a smooth projective fourfold. Assume $Y$ is symplectic deformation equivalent a smooth $4$-fold $X$ which is either

1.  a Fano $4$-fold of pseudo-index at least $2$, or

2.  a rationally connected $4$-fold whose second Betti number is $2$.

Then $Y$ is rationally connected.

Recall that the pseudo-index $i(X)$ of a smooth projective Fano variety is defined to be the minimum of the set $\{-K_X \cdot C \vert ~C \text{~is a rational curve in}~X.\}$.

The two cases are treated differently. First of all, the Fano case follows from the following theorem.

\[thm:main\] Let $X$ be a smooth projective Fano $4$-fold of pseudo-index at least $2$. Then there is a non-zero Gromov-Witten invariant of the form $\langle [pt], [pt], \ldots \rangle^X_{0, \beta}$.

Indeed, Gromov-Witten invariants are symplectic deformation invariants. Thus by Theorem \[thm:main\] there is a non-zero Gromov-Witten invariant of the form $\langle [pt], [pt], \ldots \rangle^Y_{0, \beta}$ on $Y$. Then the moduli space of rational curves containing $2$ points is non-empty otherwise the Gromov-Witten invariant is $0$. Therefore $Y$ is rationally connected.

The proof of Theorem \[thm:main\] naturally divides into two different cases. When the Fano variety has Picard number $1$, then the quotient construction of Koll[á]{}r-Miyaoka-Mori ([@KMM92RCFano], [@KMM92RC]) gives a 