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114
Stability structures, motivic DonaldsonThomas invariants and cluster transformations
, 2008
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GromovWitten/DonaldsonThomas correspondence for toric 3folds
, 2008
"... We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau ..."
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Cited by 59 (17 self)
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We prove the equivariant GromovWitten theory of a nonsingular toric 3fold X with primary insertions is equivalent to the equivariant DonaldsonThomas theory of X. As a corollary, the topological vertex calculations by Agangic, Klemm, Mariño, and Vafa of the GromovWitten theory of local CalabiYau toric 3folds are proven to be correct in the full 3leg setting.
Derevied categories of small toric CalabiYau 3folds and counting invariants
"... We first construct a derived equivalence between a small crepant resolution of an affine toric CalabiYau 3fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent ..."
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Cited by 38 (0 self)
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We first construct a derived equivalence between a small crepant resolution of an affine toric CalabiYau 3fold and a certain quiver with a superpotential. Under this derived equivalence we establish a wallcrossing formula for the generating function of the counting invariants of perverse coherent systems. As an application we provide certain equations on DonaldsonThomas, PandeharipandeThomas and Szendroi’s invariants.
Polynomial Bridgeland stability conditions and the large volume limit
 Geom. Topol
"... ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large ..."
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Cited by 38 (4 self)
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ABSTRACT. We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpsonstability, and large volume limits of Bridgeland stability conditions. We show that the PT/DTcorrespondence relating stable pairs to DonaldsonThomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wallcrossing in our family of polynomial stability conditions. Similarly, we show that the relation between stable pairs and invariants of onedimensional torsion sheaves (proven recently by the same authors) is a wallcrossing formula. CONTENTS
A mathematical theory of the topological vertex
"... Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory ..."
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Cited by 36 (19 self)
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Abstract. We have developed a mathematical theory of the topological vertex— a theory that was originally proposed by M. Aganagic, A. Klemm, M. Mariño, and C. Vafa on effectively computing GromovWitten invariants of smooth toric CalabiYau threefolds derived from duality between open string theory of smooth CalabiYau threefolds and ChernSimons theory on three manifolds. 1.
Projectivity and birational geometry of Bridgeland moduli spaces
, 2012
"... ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby gene ..."
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Cited by 35 (2 self)
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ABSTRACT. We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wallcrossing for Bridgelandstability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Giesekerstable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the “HassettTschinkel/Huybrechts/Sawon ” conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.