUP. II

13

Proof. (1) By the filtration E ′ (−ℓ∞ ) ⊂ E(−ℓ∞ ) ⊂ E ′ ⊂ E induced by E∗ , we have an exact sequence 0 → (E/E ′ ) ⊗ OP2 (−ℓ∞ ) → E|ℓ′ ∞ → E ′ /E(−ℓ∞ ) → 0.

Then this exact sequence splits and we get our claim. (2) We set L′ := E ′ /E(−ℓ∞ ). Then L′ is an Oℓ∞ -module of rank r−1 with deg L′ = −(r−1). Let T be the torsion submodule of L′ and 0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fs = L′ /T the Harder-Narasimhan filtration of L′ /T . We set Fi /Fi−1 ∼ = Oℓ∞ (ai )⊕ni . Then a1 > a2 > · · · > Ps ′ as with i=1 ni ai + deg(T ) = deg L = −(r − 1). Since we have a surjective homomorphism E|ℓ′ ∞ → L′ → Oℓ∞ (as ),

we get as ≥ −1. Then we see that T = 0, s = 1 and L′ ∼ = Oℓ∞ (−1)⊕(r−1) . Moreover we have a commutative diagram E ′ −−−→



L′  ξ y

E ′ −−−→ Hom(E, Oℓ∞ (−1))∨ ⊗ Oℓ∞ (−1),

where ξ is an isomorphism. Hence (3.18) holds. By the filtration E ′ (−ℓ∞ ) ⊂ E(−ℓ∞ ) ⊂ E ′  induced by E∗ , we see that E/E ′ ∼ = Oℓ∞ (r − 1). Therefore (2) holds. We set S := {E ′ ∈ M(r, −H, n′ )|E|ℓ′ ∞ ∼ = Oℓ∞ (r − 2) ⊕ Oℓ∞ (−1)⊕(r−1) }.

S is a locally closed subscheme of M(r, −H, n′ ), where we use the reduced scheme structure on S. By Lemma 3.17 (1), we get φ(M′) ⊂ S. Lemma 3.19. M′ → S is an isomorphism. Proof. Let E ′ be the universal family on S × P2 and qS : S × P2 → S the projection. By the ′ , Oℓ∞ (−1)) ∼ definition of S, Hom(E|{s}×ℓ = C⊕(r−1) for s ∈ S. Since S is reduced, the base ∞ ′ , OS×ℓ∞ (−1)) is a locally free sheaf on S and change theorem implies that U := HomqS (E|S×ℓ ∞ we have a family of homomorphisms ∨ ′ ⊠ Oℓ∞ (−1). f : E|S×P 2 → U

We set E := (ker f )(ℓ∞ ). Then we have a family of parabolic sheaves E(−ℓ∞ ) ⊂ E ′ ⊂ E. They are µ-stable by Lemma 3.16 (2). Thus we have a morphism ψ : S → M′ . It is easy to see that φ ◦ ψ is the identity. By Lemma 3.17 (2), we also see that ψ ◦ φ is the identity. Therefore φ is an isomorphism.  Let (E, Φ) be the universal family on M(r, n)×P2 . Then we have a surjective homomorphism φ

⊕r Ψ : E → OM (r,n)×ℓ∞ → OM (r,n)×ℓ∞ (r − 1)

where φ := (z1r−1 , z1r−2 z2 , ..., z2r−1 ). We set E ′ := ker Ψ. Then we have a family of parabolic sheaves E(−ℓ∞ ) ⊂ E ′ ⊂ E. Lemma 3.20. E(−ℓ∞ ) ⊂ E ′ ⊂ E is a family of µ-stable parabolic sheaves.

14

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

′ ∼ ⊕r Proof. For s ∈ M(r, n), we set E := E|{s}×P2 and E ′ := E|{s}×P 2 . Since Eℓ∞ = Oℓ∞ , E is µsemi-stable. Assume that there is a saturated proper subsheaf G of E with deg G = 0. Then, since E/G is torsion free, G|ℓ∞ is a subsheaf of E|ℓ∞ ∼ . Hence G|ℓ∞ is a direct summand = Oℓ⊕r ∞ ′ of E|ℓ∞ , which implies that rankℓ∞ (G/G ∩ E ) = 1. By Lemma 3.16 (1), E(−ℓ∞ ) ⊂ E ′ ⊂ E is µ-stable. 

