SURGERY OF REAL SYMPLECTIC FOURFOLDS AND WELSCHINGER INVARIANTS

A surgery of a real symplectic manifold X-R along a real Lagrangian sphere S is a modification of the symplectic and real structure on X-R in a neighborhood of S. Genus 0 Welschinger invariants of two real symplectic 4-manifolds differing by such a surgery have been related in [BP15]. In the present paper, we explore some particular situations where general formulas from [BP15] greatly simplify. As an application, we reduce the computation of genus 0 Welschinger invariants of all del Pezzo surfaces to the cases covered by [Bru15], and of all R-minimal real conic bundles to the cases covered by [HS12]. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize results from [BP15] to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as [Shu14].