Let $X=\mathbb{A}_2\cup B$ be a smooth complex projective completion of the affine plane $\mathbb{A}_2$ with boundary $B$ a simple normal crossing divisor with rational components (e.g $X$ a smooth projetive toric surface). Let $C$ be a big and nef Cartier divisor of $X$. Do we have $$ H^1(X,\Omega^1_X(\log(B))\otimes \mathcal{O}_X(-C))=0 ? $$ This is true for $X=\mathbb{P}^2$. But in general ?
It's easy to show that $H^i(X,\Omega^1_X(\log(B))=0$ for $i=0,1$ so that this question reduces to show that $H^0(X,\Omega^1_X(\log(B))\otimes \mathcal{O}_C)=0$.
