Definition:Two symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are defined to be symplectically equivalent if there exists a diffeomorphism $\phi:X\to Y$ such that $\phi^∗ω_Y$ is in the same deformation class as $\omega_X$.
A line bundle $L$ over compact complex manifold $X$ is said to be nef(in the sense of Demaily) if for every $ε>0$ there exists a smooth hermitian metric $h_\epsilon$ on $L$ such that its curvature $Θ_{h_\epsilon}(L)≥−\epsilon ω$ . The good thing is that this definition even work for Moishezon manifold due to Mihai Paun. So my following question make sense
We know if $X$ and $Y$ are symplectically equivalent compact Kähler threefolds then $X$ has an algebraically nef canonical bundle if and only if the same is true for $Y$. See also Peternell-Höring paper and Voisin paper as introductions.
In fact this is the extension of the result of Y. Ruan when $X$ and $Y$ are projective by Gromov-Witten invariant.
Question: Is there any same result when the symplectic manifolds $(X,\omega_X)$ and $(Y,\omega_Y)$ are Moishezon manifolds(here I mean nefness in numerical sense)? .
Motivation: Every Moishezon variety which is not projective contains a rational curve. Kollar conjectured : Assume $X$ is rationally connected. Let $Y$ be a compact Kahler manifold symplectically equivalent to $X$. Then $Y$ is also rationally connected. I believe that attacking to these conjectures by analytical tools correspond to relative Kahler Ricci flow which connect Gromov-Witten theory to study of Relative Kahler Ricci flow.
