Symplectic geometry of rationally connected threefolds

TL;DR

This paper investigates the symplectic properties of rationally connected threefolds, establishing invariance under deformation and linking rational connectedness with Gromov-Witten invariants, thus advancing understanding of their symplectic geometry.

Contribution

It proves that rational connectedness is a symplectic deformation invariant in three dimensions and connects rational connectedness with symplectic rational connectedness via Gromov-Witten invariants.

Findings

Rational connectedness is a symplectic deformation invariant in dimension 3.

Rationally connected Fano or with b2=2 threefolds are symplectic rationally connected.

Many rationally connected threefolds are birational to symplectic rationally connected varieties.

Abstract

We study symplectic geometry of rationally connected $3$-folds. The first result shows that rationally connectedness is a symplectic deformation invariant in dimension $3$. If a rationally connected $3$-fold $X$ is Fano or $b_2(X)=2$, we prove that it is symplectic rationally connected, i.e. there is a non-zero Gromov-Witten invariant with two insertions being the class of a point. Finally we prove that many rationally connected $3$-folds are birational to a symplectic rationally connected variety.