Let $X$ be a fake projective plane, id est $X$ is a smooth projective surface with the same Betti numbers of $\mathbb{P}^2_{\mathbb{C}}$ but not isomorphic to $\mathbb{P}^2_{\mathbb{C}}$. These projective surface there exist; in particular $K_X$ is ample (hence $X$ is a minimal surface of general type) and $\displaystyle{\int_X3c_2(X)=\int_Xc_1(X)^2=9}$, the Picard number $\rho(X)$ of $X$ is $1$. Let $\mathcal{O}_X(1)$ be the ample generator of the torsion-free part of $Pic(X)$ such that $K_X=\mathcal{O}_X(3)$ and $c_1\left(\mathcal{O}_X(1)\right)^2=1$.
I need to compute the cohomology of $K_X^{\otimes2}\otimes TX$.
The Serre duality does not help because: $$ \forall i\in\{0,1,2\},\,H^i\left(K_X^{\otimes 2}\otimes TX\right)\cong H^{2-i}\left(X,\mathcal{O}_X(-3)\otimes\Omega^1_X\right)^{\vee}. $$ On the other hand, I'm thinking to use something like a "fake Euler sequence", but I don't know whether this generalization there exists.
What can I do this?
Update. $\deg_{\mathcal{O}_X(1)}\mathcal{O}_X(-3)\otimes\Omega^1_X=-3$, hence $H^2\left(K_X^{\otimes 2}\otimes TX\right)\cong H^0\left(X,\mathcal{O}_X(-3)\otimes\Omega^1_X\right)^{\vee}=\{0\}$.