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27
CalabiYau algebras
, 2007
"... We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potentia ..."
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Cited by 151 (1 self)
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We introduce some new algebraic structures arising naturally in the geometry of CY manifolds and mirror symmetry. We give a universal construction of CY algebras in terms of a noncommutative symplectic DG algebra resolution. In dimension 3, the resolution is determined by a noncommutative potential. Representation varieties of the CY algebra are intimately related to the set of critical points, and to the sheaf of vanishing cycles of the potential. Numerical invariants, like ranks of cyclic homology groups, are expected to be given by ‘matrix integrals ’ over representation varieties. We discuss examples of CY algebras involving quivers, 3dimensional McKay correspondence, crepant resolutions, Sklyanin algebras, hyperbolic 3manifolds and ChernSimons. Examples related to quantum Del Pezzo surfaces are discussed in [EtGi].
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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Cited by 126 (5 self)
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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.
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
Moduli stacks and invariants of semistable objects on K3
, 2007
"... For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Arti ..."
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Cited by 32 (9 self)
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For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T.Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D.Joyce’s work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.
Stability conditions and crepant small resolutions
, 2007
"... In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transfor ..."
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Cited by 22 (9 self)
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In this paper, we describe the spaces of stability conditions on the triangulated categories associated to three dimensional crepant small resolutions. The resulting spaces have chamber structures such that each chamber corresponds to a birational model together with a special FourierMukai transform. We observe that these spaces are covering spaces over certain open subsets of finite dimensional vector spaces, and determine their deck transformations.
Stability conditions, torsion theories and tilting
, 2009
"... The space of stability conditions of a triangulated category is naturally partitioned into subsets U(A) of stability conditions with a given heart A. If A has finite length and n simple objects then U(A) has a simple geometry, depending only on n. Furthermore, Bridgeland has shown that if B is obtai ..."
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Cited by 14 (2 self)
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The space of stability conditions of a triangulated category is naturally partitioned into subsets U(A) of stability conditions with a given heart A. If A has finite length and n simple objects then U(A) has a simple geometry, depending only on n. Furthermore, Bridgeland has shown that if B is obtained from A by tilting at a torsion theory generated by one simple object then the intersection of the closures of U(A) and U(B) has codimension one. We show that if B is obtained from A by tilting at any finitelygenerated torsion theory then the closures of U(A) and U(B) intersect in a subset whose codimension can be determined from the torsion theory. We also give conditions under which the union of subsets U(B), where B is obtained from A by a finite sequence of finitelygenerated tilts, forms an entire component of the space of stability conditions. We illustrate this by computing the space of stability conditions of the constructible derived category of the complex projective line stratified by a point and its complement. 1
INDUCING STABILITY CONDITIONS
, 705
"... Abstract. We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived cat ..."
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Cited by 12 (2 self)
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Abstract. We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we give a derived version of the Torelli Theorem for the latter surfaces. We also study the relationship between stability conditions on projective spaces and those on their canonical bundles. 1.
Stability conditions and Stokes factors
, 2008
"... Let G be a complex algebraic group and ∇ a meromorphic connection on the trivial G–bundle over P 1 with a pole of order 2 at zero and a pole of order 1 at infinity. We give explicit formulae involving multilogarithms for the map taking the residue of ∇ at zero to the corresponding Stokes factors a ..."
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Cited by 12 (0 self)
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Let G be a complex algebraic group and ∇ a meromorphic connection on the trivial G–bundle over P 1 with a pole of order 2 at zero and a pole of order 1 at infinity. We give explicit formulae involving multilogarithms for the map taking the residue of ∇ at zero to the corresponding Stokes factors and for the Taylor series of the inverse map. We show moreover that, when G is the Ringel–Hall group of the category A of modules over a complex, finite–dimensional algebra, this Taylor series coincides with the holomorphic generating function for counting invariants in A recently constructed by D. Joyce [21]. This allows us to interpret Joyce’s construction as one of an isomonodromic family of irregular connections on P 1 parametrised by the space of stability conditions of A.