This is basically the last step of problem 5.3.7 in Huybrechts' Complex Geometry.

Let $X$ be a complex manifold, $x \in X$, $E$ a holomorphic vector bundle on $X$ and $L$ a positive line bundle. Denote by $ \hat{X} = \operatorname{Bl}_{\{x\}} X$ the blow-up of $X$ at $x$ and let $\sigma: \hat{X} \to X$ be the projection map and $F = \sigma^{-1}(x)$ the exceptional divisor. I'm trying to show that $$ H^1\left(\hat{X}, \sigma^* E \otimes \mathcal{O}_{\hat{X}} \left(-F\right) \otimes \sigma^*L^k\right) = 0 $$ for $k$ large enough. Any ideas?

This is of course almost Serre's vanishing theorem: $\sigma^*L$ is positive for every tangent vector except those of $TF$ and one can show that $\mathcal{O}_{\hat{X}} \left(-F\right) \otimes \sigma^*L^k$ is positive for $k$ large enough, but the problem is that we can't take powers of $\mathcal{O}_{\hat{X}} \left(-F\right)$...