Degenerations, transitions and quantum cohomology

TL;DR

This paper explores how degenerations and transitions in singular varieties influence quantum cohomology, emphasizing toric degenerations' role in computing Gromov-Witten invariants and their application in the Fanosearch program.

Contribution

It provides a framework linking degenerations, resolutions, and smoothings to quantum cohomology, facilitating invariant computations under geometric transitions.

Findings

Toric degenerations aid in computing Gromov-Witten invariants.

The study connects degenerations with quantum cohomology transformations.

Application to complex 3-folds and enumerative symplectic geometry.

Abstract

Given a singular variety I discuss the relations between quantum cohomology of its resolution and smoothing. In particular, I explain how toric degenerations helps with computing Gromov--Witten invariants, and the role of this story in Fanosearch programme. The challenge is to formulate enumerative symplectic geometry of complex $3$-folds in a way suitable for extracting invariants under blowups, contractions, and transitions.