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From real affine geometry to complex geometry
We construct from a real ane manifold with singularities (a tropical manifold) a degeneration of Calabi-Yau manifolds. This solves a fundamental problem in mirror symmetry. Furthermore, a strikingExpand
The tropical vertex
Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group areExpand
Logarithmic Gromov-Witten invariants
ing, now let π : Y → W be a proper morphism of schemes and let αi : Mi → OY , i = 1, 2, be two fine saturated log structures on Y . Consider the functor (2.1) LMorY/W (M1, M2) : (Sch/W ) −→ (Sets)Expand
Toric degenerations of toric varieties and tropical curves
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 isExpand
Mirror Symmetry via Logarithmic Degeneration Data I
Introduction. 1 1. Derivations and differentials 6 2. Log Calabi-Yau spaces: local structure and deformation theory 16 2.1. Local structure 16 2.2. Deformation theory 25 3. Cohomology of logExpand
On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator
We observe a general structure theorem for quantum cohomology rings, a non-homogeneous version of the usual cohomology ring encoding information about (almost holomorphic) rational curves. AnExpand
Affine manifolds, log structures, and mirror symmetry
This is an outline of work in progress concerning an algebro-geometric form of the Strominger-Yau-Zaslow conjecture. We introduce a limited type of degeneration of Calabi-Yau manifolds, which we callExpand
An invitation to toric degenerations
This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going throughExpand
Gromov-Witten invariants of general symplectic manifolds
We present an approach to Gromov-Witten invariants that works on arbitrary (closed) symplectic manifolds. We avoid genericity arguments and take into account singular curves in the very formulation.Expand
Mirror Symmetry via Logarithmic Degeneration Data II
This paper continues the authors' program of studying mirror symmetry via log geometry and toric degenerations, relating affine manifolds with singularities, log Calabi-Yau spaces, and toricExpand