Partial desingularizations of good moduli spaces of Artin toric stacks

Edidin, Dan; More, Yogesh

doi:10.48550/arxiv.1012.1301

Cited by 3 publications

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“…For categories B for which M pl is a smooth Artin stack, we could use Kirwan's algorithm [117] when M st α (τ ) = M ss α (τ ), applying repeated blow ups and deletions to M ss α (τ ) until it becomes a smooth, proper Deligne-Mumford stack Mss α (τ ), and then define [M ss α (τ )] inv = [ Mss α (τ )] fund to be its fundamental class. (See Edidin-More-Rydh [54,55] for discussion, and progress in extending this algorithm to non-smooth stacks.) Again, calculation of examples by the author shows that this gives a different answer to Theorem 5.7.…”

Section: Note In Particular Thatmentioning

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: Note In Particular Thatmentioning

confidence: 99%

Enumerative invariants and wall-crossing formulae in abelian categories

Joyce¹

2021

Preprint

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“…The following result gives expresses how the Euler characteristic increases after a stacky star subdivision [EM,Definition 4.1] of a simplicial stacky fan. Lemma 3.12.…”

Section: 2mentioning

confidence: 99%

“…Let Σ be a (not necessarily simplicial) stacky fan and let X (Σ) be the associated Artin toric stack. By [EM,Theorem 5.2] there is a simplicial stacky fan Σ ′ canonically obtained from Σ by stacky star subdivisions and a commutative diagram of stacks and toric varieties…”

Section: Integration On Artin Toric Stacksmentioning

confidence: 99%

Integration on Artin toric stacks and Euler characteristics

Edidin

2013

Proc. Amer. Math. Soc.

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There is a well-developed intersection theory on smooth Artin stacks with quasi-affine diagonal [Gil, Vis, EG98, Kre]. However, for Artin stacks whose diagonal is not quasi-finite the notion of the degree of a Chow cycle is not defined. In this paper we propose a definition for the degree of a cycle on Artin toric stacks whose underlying toric varieties are complete. As an application we define the Euler characteristic of an Artin toric stack with complete good moduli space -extending the definition of the orbifold Euler characteristic. An explicit combinatorial formula is given for 3-dimensional Artin toric stacks.

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Integration on Artin toric stacks and Euler characteristics

Edidin¹,

More²

2012

Preprint

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Partial desingularizations of good moduli spaces of Artin toric stacks

Cited by 3 publications

References 6 publications

Enumerative invariants and wall-crossing formulae in abelian categories

Enumerative invariants and wall-crossing formulae in abelian categories

Integration on Artin toric stacks and Euler characteristics

Integration on Artin toric stacks and Euler characteristics

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