The Kodaira vanishing theorem state that if $L$ be a positive line bundle on a compact Kahler manifold $X$ then $$H^q(X,\Omega^p\otimes L) = 0$$
for $p+q >n$。
We have another dual version of it if $L$ is a negative line bundle then $$H^q(X,\Omega^p\otimes L) = 0$$ for $p+q <n$.
I stated a wrong version of the Serre duality, the correct version is as follows,and the proof of it is a direct consequence of the Hodge isomorphism theorem. together with some commutation relation.
The correct version is the pairing:
$$H^{p,q}(X,E) \times H^{n-p,n-q}(X,E^*) \to \Bbb{C}$$ is non degenerate.
Therefore to non degenerate pairing induce isomorphism $$H^{p,q}(X,E) \cong H^{n-p,n-q}(X,E^*)^*$$
Best Answer
Let $\mathcal{F}:= \Omega^p$. Let $\omega_X$ denote the canonical on $X$. Serre duality tells you more generally that $H^q(X,\mathcal{F}\otimes \mathcal{L}) \cong H^{n-q}(X,\mathcal{F}^*\otimes L^{-1}\otimes \omega_X)^*$.
Now $\mathcal{F}^* \otimes \omega_X \cong \Omega^{n-p}$ and $\mathcal{L}$ being positive tells you that $\mathcal{L}^{-1}$ is negative.