Hence we have a morphism M(r, n) → M. Let M′′ be the open subscheme of M′ such that E(−ℓ∞ ) ⊂ E ′ ⊂ E satisfies E|ℓ∞ ∼ . = Oℓ⊕r ∞ Lemma 3.21. We have an isomorphism M(r, n) → M′′ .

Proof. We construct the inverse map. For the family of parabolic sheaves E(−ℓ∞ ) ⊂ E ′ ⊂ E on M′′ × P2 , we see that E/E ′ ∼ = L ⊠ Oℓ∞ (r − 1) for a line bundle L on M′′ . By taking the ∨ ′ ∼ ∗ tensor product with qM ′′ (L ) to the family, we may assume that E/E = OM′′ ⊠ Oℓ∞ (r − 1). ⊕r We set g : E → OM′′ ⊠ Oℓ∞ (r − 1). Since E|ℓ∞ is a family of Oℓ∞ , g induces an isomorphism qM′′ ∗ (E|ℓ∞ ) → OM′′ ⊗ H 0 (ℓ∞ , Oℓ∞ (r − 1)). Then we have a commutative diagram ∼

qM′′ ∗ (E|ℓ∞ ) ⊠ Oℓ∞ −−−→   y E|ℓ∞

g|ℓ∞

OM′′ ⊠ Oℓ⊕r  ∞ φ y

−−−→ OM′′ ×ℓ∞ (r − 1)

⊕r Since the left vertical map is an isomorphism, we can find a homomorphism Φ : E → OM ′′ ×ℓ ∞ such that φ ◦ Φ = g. 

By Lemma 3.19 and Lemma 3.21, we have a sequence of morphisms: ∼



M(r, n) → M′′ ֒→ M′ → S ֒→ M(r, −H, n′ ).

Thus we obtain the following proposition.

Proposition 3.22. We have an immersion ι : M(r, n) → M(r, −H, n′ ).

3.3. Construction of M0 (r, n) → M0 (r, −H, n′ ). Let π ′ : M(r, −H, n′ ) → M0 (r, −H, n′ ) be the contraction map to the Uhlenbeck’s compactification of the open subset consisting of locally free sheaves in M(r, −H, n′ ). Since π ′ is surjective, replacing M0 (r, −H, n′ ) by its normalization, we may assume that M0 (r, −H, n′ ) is normal. Then by Corollary A.2, M0 (r, −H, n′ ) is the quotient of the equivalence relation which defines the contraction to the Uhlenbeck compactification. We consider the composition ̟ : M(r, n) → M(r, −H, n′ ) → M0 (r, −H, n′ ). Then Lemma 3.23. ̟((E1 , Φ1 )) = ̟((E2 , Φ2 )) if and only if (E1∨∨ , Φ1 ) ∼ = (E2∨∨ , Φ2 ) and Supp(E1∨∨ /E1 ) = ∨∨ Supp(E2 /E2 ). Proof. We set Ei′ := ι((Ei , Φi )), i = 1, 2. Then ̟((E1 , Φ1 )) = ̟((E2 , Φ2 )) if and only if (E1′ )∨∨ ∼ = (E2′ )∨∨ and Supp((E1′ )∨∨ /E1′ ) = Supp((E2′ )∨∨ /E2′ ). Since Ei′ , i = 1, 2 fits in an exact sequence 0 → Ei′ → Ei → Oℓ∞ (r − 1) → 0 and Ei is locally free along ℓ∞ , we have an exact sequence 0 → (Ei′ )∨∨ → Ei∨∨ → Oℓ∞ (r − 1) → 0.

Moreover Ei∨∨ → Oℓ∞ (r−1) is uniquely determined by (Ei′ )∨∨ . Therefore our lemma holds. 

We thus get the diagram (3.8) except the bottom arrow. Now the morphism M0 (r, n) → M0 (r, −H, n′ ) exists as M0 (r, n) is normal by Lemma A.1.

