Bifurcations of embedded curves and towards an extension of Taubes' Gromov invariant to Calabi-Yau 3-folds

TL;DR

This paper extends Taubes' Gromov invariant to Calabi-Yau 3-folds by defining a virtual count of embedded pseudo-holomorphic curves, involving a detailed analysis of bifurcations in moduli spaces.

Contribution

It introduces an integer-valued invariant for embedded pseudo-holomorphic curves in Calabi-Yau 3-folds, generalizing Taubes' Gromov invariant through bifurcation analysis.

Findings

Defined a virtual count of embedded pseudo-holomorphic curves in Calabi-Yau 3-folds.

Analyzed bifurcations of moduli spaces of embedded curves.

Connected the construction to recent advances in super-rigidity conjectures.

Abstract

We define an integer-valued virtual count of embedded pseudo-holomorphic curves of two times a primitive homology class and arbitrary genus in symplectic Calabi--Yau $3$-folds, which can be viewed as an extension of Taubes' Gromov invariant. The construction depends on a detailed study of bifurcations of moduli spaces of embedded pseudo-holomorphic curves which is partially motivated by Wendl's recent solution of Bryan--Pandharipande's super-rigidity conjecture.