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522
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.
Introduction to Ainfinity algebras and modules
, 1999
"... These are slightly expanded notes of four introductory talks on ..."
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Cited by 117 (4 self)
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These are slightly expanded notes of four introductory talks on
Graded Lagrangian submanifolds
 Bull Math. Soc. France
"... Floer theory assigns, in favourable circumstances, an abelian group HF(L0, L1) to a pair (L0, L1) of Lagrangian submanifolds of a symplectic manifold (M, ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L0 or L1. Following Floer [7] one can equip HF(L ..."
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Cited by 117 (15 self)
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Floer theory assigns, in favourable circumstances, an abelian group HF(L0, L1) to a pair (L0, L1) of Lagrangian submanifolds of a symplectic manifold (M, ω). This group is a qualitative invariant, which remains unchanged under suitable deformations of L0 or L1. Following Floer [7] one can equip HF(L0, L1) with a canonical
Dbranes in LandauGinzburg models and algebraic geometry
 J. High Energy Phys
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Categorical mirror symmetry: the elliptic curve
 ADV. THEOR. MATH. PHYS
, 1998
"... We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M ..."
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Cited by 109 (11 self)
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We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M is an elliptic curve and ˜ M is its dual curve, exhibiting the dictionary in detail.
Strong homotopy algebras of a Kähler manifold
 math.AG/9809172, Int. Math. Res. Notices
, 1999
"... It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differ ..."
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Cited by 96 (9 self)
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It is shown that any compact Kähler manifold M gives canonically rise to two strongly homotopy algebras, the first one being associated with the Hodge theory of the de Rham complex and the second one with the Hodge theory of the Dolbeault complex. In these algebras the product of two harmonic differential forms is again harmonic. If M happens to be a CalabiYau manifold, there exists a third strongly homotopy algebra closely related to the BarannikovKontsevich extended moduli space of complex structures. 1
Mirror symmetry for weighted projective planes and their noncommutative deformations
, 2004
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Floer cohomology and disc instantons of Lagrangian torus fibers in Fano toric manifolds
, 2003
"... In this paper, we first provide an explicit description of all holomorphic discs (“disc instantons”) attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute FukayaOhOhtaOno’s (FOOO’s) obstruction (co)chains and the Flo ..."
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Cited by 91 (13 self)
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In this paper, we first provide an explicit description of all holomorphic discs (“disc instantons”) attached to Lagrangian torus fibers of arbitrary compact toric manifolds, and prove their Fredholm regularity. Using this, we compute FukayaOhOhtaOno’s (FOOO’s) obstruction (co)chains and the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. In particular specializing to the formal parameter T 2π = e −1, our computation verifies the folklore that FOOO’s obstruction (co)chains correspond to the LandauGinzburg superpotentials under the mirror symmetry correspondence, and also proves the prediction made by K. Hori about the Floer cohomology of Lagrangian torus fibers of Fano toric manifolds. The latter states that the Floer cohomology (for the parameter value T 2π = e −1) of all the fibers vanish except at a finite number, the Euler characteristic of the toric manifold, of base points in the momentum polytope that are critical points of the superpotential of the LandauGinzburg mirror to the toric manifold. In the latter cases, we also prove that the Floer cohomology of the corresponding fiber is isomorphic to its singular cohomology. We also introduce a restricted version of the Floer cohomology of Lagrangian submanifolds, which is a priori more flexible to define in general, and which we call the adapted Floer cohomology. We then prove that the adapted Floer cohomology of any nonsingular torus fiber of Fano toric manifolds is welldefined, invariant under the Hamiltonian isotopy and isomorphic to the BottMorse Floer cohomology of the fiber.
Derived Categories of Twisted Sheaves on CalabiYau Manifolds
, 2000
"... This dissertation is primarily concerned with the study of derived categories of twisted sheaves on CalabiYau manifolds. Twisted sheaves occur naturally in a variety of problems, but the most important situation where they are relevant is in the study of moduli problems of semistable sheaves on var ..."
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Cited by 86 (3 self)
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This dissertation is primarily concerned with the study of derived categories of twisted sheaves on CalabiYau manifolds. Twisted sheaves occur naturally in a variety of problems, but the most important situation where they are relevant is in the study of moduli problems of semistable sheaves on varieties. Although universal sheaves may not exist as such, in many cases one can construct them as twisted universal sheaves. In fact, the twisting is an intrinsic property of the moduli problem under consideration. A fundamental construction due to Mukai associates to a universal sheaf a transform between the derived category of the original space and the derived category of the moduli space, which often turns out to be an equivalence. In the present work we study what happens when the universal sheaf is replaced by a twisted one. Under these circumstances we obtain a transform between the de