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From real affine geometry to complex geometry

- M. Gross, Bernd S Siebert
- Mathematics
- 28 March 2007

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 striking… Expand

The tropical vertex

- M. Gross, R. Pandharipande, Bernd S Siebert
- Mathematics
- 4 February 2009

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 are… Expand

Logarithmic Gromov-Witten invariants

- M. Gross, Bernd S Siebert
- Mathematics
- 21 February 2011

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

- T. Nishinou, Bernd S Siebert
- Mathematics
- 4 September 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… Expand

Mirror Symmetry via Logarithmic Degeneration Data I

- M. Gross, Bernd S Siebert
- Mathematics
- 4 September 2003

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 log… Expand

On Quantum Cohomology Rings of Fano Manifolds and a Formula of Vafa and Intriligator

- Bernd S Siebert, G. Tian
- Mathematics
- 12 March 1994

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. An… Expand

Affine manifolds, log structures, and mirror symmetry

- M. Gross, Bernd S Siebert
- Mathematics
- 5 November 2002

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 call… Expand

An invitation to toric degenerations

- M. Gross, Bernd S Siebert
- Mathematics
- 20 August 2008

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 through… Expand

Gromov-Witten invariants of general symplectic manifolds

- Bernd S Siebert
- Mathematics
- 12 August 1996

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

- M. Gross, Bernd S Siebert
- Mathematics
- 14 September 2007

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 toric… Expand

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