Gromov-Witten invariants of varieties with holomorphic 2-forms

TL;DR

This paper introduces a localization technique for Gromov-Witten invariants of varieties with holomorphic 2-forms, enabling the computation of these invariants in cases where the form degenerates, and proves related formulas for certain surfaces.

Contribution

It develops a new algebro-geometric localization method for Gromov-Witten invariants using holomorphic 2-forms, extending the understanding of these invariants in non-proper cases.

Findings

Defined localized GW-invariants that coincide with ordinary invariants for proper varieties.

Proved formulas for low degree GW-invariants of minimal general type surfaces with p_g>0.

Established deformation invariance of the localized GW-invariants.

Abstract

We show that a holomorphic two-form $\theta$ on a smooth algebraic variety X localizes the virtual fundamental class of the moduli of stable maps $\mgn(X,\beta)$ to the locus where $\theta$ degenerates; it then enables us to define the localized GW-invariant, an algebro-geometric analogue of the local invariant of Lee and Parker in symplectic geometry, which coincides with the ordinary GW-invariant when X is proper. It is deformation invariant. Using this, we prove formulas for low degree GW-invariants of minimal general type surfaces with p_g>0 conjectured by Maulik and Pandharipande.