---
abstract: 'We study some symplectic geometric aspects of rationally connected $4$-folds. As a corollary, we prove that any smooth projective $4$-fold symplectic deformation equivalent to a Fano $4$-fold of pseudo-index at least $2$ or a rationally connected $4$-fold whose second Betti number is $2$ is rationally connected.'
address: |
     Department of Mathematics 253-37\
    California Institute of Technology\
    Pasadena, CA, 91125
author:
- Zhiyu Tian
bibliography:
- 'MyBib.bib'
title: 'Symplectic geometry and rationally connected $4$-folds'
---

Introduction
============

This paper is devoted to the study of some symplectic geometric aspects of rationally connected $4$-folds. The key motivation is the following conjecture.

\[conj:Kollar\] Let $X$ and $Y$ be two smooth projective varieties which are symplectic deformation equivalent. Then $X$ is rationally connected if and only if $Y$ is.

Here two symplectic manifolds are symplectic deformation equivalent if there is a family of smooth symplectic manifolds $\{(V_t, \omega_t), 0 \leq t \leq 1\}$ that connects the two given ones. This conjecture is known to be true in the second Betti number $1$ case ([@KollarUni]), in dimension $1, 2$ (which follow easily from classification), and $3$ ([@VoisinRC], [@SymRC]).

One of the main results of this paper is the following.

\[thm:SymRC\] Let $Y$ be 