Fundamental groups of rationally connected symplectic manifolds

TL;DR

This paper proves that symplectic manifolds with certain rational connectivity properties have finite fundamental groups, linking Gromov-Witten invariants to topological finiteness and homology structure.

Contribution

It establishes a connection between enumerative rational connectivity, Gromov-Witten invariants, and the finiteness of the fundamental group in symplectic manifolds.

Findings

Fundamental groups are finite for enumeratively rationally connected symplectic manifolds.

Non-zero Gromov-Witten invariants imply finite fundamental group.

Holomorphic indecomposability of certain classes constrains the second homology rank.

Abstract

We show that the fundamental group of every enumeratively rationally connected closed symplectic manifold is finite. In other words, if a closed symplectic manifold has a non-zero Gromov-Witten invariant with two point insertions, then it has finite fundamental group. We also show that if the spherical homology class associated to such a non-zero Gromov-Witten invariant is holomorphically indecomposable, then the rational second homology of the symplectic manifold has rank one.