Behavior of Welschinger Invariants under Morse Simplifications

TL;DR

This paper investigates how Welschinger invariants change under Morse simplifications in rational real symplectic 4-manifolds, using a real Abramovich-Bertram formula and exploring implications like invariant vanishing.

Contribution

It establishes a relation between Welschinger invariants before and after Morse simplifications via a real Abramovich-Bertram formula, providing new insights into their behavior.

Findings

Relation between Welschinger invariants pre- and post-Morse simplification

Derivation of a real Abramovich-Bertram formula for Gromov-Witten invariants

Identification of cases where Welschinger invariants vanish

Abstract

We relate Welschinger invariants of a rational real symplectic 4-manifold before and after a Morse simplification (i.e deletion of a sphere or a handle of the real part of the surface). This relation is a consequence of a real version of Abramovich-Bertram formula which computes Gromov-Witten invariants by means of enumeration of $J$-holomorphic curves with a non-generic almost complex structure $J$. In addition, we give some qualitative consequences of our study, for example the vanishing of Welschinger invariants in some cases.