INSTANTON COUNTING ON BLOWUP. II

15

c k, n 3.4. Proof of Proposition 3.7. In the same way, we have an immersion M(r, b) → ′ c(r, kC − H, n M b ) and the commutative diagram (3.9). As we already mentioned before, this implies that −µ(C) is π b′ -nef and π b′ -big. Here µ is b2 × M c(r, kC − H, n defined as µ(•) := ∆(E ′ )/[•] by the universal family E ′ on P b′ ). This is ′ enough for our purpose as two universal sheaves E and E are isomorphic outside ℓ∞ , so µ(C) turns out to be the same for E and E ′ . Thus we complete the proof of Proposition 3.7.  4. Behavior at ε1 , ε2 = 0 We prove a part of Nekrasov’s conjecture and derive an equation satisfied by the ‘genus 0’ part of the partition function in this section. ~

4.1. Nekrasov’s conjecture. We collect some properties of lαk~ . ~

~

Lemma 4.1. (1) lαk~ (ε1 , ε2 ,~a) = lαk~ (ε2 , ε1 ,~a). 2 ~ ~ ~ ~ ~ −~k a). (2) lαk~ (ε1 , ε2 ,~a) = (−e−βh~a,αi )hk,αi(hk,αi−1)/2 (eβ(ε1 +ε2 ) )hk,αi(hk,αi −1)/6 l−~ α (−ε1 , −ε2 ,~ ~k (3) lα~ (ε1 , ε2 ,~a) is regular at (ε1 , ε2 ) = 0 and ~

~

~

lαk~ (0, 0,~a) = (1 − e−βh~a,~αi )hk,αi(hk,~αi−1)/2 . Since Y

~

~

(−e−βh~a,~αi )hk,~αi(hk,~αi−1)/2 =

Y

~

~

~

~

~

eβh~a,~αihk,~αi (−1)hk,~αi = erβ(k,~a) (−1)2hk,ρi = erβ(k,~a) ,

α∈∆+

α ~ ∈∆

we get

Y

(4.2)

~

~

lαk~ (ε1 , ε2 ,~a) = erβ(k,~a)

α ~ ∈∆

Y

α ~ ∈∆

~

−k l−~ a). α (−ε1 , −ε2 ,~

The following will be used only afterwards, but it illustrates a usage of the blowup equations in Theorem 2.4. Lemma 4.3. Z inst (ε1 , −2ε1 ,~a; q, β) = Z inst (2ε1 , −ε1 ,~a; q, β).

Proof. We apply the relations (2.2) and (2.5) for d = r, 0 after setting ε2 = −ε1 . We omit β by letting it be 1 for brevity. Then ~~  X q(k,k)/2 ~ r/2 −r/2 e(k,~a)r Z inst (ε1 , −2ε1 ,~a + ε1~k; t1 q)Z inst (2ε1 , −ε1 ,~a − ε1~k; t1 q) Q ~k a) α ~ ∈∆ lα ~ (ε1 , −ε1 ,~ ~k  −r/2 r/2 − Z inst (ε1 , −2ε1 ,~a + ε1~k; t1 q)Z inst (2ε1 , −ε1 ,~a − ε1~k; t1 q) = 0. Let us expand Z inst as in (1.3). Then the above equation implies rn/2

(Zn (ε1 , −2ε1 ,~a) − Zn (2ε1 , −ε1 ,~a)) (t1 =−

X

(~k,~k)/2+l+m=n l6=n,m6=n

−rn/2

− t1

)

~ er(k,~a)/2 Zm (ε1 , −2ε1 ,~a + ε1~k)Zl (2ε1 , −ε1 ,~a − ε1~k) Q ~k a) α ~ ∈∆ lα ~ (ε1 , −ε1 ,~   r(~k,~a)/2 r(m−l)/2 r(−~k,~a)/2 −r(m−l)/2 × e t1 −e t1 .

Let us show that Zn (ε1 , −2ε1 ,~a) = Zn (2ε1 , −ε1 ,~a) by using the induction on n. It holds for n = 0 as Z0 = 1. Suppose that it is true for l, m < n. Then the right hand side of the above

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

16

equation vanishes, as terms with (~k, l, m) and (−~k, m, l) cancel thanks to Lemma 4.1 (1) and (4.2), and the term (0, l, l) is 0. Therefore it is also true for n.  Now we prove a part of Nekrasov’s conjecture: Theorem 4.4. We set def.

