Localizing Virtual Structure Sheaves for Almost Perfect Obstruction Theories

Kiem, Young-Hoon; Savvas, Michail

doi:10.48550/arxiv.2007.06820

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“…This exists automatically if X a is a projective C-scheme. It is generally hoped that this assumption can be eliminated (see [115,discussion of Ass. 5.3], for instance), but this does not seem to be available in the literature at present.…”

Section: G M -Localization Of Virtual Classesmentioning

confidence: 99%

Enumerative invariants and wall-crossing formulae in abelian categories

Joyce¹

2021

Preprint

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An enumerative invariant theory in Algebraic Geometry is the study of invariants which 'count' τ -(semi)stable objects E with fixed topological invariants E = α in some geometric problem, by means of a virtual class [M ss α (τ )]virt in some homology theory, for the moduli spacesWe can obtain numbers by taking integrals [M ss α (τ )] virt Υ for suitable universal cohomology classes Υ. Examples include Mochizuki's invariants for coherent sheaves on surfaces [146], and Donaldson-Thomas type invariants for coherent sheaves on Calabi-Yau 3-and 4-folds and Fano 3-folds, [28,108,118,152,176].Let A be a C-linear abelian category coming from Algebraic Geometry. There are two moduli stacks of objects E in A: the usual moduli stack M, and the 'projective linear' moduli stack M pl modulo projective linear isomorphisms, that is, we quotient out by λ idE : E E for λ ∈ Gm. Both are Artin C-stacks. Previous work by the author [106] gives H * (M) the structure of a graded vertex algebra, and H * (M pl ) a graded Lie algebra, closely related to H * (M). Virtual classes [M ss α (τ )]virt lie in H * (M pl ). We develop a universal theory of enumerative invariants in such categories A, which includes and extends many cases of interest. Virtual classes [M ss α (τ )]virt are only defined when M st α (τ ) = M ss α (τ ). We give a systematic way to define invariants [M ss α (τ )]inv in H * (M pl ) for all classes α ∈ C(A), with [M ss α (τ )]inv = [M ss α (τ )]virt when M st α (τ ) = M ss α (τ ). If (τ, T, ) and (τ , T , ) are two suitable (weak) stability conditions on A, we prove a wall-crossing formula which expresses [M ss α (τ )]inv in terms of the [M ss β (τ )]inv, using the Lie bracket on H * (M pl ). We apply our results when A is a category mod-CQ or mod-CQ/I of representations of a quiver Q or quiver with relations (Q, I), and when A = coh(X) for X a curve, surface, or Fano 3-fold, and when A is a category of 'pairs' ρ : V ⊗ C L E in coh(X) for X a curve or surface, where V is a vector space, L X is a fixed line bundle, and E ∈ coh(X). We also speculate on extensions of our theory to 3-Calabi-Yau categories, which would give an alternative approach to Donaldson-Thomas theory to [108,176], and to 4-Calabi-Yau categories, which would give a theory of Donaldson-Thomas type invariants of Calabi-Yau 4-folds.Our results prove conjectures made in Gross-Joyce-Tanaka [81].

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Section: G M -Localization Of Virtual Classesmentioning

confidence: 99%

Enumerative invariants and wall-crossing formulae in abelian categories

Joyce¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Localizing Virtual Structure Sheaves for Almost Perfect Obstruction Theories

Cited by 1 publication

References 0 publications

Enumerative invariants and wall-crossing formulae in abelian categories

Enumerative invariants and wall-crossing formulae in abelian categories

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