Characteristic classes and stability conditions for projective Kleinian orbisurfaces

Lim, Bronson; Rota, Franco

doi:10.1007/s00209-021-02805-8

Cited by 2 publications

(2 citation statements)

References 24 publications

(31 reference statements)

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“…The basic idea is the following. Using the results of Subsection 5.2 and the projection X → C we show that all ı-stable MT sheaves on X are given by Nironi δ-stable sheaves on [S/ı] × {t} for some t ∈ C. We then show that Nironi δ-stability on [S/ı] is the large volume limit of a certain Bridgeland stability condition on [S/ı] constructed by Lim and Rota [15]. We then apply the derived Fourier-Mukai correspondence to show that our moduli spaces are given by, up to a factor of C, moduli spaces of objects in the derived category of S which are stable with respect to the large volume limit of one of the stability conditions on K3 surfaces constructed by Bridgeland [6].…”

Section: Local Abelian and Nikulin Surfaces (Mt Theory)mentioning

confidence: 85%

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Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Bryan¹,

Pietromonaco²

2022

Preprint

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We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) X which is equipped with an involution ı preserving the holomorphic volume form. We define integers n g,h (β) which give a virtual count of the number of genus g curves C on X, in the class β ∈ H 2 (X), which are invariant under ı, and whose quotient C/ı has genus h. We give two definitions of n g,h (β) which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory.We compute our invariants and give evidence for our conjecture in several cases. In particular, we compute our invariants when X = S × C where S is an Abelian surface with ı(a) = −a or a K3 surface with a symplectic involution (a Nikulin K3 surface). For these cases, we give formulas for our invariants in terms of Jacobi modular forms. For the Abelian surface case, the specialization of our invariants n g,h (β) to h = 0 recovers the count of hyperelliptic curves on an Abelian surface first computed in [9].

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Section: Local Abelian and Nikulin Surfaces (Mt Theory)mentioning

confidence: 85%

“…In [15], Lim and Rota construct Bridgeland stability conditions on orbifold surfaces with Kleinian orbifold points. For notational simplicity, they assume that their orbifold surface has a single orbifold point, but their method easily applies to orbifold surfaces with multiple Kleinian orbifold points such as [S/ı].…”

Section: Nironi Stabilitymentioning

confidence: 99%

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Bryan¹,

Pietromonaco²

2022

Preprint

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Riemann–Roch coefficients for Kleinian orbisurfaces

Lim

Rota

2023

Boll Unione Mat Ital

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Suppose $$\mathcal {S}$$ S is a smooth, proper, and tame Deligne–Mumford stack. Toën’s Grothendieck–Riemann–Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of Toën’s Grothendieck–Riemann–Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.

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Characteristic classes and stability conditions for projective Kleinian orbisurfaces

Cited by 2 publications

References 24 publications

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Counting Invariant Curves: a theory of Gopakumar-Vafa invariants for Calabi-Yau threefolds with an involution

Riemann–Roch coefficients for Kleinian orbisurfaces

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