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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 25 (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.
Cluster tilting for onedimensional hypersurface singularities
 Adv. Math
"... Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete d ..."
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Cited by 8 (7 self)
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Abstract. In this article we study CohenMacaulay modules over onedimensional hypersurface singularities and the relationship with representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological method using higher almost split sequences and results from birational geometry. We obtain a large class of 2CY tilted algebras which are finite dimensional symmetric and satisfies τ 2 = id. In particular, we compute 2CY tilted algebras for simple/minimally elliptic curve singuralities.
On a relative FourierMukai transform on genus one fibrations
"... We study relative FourierMukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and nonprojective. Grothendieck duality is used to prove a skewcommutativity relation between this equivalence of categories and certain duality fu ..."
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Cited by 7 (3 self)
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We study relative FourierMukai transforms on genus one fibrations with section, allowing explicitly the total space of the fibration to be singular and nonprojective. Grothendieck duality is used to prove a skewcommutativity relation between this equivalence of categories and certain duality functors. We use our results to explicitly construct examples of semistable sheaves on degenerating families of elliptic curves.
Brane Tilings for Parallelograms with Application to Homological Mirror Symmetry
, 2006
"... We discuss the relation between quivers obtained by the algorithm of Hanany and Vegh [12] and the derived category of coherent sheaves on toric varieties in the case of lattice parallelograms, emphasizing the role of algae introduced by Feng, He, Kennaway and Vafa [6]. We also discuss the homologica ..."
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Cited by 7 (5 self)
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We discuss the relation between quivers obtained by the algorithm of Hanany and Vegh [12] and the derived category of coherent sheaves on toric varieties in the case of lattice parallelograms, emphasizing the role of algae introduced by Feng, He, Kennaway and Vafa [6]. We also discuss the homological mirror symmetry for some orbifolds of P 1 ×P 1. 1
Stability Conditions on AnSingularities
, 2006
"... We study the spaces of locallyfinite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of Ansingularities supported at the exceptional sets. Our main theorem is that they are connected and simplyconnected. The proof is based on the study of spherical ob ..."
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Cited by 4 (0 self)
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We study the spaces of locallyfinite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of Ansingularities supported at the exceptional sets. Our main theorem is that they are connected and simplyconnected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for Ansingularities. 1
EXCEPTIONAL SEQUENCES OF INVERTIBLE SHEAVES ON RATIONAL SURFACES
, 2008
"... In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongl ..."
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Cited by 3 (1 self)
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In this article we consider exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that to every such sequence one can associate a smooth complete toric surface in a canonical way. We use this structural result to prove various theorems on exceptional and strongly exceptional sequences of invertible sheaves on rational surfaces. We construct full strongly exceptional sequences for a large class of rational surfaces. For the case of toric surfaces we give a complete classification of full strongly exceptional sequences of invertible sheaves.
Tilting generators via ample line bundles
, 804
"... It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain noncommutative algebra. In this paper, we explain a method to construct a tilting generator from an ample line bundle, and construct it in several examples. ..."
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It is known that a tilting generator on an algebraic variety X gives a derived equivalence between X and a certain noncommutative algebra. In this paper, we explain a method to construct a tilting generator from an ample line bundle, and construct it in several examples.
unknown title
, 2009
"... Counting invariant of perverse coherent sheaves and its wallcrossing ..."
unknown title
, 2009
"... Counting invariant of perverse coherent sheaves and its wallcrossing ..."