Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants

Abstract

Let X be a real algebraic convex 3-manifold whose real part is equipped with a Pin− structure. We show that every irreducible real rational curve with nonempty real part has a canonical spinor state belonging to {± 1}. The main result is then that the algebraic count of the number of real irreducible rational curves in a given numerical equivalence class passing through the appropriate number of points does not depend on the choice of the real configuration of points, provided that these curves are counted with respect to their spinor states. These invariants provide lower bounds for the total number of such real rational curves independently of the choice of the real configuration of points.