Kota Yoshioka

Instanton counting on blowup. I. 4-dimensional pure gauge theory

We give a mathematically rigorous proof of Nekrasov’s conjecture: the integration in the equivariant cohomology over the moduli spaces of instantons on ℝ4 gives a deformation of the Seiberg-Witten prepotential for N=2 SUSY Yang-Mills theory. Through a study of moduli spaces on the blowup of ℝ4, we derive a differential equation for the Nekrasov’s partition function. It is a deformation of the equation for the Seiberg-Witten prepotential, found by Losev et al., and further studied by Gorsky et al.

Euler characteristics of SU(2) instanton moduli spaces on rational elliptic surfaces

Abstract:Recently, Minahan, Nemeschansky, Vafa and Warner computed partition functions for N = 4 topological Yang–Mills theory on rational elliptic surfaces. In particular they computed generating functions of Euler characteristics of SU(2)-instanton moduli spaces. In mathematics, they are expected to coincide with those of Gieseker compactifications. In this paper, we compute Euler characteristics of these spaces and show that our results coincide with theirs. We also check the modular property of ZSU(2) and ZSO(3) conjectured by Vafa and Witten.

Twisted Stability and Fourier–Mukai Transform I

In this paper, we consider the preservation of stability by using the notion of twisted stability. As applications, (1) we show that moduli spaces of stable sheaves on K3 and abelian surfaces are irreducible and (2) we compute Hodge polynomials of some moduli spaces of stable sheaves on Enriques surfaces.

Donaldson = Seiberg–Witten from Mochizuki's Formula and Instanton Counting

We propose an explicit formula connecting Donaldson invariants and Seiberg-Witten invariants of a 4-manifold of simple type via Nekrasovs deformed partition function for the N=2 SUSY gauge theory with a single fundamental matter. This formula is derived from Mochizukis formula, which makes sense and was proved when the 4-manifold is complex projective. Assuming our formula is true for a 4-manifold of simple type, we prove Wittens conjecture and sum rules for Seiberg-Witten invariants (superconformal simple type condition), conjectured by Mari\~no, Moore and Peradze.

Perverse coherent sheaves and Fourier–Mukai transforms on surfaces, I

We study perverse coherent sheaves on the resolution of rational double points. As examples, we consider rational double points on 2-dimensional moduli spaces of stable sheaves on K3 and elliptic surfaces. Then we show that perverse coherent sheaves appears in the theory of Fourier-Mukai transforms. As an application, we generalize the Fourier-Mukai duality for K3 surfaces to our situation.

PERVERSE COHERENT SHEAVES ON BLOW-UP. II. WALL-CROSSING AND BETTI NUMBERS FORMULA

This is the second of series of papers studyig moduli spaces of a certain class of coherent sheaves, which we call stable perverse coherent sheaves, on the blow-up p: b X → X of a projective surface X at a point 0. The followings are main results of this paper: a) We describe the wall-crossing between moduli spaces caused by twisting of the line bundle O(C) associated with the exceptional divisor C. b) We give the formula for virtual Hodge numbers of moduli spaces of stable perverse coherent sheaves. Moreover we also give proofs of the followings which we observed in a special case in (24): c) The moduli space of stable perverse coherent sheaves is isomorphic to the usual moduli space of stable coherent sheaves on the original surface if the first Chern class is orthogonal to (C). d) The moduli space becomes isomorphic to the usual moduli space of stable coherent sheaves on the blow-up after twisting by O(−mC) for sufficiently largem. Therefore usual moduli spaces of stable sheaves on the blow-up and the original surfaces are connected via wall-crossings.

Perverse coherent sheaves on blowup, III: Blow-up formula from wall-crossing

In earlier papers (21, 22) of this series we constructed a sequence of interme- diate moduli spaces fc M m (c)gm=0,1,2,... connecting a moduli space M(c) of stable torsion free sheaves on a nonsingular complex projective surface X and c M(c) on its one point blow-up b X. They are moduli spaces of perverse coherent sheaves on b X. In this pa- per we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from c M m (c) to c M m+1 (c), and then from M(c) to c M(c). As an application we prove that Nekrasov-type partition functions satisfy certain equations which determine invariants recursively in second Chern classes. They are generalization of the blow-up equation for the original Nekrasov deformed partition function for the pure N = 2 SUSY gauge theory, found and used to derive the Seiberg-Witten curves in (18).

Singularities on the 2-Dimensional Moduli Spaces of Stable Sheaves on K3 Surfaces

We consider the singuralities of 2-dimensional moduli spaces of semi-stable sheaves on K3 surfaces. We show that the moduli space is normal, in particular the singuralities are rational double points. We also describe the exceptional locus on the resolution in terms of exceptional sheaves.

A note on stable sheaves on Enriques surfaces II

We shall give a necessary and sufficient condition for the existence of stable sheaves on non-classical Enriques surfaces.

Moduli spaces of stable sheaves on Enriques surfaces

We shall study the existence condition of slope stable sheaves on Enriques surfaces. We also gives a different proof of the irreducibility of the moduli spaces of rank 2 stable sheaves.