Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

Huybrechts, Daniel; Thomas, Robert Paul

doi:10.1007/s00208-009-0397-6

Cited by 126 publications

(193 citation statements)

References 22 publications

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“…obtained as in Example 2.1, the theorem can also be deduced from the more general results in [22]. (The assumptions on the non-positive Ext's are not needed in [22].) Our main application of this theorem concerns the case where E n is a Fourier-Mukai kernel.…”

Section: Deformation Of the Fourier-mukai Kernelmentioning

confidence: 92%

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Derived equivalences of K3 surfaces and orientation

2009

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Abstract. Every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms.

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Section: Deformation Of the Fourier-mukai Kernelmentioning

confidence: 92%

“…When the deformation X n is integrable, e.g. obtained as in Example 2.1, the theorem can also be deduced from the more general results in [22]. (The assumptions on the non-positive Ext's are not needed in [22].)…”

Section: Deformation Of the Fourier-mukai Kernelmentioning

confidence: 99%

Derived equivalences of K3 surfaces and orientation

2009

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“…is shown in [26,13] to define a perfect obstruction theory for P n (X, β) of virtual dimension zero. A virtual cycle is then obtained by [4,22].…”

Section: Introductionmentioning

confidence: 99%

Stable pairs and BPS invariants

Pandharipande

Thomas

2009

J. Amer. Math. Soc.

155

179

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We define the BPS invariants of Gopakumar-Vafa in the case of irreducible curve classes on Calabi-Yau 3-folds. The main tools are the theory of stable pairs in the derived category and Behrend's constructible function approach to the virtual class. We prove that for irreducible classes the stable pairs generating function satisfies the strong BPS rationality conjectures. We define the contribution of each curve to the BPS invariants. A curve $C$ only contributes to the BPS invariants in genera lying between the geometric genus and arithmetic genus of $C$. Complete formulae are derived for nonsingular and nodal curves. A discussion of primitive classes on K3 surfaces from the point of view of stable pairs is given in the Appendix via calculations of Kawai-Yoshioka. A proof of the Yau-Zaslow formula for rational curve counts is obtained. A connection is made to the Katz-Klemm-Vafa formula for BPS counts in all genera.Comment: Fixed typo pointed out by Filippo Vivian

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“…X is a Calabi-Yau threefold, MNOP and stable pair invariants take a particularly simple form. Then the virtual dimension is zero and we get invariants by taking the length of the 0-dimensional virtual cycle: In this case the deformation-obstruction theories [24,21,9] used to define the virtual cycles are self dual in the sense of [2]. This implies that I .…”

Section: Introductionmentioning

confidence: 99%

Hilbert schemes and stable pairs: GIT and derived category wall crossings

Stoppa¹,

Thomas²

2011

Bul. Soc. Math. France

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Abstract. -We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations.We explain why this is not enough to prove the "DT/PT wall crossing conjecture" relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristics of these moduli spaces.When the 3-fold is Calabi-Yau the identity is the Euler-characteristic analogue of the DT/PT wall crossing conjecture, but for general 3-folds it is something different, as we discuss. Résumé (Schémas de Hilbert et paires stables : GIT et croisements de murs de caté-gories dérivées)Nous montrons que le schéma de Hilbert de courbes et l'espace de modules de Le Potier de paires stables à support à une dimension, ont une construction GIT commune. Les deux espaces correspondents aux chambres de par et d'autre d'un mur dans l'espace de linéarisations GIT.Nous expliquons pourquoi cela ne suffit pas pour prouver la « conjecture de croisement de murs DT/PT » qui relie les invariants dérivés de ces espaces de modules quand la variété sous-jacente est un 3-fold. Nous donnons, ensuite, une introduction simple à une petite partie de la théorie de Joyce sur les croisements de murs de ce type, Quand le 3-fold est de type Calabi-Yau, l'identité est le pendant, pour la caractéristique d'Euler, de la conjecture de croisement de murs DT/PT, mais dans le cas général elle s'avère être différente de celle-ci, comme nous l'expliquons.

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Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

Cited by 126 publications

References 22 publications

Derived equivalences of K3 surfaces and orientation

Derived equivalences of K3 surfaces and orientation

Stable pairs and BPS invariants

Hilbert schemes and stable pairs: GIT and derived category wall crossings

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