F inst (ε1 , ε2 ,~a; q, β) = ε1 ε2 log Z inst (ε1 , ε2 ,~a; q, β). Then F inst (ε1 , ε2 ,~a; q, β) is regular at (ε1 , ε2) = (0, 0). Proof. We omit β by letting it be 1 for brevity. inst inst By Theorem 2.4(1), we have Zb0,d+1 − Zb0,d = 0 for 0 ≤ d < r. We divide this equation by d−r/2 d−r/2 inst inst Z (ε1 , ε2 − ε1 ,~a, t1 q)Z (ε1 − ε2 , ε2,~a, t2 q). Let us expand F inst as X Fn (ε1 , ε2 ,~a)qn . F inst (ε1 , ε2 ,~a; q) = n

We note that

d+1−r/2 d+1−r/2 Z inst (ε1 , ε2 − ε1 ,~a + ε1~k; t1 q)Z inst (ε1 − ε2 , ε2,~a + ε2~k; t2 q) d−r/2

=

d−r/2

Z inst (ε1 , ε2 − ε1 ,~a; t1 q)Z inst (ε1 − ε2 , ε2,~a; t2 q) d+1−r/2 d+1−r/2 inst inst Z (ε1 , ε2 − ε1 ,~a + ε1~k; t q)Z (ε1 − ε2 , ε2,~a + ε2~k; t q) Z inst (ε

= exp

" X

1 2 d+1−r/2 d+1−r/2 inst a; t1 q)Z (ε1 − ε2 , ε2,~a; t2 q) 1 , ε2 − ε1 ,~ d+1−r/2 d+1−r/2 Z inst (ε1 , ε2 − ε1 ,~a; t1 q)Z inst(ε1 − ε2 , ε2 ,~a; t2 q) × d−r/2 d−r/2 Z inst (ε1 , ε2 − ε1 ,~a; t1 q)Z inst(ε1 − ε2 , ε2 ,~a; t2 q)

Fninst (ε1 , ε2 − ε1 ,~a + ε1~k) − Fninst (ε1 , ε2 − ε1 ,~a) (d+1−r/2)n t1 ε1 (ε2 − ε1 ) n≥1 ! Fninst (ε1 − ε2 , ε2,~a + ε1~k) − Fninst (ε1 − ε2 , ε2 ,~a) (d+1−r/2)n t2 qn + ε2 (ε2 − ε1 ) +

X n≥1

(d−r/2)n

Fninst (ε1 , ε2 − ε1 ,~a)(tn1 − 1)t1 ε1 (ε2 − ε1 )

(d−r/2)n

F inst (ε1 − ε2 , ε2 ,~a)(tn2 − 1)t2 − n ε2 (ε2 − ε1 )

!

qn .

By (2.2) we get the following relations for d = 0, 1: X n≥1

X n≥1

−rn/2

Fninst (ε1 , ε2 − ε1 ,~a)(tn1 − 1)t1 ε1 (ε2 − ε1 )

(1−r/2)n

Fninst (ε1 , ε2 − ε1 ,~a)(tn1 − 1)t1 ε1 (ε2 − ε1 )

−rn/2

F inst (ε1 − ε2 , ε2 ,~a)(tn2 − 1)t2 − n ε2 (ε2 − ε1 )

(1−r/2)n



Fninst (ε1 − ε2 , ε2 ,~a)(tn2 − 1)t2 ε2 (ε2 − ε1 )

!

!

qn =

qn =

X

An qn

n

X

Bn qn ,

n

where the right hand sides comes from terms with ~k 6= 0. In particular, An and Bn are written by Fm with m < n. Solving the above, we get Fn (ε1 , ε2 − ε1 ,~a) =

(tn1

ε1 (ε2 − ε1 ) rn/2 n t (t2 An − Bn ). n n 1 − 1)(t2 − t1 )

#

INSTANTON COUNTING ON BLOWUP. II

17

Hence if Fm , m < n are regular at (ε1 , ε2 ) = (0, 0), then Fn is also regular. As F0 = 0, we get the assertion by the induction on n.  4.2. The perturbative part. We set !  3 X1 1 β 1 e−βnx def. γε1 ,ε2 (x|β; Λ) = − x + (ε1 + ε2 ) + x2 log(βΛ) + , βnε1 − 1)(eβnε2 − 1) 2ε1ε2 6 2 n (e n≥1 γε1 ,ε2 (x|β; Λ) e

1 = γε1 ,ε2 (x|β; Λ) + ε1 ε2

def.



