On Welschinger invariants of symplectic 4-manifolds

TL;DR

This paper investigates Welschinger invariants in real symplectic 4-manifolds, proving their vanishing in many cases, determining their signs in some instances, and revealing divisibility properties through symplectic sum relations.

Contribution

It introduces a real symplectic sum formula to derive relations among Welschinger invariants, leading to new vanishing and divisibility results.

Findings

Many Welschinger invariants vanish in real symplectic 4-manifolds.

Some invariants have determined signs and are divisible by large powers of 2.

Relations among invariants are established via a real symplectic sum formula.

Abstract

We prove the vanishing of many Welschinger invariants of real symplectic $4$-manifolds. In some particular instances, we also determine their sign and show that they are divisible by a large power of 2. Those results are a consequence of several relations among Welschinger invariants obtained by a real version of symplectic sum formula. In particular, this note contains proofs of results announced in [BP13].