Blowing-Ups Describing the Polarization Change of Moduli Schemes of Semistable Sheaves of General Rank

Yamada, Kimiko

doi:10.1080/00927872.2010.481776

Cited by 2 publications

(1 citation statement)

References 19 publications

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“…We remark that Yamada also used moduli spaces of pairs of a sheaf and a full flag in the sheaf cohomology group to study wall-crossing of moduli spaces of higher rank sheaves [29].…”

Section: Enhanced Master Spacesmentioning

confidence: 99%

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

Nakajima¹,

Yoshioka²

2011

Kyoto J. Math.

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In earlier papers [21,22] of this series we constructed a sequence of intermediate moduli spaces { M m (c)} m=0,1,2,... connecting a moduli space M (c) of stable torsion free sheaves on a nonsingular complex projective surface X and M (c) on its one point blow-up X. They are moduli spaces of perverse coherent sheaves on X. In this paper we study how Donaldson-type invariants (integrals of cohomology classes given by universal sheaves) change from M m (c) to M m+1 (c), and then from M (c) to 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].2000 Mathematics Subject Classification. Primary 14D21; Secondary 16G20. Main resultNotations. Let [z 0 : z 1 : z 2 ] be the homogeneous coordinates on P 2 and ℓ ∞ = {z 0 = 0} the line at infinity. Let p : P 2 → P 2 be the blow-up of P 2 at [1 : 0 : 0]. Then P 2 is the closed subvariety of P 2 × P 1 defined bywhere the map p is the projection to the first factor. We denote p −1 (ℓ ∞ ) also by ℓ ∞ for brevity. Let C denote the exceptional divisor given by z 1 = z 2 = 0. Let O denote the structure sheaf of P 2 , O(C) the line bundle associated with the divisor C, and O(mC) its m th tensor product O(C) ⊗m when m > 0, (O(C) ⊗−m ) ∨ if m < 0, and O if m = 0. And1 The proof in [25] can be generalized to those variants. See [26] for the theory with 5-dimensional Chern-Simons terms, and [13] for higher Casimir operators, but not with Todd genus.

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“…We remark that Yamada also used moduli spaces of pairs of a sheaf and a full flag in the sheaf cohomology group to study wall-crossing of moduli spaces of higher rank sheaves [29].…”

Section: Enhanced Master Spacesmentioning

confidence: 99%

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

Nakajima¹,

Yoshioka²

2011

Kyoto J. Math.

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Variation of Gieseker moduli spaces via quiver GIT

2016

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We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker-stability. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class $\omega \in N^1(X)_\mathbb{R}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker-semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus-flips.Comment: 51 pages; v2: added discussion concerning characteristic of the base field, fixed some typos and inconsistencies; removed "I" from title as second part has a new title; to appear in Geometry & Topolog

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Blowing-Ups Describing the Polarization Change of Moduli Schemes of Semistable Sheaves of General Rank

Cited by 2 publications

References 19 publications

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

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

Variation of Gieseker moduli spaces via quiver GIT

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