π 2 x ζ(3) − 2 6β β



ε1 + ε2 + 2ε1 ε2

  π2 ε2 + ε22 + 3ε1 ε2 x log(βΛ) + + 1 log(βΛ) 6β 12ε1 ε2

for βx > 0. Here Λ = q1/2r . If we formally expand ε1 ε2 e γε1 ,ε2 (x|β; Λ) as a power series of ε1 , ε2 around ε1 = ε2 = 0, then each coefficient is a holomorphic function of x. Indeed if we expand X cm 1 = β m−2 , ε β ε β 1 2 (e − 1)(e − 1) m≥0 m! then

X cm X1 e−βnx = β m−2 Li3−m (e−βx ), βnε βnε 1 2 n (e − 1)(e − 1) m≥0 m! n≥1

where Li3−m (e−βx ) are polylogarithms. We extend the definition of γε1,ε2 (x|β; Λ) to the range βx < 0 by analytic continuation along a circle counter-clockwise way. Proposition 4.5. If ~a is in a neighborhood of the region βh~a, α ~ i > 0 for all α ~ ∈ ∆+ , X ~ i|β; Λe(d−r/2)βε1 /2r ) γε1 ,ε2−ε1 (h~a + ~kε1 , α e  α ~ ∈∆ (d−r/2)βε2 /2r ~ + γeε1 −ε2 ,ε2 (h~a + kε2 , α ~ i|β; Λe ) − γeε1 ,ε2 (h~a, α ~ i|β; Λ)  r (4d − r)(r − 1) (~k, ~k)  2r log((βΛ) ) − (d − )(ε1 + ε2 )β =− (ε1 + ε2 )β − 48 2 2 X ~k ~ − d(k,~a)β + log(lα~ (ε1 , ε2 ,~a)). α ~ ∈∆

Proof. If k = −l with l > 0, then

(4.6)

e−(x+kε1) e−(x+kε2) e−x + − (eε1 − 1)(eε2 −ε1 − 1) (eε2 − 1)(eε1 −ε2 − 1) (eε1 − 1)(eε2 − 1)   ε1 +ε2 lε1 1 e (e − elε2 ) − (e(l+1)ε1 − e(l+1)ε2 ) −x − ε1 =e (eε1 − 1)(eε2 − 1)(eε2 − eε1 ) (e − 1)(eε2 − 1) ! Pl−1 (i+1)ε1 (l−i)ε2 Pl iε1 (l−i)ε2 e e e e + − 1 i=0 i=0 = e−x − ε1 (eε1 − 1)(eε2 − 1) (e − 1)(eε2 − 1) Pl−1 iε1 (l−i)ε2 Pl−1 iε1 X + i=0 e −x −x − i=0 e e eiε1 ejε2 . = −e =e ε 2 (e − 1) 0≤i,j i+j≤l−1

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

18

Since (etε1

ε1 ε2 e−tx − 1)(etε2 − 1)

is holomorphic at (ε1 , ε2 ) = (0, 0), (4.6) holds in C[e−x ][[ε1 , ε2 ]][ ε1 ε2 (ε11 −ε2 ) ]. If k > 0, then e−(x+kε2 ) e−x e−(x+kε1 ) + − (eε1 − 1)(eε2 −ε1 − 1) (eε2 − 1)(eε1 −ε2 − 1) (eε1 − 1)(eε2 − 1)   −kε1 1 (e − e−kε2 ) − (e−(k−1)ε1 − e−(k−1)ε2 )e−ε1 −ε2 −x−ε1 −ε2 − −ε1 =e (e−ε1 − 1)(e−ε2 − 1)(e−ε1 − e−ε2 ) (e − 1)(e−ε2 − 1) ! Pk−1 −iε1 −(k−1−i)ε2 Pk−2 −(i+1)ε1 −(k−1−i)ε2 e e e e − 1 i=0 i=0 = e−x − −ε1 (e−ε1 − 1)(e−ε2 − 1) (e − 1)(e−ε2 − 1) Pk−2 iε1 −(k−1−i)ε2 Pk−2 −iε1 X + i=0 e −x − −x i=0 e e =e e−(i+1)ε1 e−(j+1)ε2 . = −e (e−ε2 − 1) 0≤i,j i+j≤k−2

