Bernd Siebert

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 is algebraic-geometric and relies on degeneration techniques and log deformation theory. This generalizes results of Mikhalkin obtained by different methods in the surface case to arbitrary dimensions.

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 are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.

Virtual fundamental classes, global normal cones and Fulton’s canonical classes

This note, written in January 19971, grew out of an attempt to understand references [Be], [BeFa] and [LiTi]. In these papers two related but different methods are presented for the construction of a certain Chow class on moduli spaces of stable (parametrized) curves in a projective manifold V, called virtual fundamental class. This class replaces the usual fundamental class of these spaces in the definition of basic enumerative invariants of V involving curves, called Gromov-Witten (GW-) invariants. They are invariant under smooth deformations of V.

Irreducible Degenerations of Primary Kodaira Surfaces

We classify irreducible d-semistable degenerations of primary Kodaira surfaces. As an application we construct a canonical partial completion for the moduli space of primary Kodaira surfaces.

Lectures on Pseudo-Holomorphic Curves and the Symplectic Isotopy Problem

The purpose of these notes is a more self-contained presentation of the results of the authors in [SiTi3]. Some applications are also given.