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91
Stability structures, motivic DonaldsonThomas invariants and cluster transformations
, 2008
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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.
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.
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.
From real affine geometry to complex geometry
, 2007
"... We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder description of ..."
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Cited by 57 (6 self)
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We construct from a real affine manifold with singularities (a tropical manifold) a degeneration of CalabiYau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a striking feature of our approach is that it yields an explicit and canonical orderbyorder description of the degeneration via families of tropical trees. This gives complete control of the Bmodel side of mirror symmetry in terms of tropical geometry. For example, we expect our deformation parameter is a canonical coordinate, and expect period calculations to be expressible in terms of tropical curves. We anticipate this will lead to a proof of mirror symmetry via tropical methods. This
Mirror symmetry and Tduality in the complement of an anticanonical divisor
 J. GÖKOVA GEOM. TOPOL
"... We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kähler manifold. In particular, we explore the connections between Tduality and mirror symmetry in concrete examples, and show how quantum corrections ar ..."
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Cited by 55 (4 self)
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We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kähler manifold. In particular, we explore the connections between Tduality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.
Analytification is the limit of all tropicalizations
"... Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1. ..."
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Cited by 51 (7 self)
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Abstract. We introduce extended tropicalizations for closed subvarieties of toric varieties and show that the analytification of a quasprojective variety over a nonarchimedean field is naturally homeomorphic to the inverse limit of the tropicalizations of its quasiprojective embeddings. 1.
MIRROR SYMMETRY FOR LOG CALABIYAU SURFACES I
, 2011
"... We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anticanonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is con ..."
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Cited by 36 (7 self)
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We give a canonical synthetic construction of the mirror family to a pair (Y,D) of a smooth projective surface with an anticanonical cycle of rational curves, as the spectrum of an explicit algebra defined in terms of counts of rational curves on Y meeting D in a single point. In the case D is contractible, the family gives a smoothing of the dual cusp, and thus a proof of Looijenga’s 1981 cusp conjecture.