Some examples of symplectic rationally connected 4-folds

TL;DR

This paper provides examples of symplectic rationally connected 4-folds, including all Fano 4-folds with pseudo-index at least 2, supporting the conjecture relating rational connectedness and symplectic rational connectedness.

Contribution

It demonstrates that certain rationally connected 4-folds are symplectic rationally connected without relying on explicit descriptions.

Findings

All Fano 4-folds with pseudo-index ≥ 2 are symplectic rationally connected.

Provides examples supporting the conjecture relating rational and symplectic rational connectedness.

Proof does not depend on explicit descriptions of the varieties.

Abstract

A symplectic manifold is called symplectic rationally connected if there is a non-zero genus zero Gromov-Witten invariant with two point insertions. It is conjectured that every smooth projective rationally connected variety is symplectic rationally connected. In this short note we give some examples of rationally connected 4-folds which are symplectic rationally connected. In particular, all Fano 4-folds of pseudo-index at least 2 are symplectic rationally connected. The proof does not use any explicit description of these varieties.