My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort of Surgery and later Gang Tian extended Thurston's Geometrization Conjecture for symplectic manifolds by using Symplectic Minimal Model Program.
Let $X$ be a smooth compact symplectic manifold and $H(X,\mathbb Q)$ be its cohomology and there are two standard ways to find the basis of cohomology.
I. We can use Morse theory. Let $f:X\to \mathbb R$ be a Morse function then there is a Morse complex $CM(X)=\mathbb Z[Crit(f)]$ and the boundary operator $\partial$ counts isolated gradient trajectories and we can compute the cohomology and we can get the basis of cohomology
II. The second strategy to compute the basis of $H(X,\mathbb Q)$ is to use Symplectic Minimal Model Program.
Let $(X,\omega)$ be a symplectic manifold
The naive idea is to try to vary $\omega$ in the direction of negative first Chern class $-c_1(X)$ and whenever $\omega$ becomes degenerate, we do use some sort of surgery as the symplectic version of Minimal Model Program which is flips or blow-downs.
If we start up with $X$ and inside $X$ we find locus which is non-degenerate and this turns out to be projective bundle
$$X\supset\mathbb P(N)\to Z\to \mathbb P(N')\subset X'$$ which $Z$ called center and $X'$ is flips. The good example is $X=\text{blow up}^2(S^2\times S^2)$. In this case $\dim H(X)=6$
Now the idea of varying symplectic form in direction of negative first Chern class is as same as finding Kahler cone of solutions of Kahler-Ricci flow. like $[\omega]-tc_1(X)$. In fact the idea of second strategy work when $c_1(X)<0$ or $Kod(X)=dim X$. So in my analytical point of view when we are facing with $0<Kod(X)<dim X$, then the cone of solutions of Ricci flow changes and we need to use first Chern class of relative canonical bundle along Iitaka fibration $c_1(K_{X/X_{can}})$ where $X_{can}$ is canonical model of $X$. So it is unlikely for such cases varying symplectic form in negative direction of first Chern class work!. Is there any example such that confirm the paper of François Charest, Chris T. Woodward, Floer theory and flips for finding basis of cohomology of $X$ when $0<kod(X)<dimX$(take X is Kahler manifold)