Hence

(4.7)

−βn(~a,~ α) X X1 1 e−βn(~a,~α) e + βnε βnε βnε tnε 1 2 1 2 n (e − 1)(e − 1) ~a→~a+~kε1 n (e − 1)(e − 1) ~a→~a+~kε2 n≥1 n≥1 ε1 →ε1 ε2 →ε2 −ε1



X1 e−βn(~a,~α) ~ = log(lαk~ (ε1 , ε2 ,~a)) βnε βnε 1 2 n (e − 1)(e − 1) n≥1

ε1 →ε1 −ε2 ε2 →ε2

in O[[ε1 , ε2 ]][ ε1 ε2 (ε11 −ε2 ) ], where O is the ring of holomorphic functions of β~a in a neighborhood of βh~a, α ~ i > 0. By the analytic continuation, (4.7) also holds for the range βh~a, α ~ i < 0. Combining the following equalities, we get our claim. (4.8)

(4.9)

(4.10) (4.11)

−(x + ε1 k + 21 ε2 )3 −(x + ε2 k + 21 ε1 )3 + 12ε1(ε2 − ε1 ) 12ε2(ε1 − ε2 ) 1 3 −(x + 2 (ε1 + ε2 )) 4k 2 − 4k + 1 8k 3 − 6k + 2 + x+ (ε1 + ε2 ), = 12ε1 ε2 16 96 (x + kε1 )2 (x + kε2 )2 log(βΛe(d−r/2)βε1 /2r ) + log(βΛe(d−r/2)βε2 /2r ) 2ε1 (ε2 − ε1 ) 2ε2 (ε1 − ε2 )  r  2 2 2 d− 2 k x k = log(βΛ) − log(βΛ) − (ε1 + ε2 ) + kx , 2ε1 ε2 2 2r 2 π2x π 2 (x + kε2 ) π 2 (x + kε1 ) = , + 6βε1 (ε2 − ε1 ) 6β(ε1 − ε2 )ε2 6βε1 ε2 ε2 (x + kε1 ) log(βΛe(d−r/2)βε1 /2r ) ε1 (x + kε2 ) log(βΛe(d−r/2)βε2 /2r ) + 2ε1 (ε2 − ε1 ) 2(ε1 − ε2 )ε2 r (d − 2 )βx (ε1 + ε2 )x log(βΛ) k + log(βΛ) + , = 2ε1 ε2 2 4r

INSTANTON COUNTING ON BLOWUP. II

(4.12)

19

ε2 − ε22 + ε1 ε2 −ε21 + ε22 + ε1 ε2 log(βΛe(d−r/2)βε1 /2r ) + 1 log(βΛe(d−r/2)βε2 /2r ) 12ε1 (ε2 − ε1 ) 12(ε1 − ε2 )ε2 r 2 2 (d − 2 )β(ε1 + ε2 ) ε + ε2 + 3ε1 ε2 log(βΛ) + . = 1 ε1 ε2 12r 

By using (B.6) and (B.8), we see that ε1 ε2 (e γ (x; β) + e γ (−x; β)) √   2 2 2 −1 βx3 x π π x 1 x −βx − log(βΛ) + − =2 (Li (e ) − ζ(3)) + 3 2 6β 2 6 β2 √     2 2 −βx ε1 + ε2 + 3ε1ε2 x √ 1−e βx + π −1 + (ε1 + ε2 ) π −1 − log + +··· . 2 6 βΛ 2

By §B.3, we get

ε1 ε2 (e γ (x; β) + γe(−x; β))     x 3   1 3 2 −x 1 2 2 2 + x + − x log + x −→ − x log β→0 2 Λ 4 2 Λ 4 √     2 2 x √ π −1 x ε1 + ε2 + 3ε1 ε2 + (ε1 + ε2 ) π −1 − log + +··· . 2 6 Λ 2

Therefore the perturbative part of the K-theoretic partition function (see below) converges to that of the homological partition function as β → 0. 4.3. The full partition functions. By adding the perturbative term, we define the full partition functions as ! X def. Z(ε1 , ε2,~a; q, β) = exp −e γε1 ,ε2 (h~a, α ~ i|β; Λ) Z inst (ε1 , ε2 ,~a; q, β) α ~ ∈∆

(4.13)

def. Zbk,d (ε1 , ε2,~a; q, β) = exp

X

α ~ ∈∆

!

inst −e γε1 ,ε2 (h~a, α ~ i|β; Λ) Zk,d (ε1 , ε2 ,~a; q, β).

