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252
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].
Tropical geometry and its applications
 International Congress of Mathematicians vol. II, 827–852, Eur. Math. Soc
, 2006
"... Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the pr ..."
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Cited by 141 (6 self)
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Abstract. From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewiselinear objects that take over the rôle of classical algebraic varieties. This talk outlines some basic notions of this area and surveys some of its applications for the problems in classical (real and complex) geometry.
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
Tropical curves, their Jacobians and theta functions
, 2006
"... We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems. ..."
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Cited by 92 (4 self)
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We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the AbelJacobi, RiemannRoch and Riemann theta divisor theorems.
Affine structures and nonarchimedean analytic spaces
"... In this paper we propose a way to construct an analytic space over a nonarchimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric StromingerYauZaslow conjectu ..."
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Cited by 91 (8 self)
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In this paper we propose a way to construct an analytic space over a nonarchimedean field, starting with a real manifold with an affine structure which has integral monodromy. Our construction is motivated by the junction of Homological Mirror conjecture and geometric StromingerYauZaslow conjecture. In particular, we glue from “flat pieces ” an analytic K3 surface. As a byproduct of our approach we obtain an action of an arithmetic subgroup of the group SO(1,18) by piecewiselinear transformations on the 2dimensional sphere S 2 equipped with naturally defined singular affine structure.
Lagrangian Floer theory on compact toric manifolds: Survey
, 2010
"... This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed. ..."
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Cited by 79 (8 self)
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This is a survey of a series of papers [FOOO3, FOOO4, FOOO5]. We discuss the calculation of the Floer cohomology of Lagrangian submanifold which is a T n orbit in a compact toric manifold. Applications to symplectic topology and to mirror symmetry are also discussed.
Toric degenerations of toric varieties and tropical curves
 DUKE MATH. J
, 2004
"... We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory. ..."
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Cited by 63 (5 self)
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We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraicgeometric and relies on degeneration techniques and log deformation theory.