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Stability conditions on K3 surfaces
"... Abstract. This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. 1. ..."
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Abstract. This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface. 1.
Curve counting via stable pairs in the derived category
"... Abstract. For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resu ..."
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Cited by 116 (22 self)
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Abstract. For a nonsingular projective 3fold X, we define integer invariants virtually enumerating pairs (C,D) where C ⊂ X is an embedded curve and D ⊂ C is a divisor. A virtual class is constructed on the associated moduli space by viewing a pair as an object in the derived category of X. The resulting invariants are conjecturally equivalent, after universal transformations, to both the GromovWitten and DT theories of X. For CalabiYau 3folds, the latter equivalence should be viewed as a wallcrossing formula in the derived category. Several calculations of the new invariants are carried out. In the Fano case, the local contributions of nonsingular embedded curves are found. In the local toric CalabiYau case, a completely new form of the topological vertex is described. The virtual enumeration of pairs is closely related to the geometry underlying the BPS state counts of Gopakumar and Vafa. We
AN INTEGRAL STRUCTURE IN QUANTUM COHOMOLOGY AND MIRROR SYMMETRY FOR TORIC ORBIFOLDS
, 2009
"... We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the ..."
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Cited by 73 (5 self)
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We introduce an integral structure in orbifold quantum cohomology associated to the Kgroup and the b Γclass. In the case of compact toric orbifolds, we show that this integral structure matches with the natural integral structure for the LandauGinzburg model under mirror symmetry. By assuming the existence of an integral structure, we give a natural explanation for the specialization to a root of unity in Y. Ruan’s crepant resolution conjecture [66].
Noncommutative DonaldsonThomas theory and the conifold
, 2008
"... Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the th ..."
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Cited by 64 (0 self)
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Given a quiver algebra A with relations defined by a superpotential, this paper defines a set of invariants of A counting framed cyclic Amodules, analogous to rank1 Donaldson–Thomas invariants of Calabi–Yau threefolds. For the special case when A is the noncommutative crepant resolution of the threefold ordinary double point, it is proved using torus localization that the invariants count certain pyramidshaped partitionlike configurations, or equivalently infinite dimer configurations in the square dimer model with a fixed boundary condition. The resulting partition function admits an infinite product expansion, which factorizes into the rank1 Donaldson–Thomas partition functions of the commutative crepant resolution of the singularity and its flop. The different partition functions are speculatively interpreted as counting stable objects in the derived category of Amodules under different stability conditions; their relationship should then be an instance of wall crossing in the space of stability conditions on this triangulated category.
Stability conditions on a noncompact CalabiYau threefold
"... Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the ..."
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Cited by 38 (1 self)
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Abstract. We study the space of stability conditions on the noncompact CalabiYau threefold X which is the total space of the canonical bundle of P 2. We give a combinatorial description of an open subset of Stab(X) and state a conjecture relating Stab(X) to the Frobenius manifold obtained from the quantum cohomology of P 2. We give some evidence from mirror symmetry for this conjecture. 1.
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
Central charges, symplectic forms, and hypergeometric series in local mirror symmetry
 hepth/0404043 46 A. Iqbal and A.K. KashaniPoor, The Vertex on a Strip. hepth/0410174
"... Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of loc ..."
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Cited by 36 (2 self)
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Abstract. We study a cohomologyvalued hypergeometric series which naturally arises in the description of (local) mirror symmetry. We identify it as a central charge formula for BPS states and study its monodromy property from the viewpoint of Kontsevich’s homological mirror symmetry. In case of local mirror symmetry, we will identify a symplectic form, and will conjecture an integral and symplectic monodromy property of a relevant hypergeometric series of Gel’fandKapranovZelevinski type.
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.
MMP FOR MODULI OF SHEAVES ON K3S VIA WALLCROSSING: NEF AND MOVABLE CONES, LAGRANGIAN FIBRATIONS
"... ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) ..."
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Cited by 35 (2 self)
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ABSTRACT. We use wallcrossing with respect to Bridgeland stability conditions to systematically study the birational geometry of a moduli space M of stable sheaves on a K3 surface X: (a) We describe the nef cone, the movable cone, and the effective cone of M in terms of the Mukai lattice of X. (b) We establish a longstanding conjecture that predicts the existence of a birational Lagrangian fibration on M whenever M admits an integral divisor class D of square zero (with respect to the BeauvilleBogomolov form). These results are proved using a natural map from the space of Bridgeland stability conditions Stab(X) to the cone Mov(X) of movable divisors on M; this map relates wallcrossing in Stab(X) to birational transformations of M. In particular, every minimal model of M appears as a moduli space of Bridgelandstable objects on X. CONTENTS