Towards the symplectic Graber-Harris-Starr theorems

TL;DR

This paper explores the existence of sections with non-zero Gromov-Witten invariants in symplectic fibrations over curves, extending classical algebraic geometry results to symplectic geometry and providing new examples and lifting results.

Contribution

It introduces symplectic analogues of Graber-Harris-Starr theorems, showing the dependence on generic fibers and providing examples and lifting theorems for Gromov-Witten invariants.

Findings

Existence of sections with non-zero Gromov-Witten invariants depends only on the generic fiber.

All rational surface fibrations have this property.

Certain cases allow lifting Gromov-Witten invariants from base to total space.

Abstract

A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given a rationally connected fibration over a curve, can one find a section which gives a non-zero Gromov-Witten invariant? We observe that for any fibration, the existence of a section which gives a non-zero Gromov-Witten invariant only depends on the generic fiber, i.e. a variety defined over the function field of a curve. Some examples of rationally connected fibrations with this property are given, including all rational surface fibrations. We also prove some results, which says that in certain cases we can "lift" Gromov-Witten invariants of the base to the total space of a rationally connected fibration.