2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/157478We study the following question, asked to us By Pandharipande and Starr: Let $X$ be a rationally connected $3$-fold, and $Y$ be a compact Kaehler $3$-fold symplectically equivalent to it. Is $Y$ rationally connected? We show that the answer is positive if $X$ is Fano or $b_2(X)\leq2$.New final version. The statement is improved, thanks to the help of Jason Starr. Indeed, the result holds now for general syplectic equivalence, and not the restricted notion of symplectic equivalence we used beforeAlgebraic GeometrySymplectic Geometrytext