By (2.2) and Proposition 4.5, we get   X (4d − r)(r − 1) Zbk,d(ε1 , ε2 ,~a; q, β) = exp − β(ε1 + ε2 ) 48 ~ (4.14) {k}=−k/d r

r

(d− ) (d− ) × Z(ε1 , ε2 − ε1 ,~a + ε1~k; t1 2 q, β)Z(ε1 − ε2 , ε2 ,~a + ε2~k; t2 2 q, β).

If we set ε2 = −ε1 , then Zbk,d(ε1 , −ε1 ,~a; q, β) =

X

{~k}=−k/d

r

r

−(d− ) (d− ) Z(ε1 , −2ε1 ,~a + ε1~k; t1 2 q, β)Z(2ε1 , −ε1 ,~a − ε1~k; t1 2 q, β).

We expand F as

F (ε1 , ε2 ,~a; q, β) = ε1 ε2 log Z(ε1 , ε2 ,~a; q, β) = F0 (~a; q, β) + (ε1 + ε2 )H(~a; q, β) + (ε1 + ε2 )2 G(~a; q, β) + ε1 ε2 F1 (~a; q, β) + · · · .

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

20

By Lemma 4.3, H(~a; q, β) comes from the perturbative part:  X 1 β 1 π2 −β(aα −aβ ) 2 Li2 (e ) + (aα − aβ ) − (aα − aβ ) log(βΛ) − H(~a; q, β) = 2β 8 2 12β α6=β =−

X √ (aα − aβ ) √ π −1 = −π −1h~a, ρi. 2 α 0, satisfying the recursive system Pk+1(w) = (1 + kw)Pk (w) + w(1 − w)Pk′ (w). In particular, Li−k (w) is defined for w 6= 1. For |w| < 1, Lik can be expressed by a series ∞ X wn Lik (w) = . nk n=1

B.2. Inversion formulas. We need to relate Lik (1/w) to Lik (w). For k = −l with l > 0, we have 1 1/w = = −1 − Li0 (w), Li0 (1/w) = (1 − 1/w) w−1 l l   1 d d 1 1 Li−l ( ) = Li0 ( ) = − −w Li01 (w) = (−1)l−1 Li−l (w). w w d(1/w) w dw Next consider Li1 (w) = − log(1 − w). We need to specify how we take the branch of log. We set w = e−y with y > 0. Then Li1 (e−y ) is defined. We define Li1 (e−y ) with y < 0 by analytic continuation. Then √ (B.1) Li1 (ey ) = Li1 (e−y ) − y − π −1. If 2π > y > 0, then by integrating

(B.2) we have (B.3)

1 B1 2 B2 4 log(1 − e−y ) = log y − y + y − y −··· , 2 2 · 2! 4 · 4! Li2 (e−y ) =

y2 B1 B2 π2 + (y log y − y) − + y3 − y5 − · · · . 6 4 2 · 3 · 2! 4 · 5 · 4!

INSTANTON COUNTING ON BLOWUP. II

25

Note that B1 B2 y2 + y3 − y5 − · · · . 4 2 · 3 · 2! 4 · 5 · 4! converges for |y| < 2π. We define Li2 (e−y ), y < 0 by analytic continuation. So −

(B.4)

(B.5)

Li2 (e−y ) =

for |y| < 2π. Hence

π2 y2 B1 B2 + (y log y − y) − + y3 − y5 − · · · . 6 4 2 · 3 · 2! 4 · 5 · 4!

y2 π2 − y log(−y) + y log y − 3 2 (B.6) √ y2 π2 = − yπ −1 − 3 2 for y > 0. By integrating (B.5), we have   y2 3 2 y3 B2 B1 π2 4 6 −y − y − y −··· (B.7) Li3 (e ) = ζ(3) − y − log y + y + 6 2 4 12 2 · 3 · 4 · 2! 4 · 5 · 6 · 4! Li2 (ey ) + Li2 (e−y ) =

for |y| < 2π. Hence we have

π2 y+ 3 π2 = Li3 (e−y ) + y − 3

Li3 (ey ) = Li3 (e−y ) + (B.8)

