Toric degenerations of toric varieties and tropical curves

Abstract

Abstract. 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 algebraic-geometric and relies on degeneration techniques and log deformation theory.

Mirror Symmetry via Logarithmic Degeneration Data I

“…For example, we hope it should be possible to define log Gromov-Witten invariants [44] which will coincide with those of a smoothing. The results of [34] and [36] suggest that this should be possible by counting certain piecewise straight graphs on B ("tropical curves"). In addition, the philosophy presented in this paper is that invariants of toric log CalabiYau spaces can be computed doing calculations on B.…”

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

“…In particular, G. Mikhalkin presented and proved in [127,129] a formula enumerating curves of arbitrary genus in toric surfaces. See also the papers [60,72,73,131,155].…”

“…Last time many other papers on tropical algebraic geometry and its applications to the conventional (e.g., complex) algebraic geometry and other areas appeared, see, e.g., [7,55,56,76,131,134,135,168,169,171]. The thing is that some difficult traditional problems can be reduced to their tropical versions which are hopefully not so difficult.…”

From real affine geometry to complex geometry

“…Furthermore, our description of the B-model side is tropical in nature. It is well known [Mik05], [NS06] that one should expect a correspondence between tropical curves on B and families of holomorphic curves on the corresponding degeneration X → T . Thus it is an important point for understanding mirror symmetry that the construction of the corresponding complex manifold is controlled by tropical curves onB, hence by holomorphic curves on the mirror degenerationX → T corresponding toB.…”