Real Gromov-Witten Theory in All Genera and Real Enumerative Geometry: Construction

TL;DR

This paper develops a comprehensive real Gromov-Witten theory for all genera, constructing invariants for various real symplectic manifolds and establishing foundational orientation results using topological methods.

Contribution

It introduces a new topological approach to orienting moduli spaces of real maps, enabling the construction of invariants in all genera for a wide class of real symplectic manifolds.

Findings

Constructed positive-genus invariants for real symplectic manifolds.

Established orientations on moduli spaces extending across boundaries.

Derived vanishing results and connections with real enumerative geometry.

Abstract

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for counts of real positive-genus curves in real algebraic varieties. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces; the previous attempts involved direct computations for the determinant lines of Fredholm operators over bordered surfaces. We use the notion of real orientation introduced in this paper to obtain isomorphisms of real bundle pairs over families of symmetric surfaces and then apply the determinant functor to these isomorphisms. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis for a fully fledged real Gromov-Witten theory. The second and third parts of this work concern applications: they describe important properties of our orientations on the moduli spaces, establish some connections with real enumerative geometry, provide the relevant equivariant localization data for projective spaces, and obtain vanishing results in the spirit of Walcher's predictions.