1 2 1 y3 y log y − y 2 log(−y) − 2 2 6 1 2 √ y3 y π −1 − 2 6

for y > 0. B.3. Limit. We have lim β k+1 Li−k (e−βx ) = x−k−1 Pk (1) = k!x−k−1

β→0

for k ∈ Z≥0 . We have −βx

Li1 (e Then

Furthermore

) + log(βΛ) = − log



1 − e−βx βΛ



−−→ − log β→0

x Λ

.

  π2 1 −βx Li2 (e ) − + x log(βΛ) Li1 (e ) + log(βΛ)dx = − β 6 0  ′ Z x x x −−→ − log dx′ = −x log + x. β→0 Λ Λ 0

Z

x

−βx′



  π2 −βx′ Li2 (e )− + x′ log(βΛ)dx′ 6 0  x2 π2x 1 −βx log(βΛ) + = 2 Li3 (e ) − ζ(3) + 2 6β β  ′  Z x  x 1 x 3 2 −x′ log −−→ + x′ dx′ = − x2 log + x. β→0 Λ 2 Λ 4 0

Z

x

1 − β

26

¯ HIRAKU NAKAJIMA AND KOTA YOSHIOKA

References [1] M.F. Atiyah and G.B. Segal, The index of elliptic operators, II, Ann. of Math., 87 (1968), 531–545. 2 [2] J. Bryan, Symplectic geometry and the relative Donaldson invariants of CP , Forum Math. 9 (1997), 325–365. [3] W. Crawley-Boevey, Normality of Marsden-Weinstein reductions for representations of quivers, Math. Ann. 325 (2003), 55–79. [4] D. Edidin and W. Graham, Equivariant intersection theory, Invent. Math. 131 (1998), 595–634. [5] L. G¨ottsche, H. Nakajima and K. Yoshioka, in preparation. [6] A. Gorsky, A. Marshakov, A. Mironov and A. Morozov, RG equations from Whitham hierarchy, Nucl. Phys. B 527 (1998), 690–716; hep-th/9802007. [7] Y. Kawamata, K. Matsuda and K. Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, 283–360, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, 1987 [8] J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Differential Geom. 37 (1993) 417–466. , Compactification of moduli of vector bundles over algebraic surfaces, Collection of papers on [9] geometry, analysis and mathematical physics, World Sci. Publishing, River Edge, NJ, (1997) 98–113. [10] M. Mari˜ no, The uses of Whitham hierarchies, Progr. Theoret. Phys. Suppl. No. 135 (1999), 29–52; hep-th/9905053. [11] M. Maruyama, Instantons and parabolic sheaves, Geometry and analysis (Bombay, 1992), 245–267, Tata Inst. Fund. Res., Bombay, 1995. [12] M. Maruyama and K. Yokogawa, Moduli of parabolic stable sheaves, Math. Ann. 293 (1992), 77–99 [13] H. Nakajima and K. Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, math.AG/0306198, Invent. Math. to appear [14] , Lectures on instanton counting, in Algebraic Structures and Moduli Spaces, CRM Proceedings & Lecture Notes 38, AMS, 2004, 31–101; math.AG/0311058. [15] N. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5, 831–864; hep-th/0206161. [16] N. Nekrasov and A. Okounkov, Seiberg-Witten prepotential and random partitions, preprint, hep-th/0306238. [17] N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory Nuclear Phys. B 426 (1994), 19–52; Erratum, Nuclear Phys. B 430 (1994), 485–486. [18] R. Thomason, Une formule de Lefschetz en K-th´eorie ´equivariante alg´ebrique, Duke Math. 68 (1992), 447–462. [19] K. Yoshioka: A note on a paper of J.-M. Dr´ezet on the local factoriality of some moduli spaces, Internat. J. Math. 7 (1996), 843–858 Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan E-mail address: [email protected] Department of Mathematics, Faculty of Science, Kobe University, Kobe 657-8501, Japan E-mail address: [email protected]