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.
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