Counting real rational curves on real symplectic 4-manifolds with vanishing first Chern class

TL;DR

This paper introduces a new invariant counting real rational pseudo-holomorphic curves on real Spin symplectic K3 surfaces, which remains unchanged under deformations and generalizes counts on real projective K3 surfaces.

Contribution

It defines a signed count of real rational curves on real Spin symplectic K3 surfaces and proves its invariance under deformation, extending previous enumerative geometry results.

Findings

The count is invariant under deformation of the family.

Specializes to count real rational curves on real projective K3 surfaces.

Provides a new enumerative invariant in symplectic geometry.

Abstract

We define a signed count of real rational pseudo-holomorphic curves appearing in a one-parameter family of real Spin symplectic K3 surfaces. We show that this count is an invariant of the deformation class of the family. In the case of a real projective K3 surface, this invariant specializes to count real rational curves appearing in a linear system on the surface.