Pencils of quadrics and Gromov-Witten-Welschinger invariants of CP3\mathbb C P^3

We establish a formula for the Gromov-Witten-Welschinger invariants of $\mathbb CP^3$ with mixed real and conjugate point constraints. The method is based on a suggestion by J. Koll\'ar that, considering pencils of quadrics, some real and complex enumerative invariants of $\mathbb CP^3$ could be computed in terms of enumerative invariants of $\mathbb CP^1\times\mathbb CP^1$ and of elliptic curves.Comment: 14 pages, 4 figures, minor corrections following referee's suggestion

On the Topology of Real Bundle Pairs over Nodal Symmetric Surfaces

We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation.Comment: 19 pages, 3 figure

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

The first part of this work constructs positive-genus real Gromov-Witten invariants of real-orientable symplectic manifolds of odd "complex" dimensions; the present part focuses on their properties that are essential for actually working with these invariants. We determine the compatibility of the orientations on the moduli spaces of real maps constructed in the first part with the standard node-identifying immersion of Gromov-Witten theory. We also compare these orientations with alternative ways of orienting the moduli spaces of real maps that are available in special cases. In a sequel, we use the properties established in this paper to compare real Gromov-Witten and enumerative invariants, to describe equivariant localization data that computes the real Gromov-Witten invariants of odd-dimensional projective spaces, and to establish vanishing results for these invariants in the spirit of Walcher's predictions.Comment: 56 pages; some expositional changes in the introductio