Norimatsu's lemma says that on a smooth projective complex variety $X$ of dimension $n$, then we have $H^i(X,\mathcal O\_X(-A-E))=0$ for $i < n$ when $A$ is an ample divisor and $E$ is a simple normal crossings (SNC) divisor. Does this statement remain true if $E$ is an effective divisor with SNC support? In other words, if $\sum E\_i$ is SNC, do we still have these vanishings for $\mathcal O\_X(-A-\sum a\_i E\_i)$ where $a\_i >0$? Or is there a counterexample?



Kodaira type vanishing for an ample divisor + effective divisor with SNC support
Here is a counterexample.


Let $X$ be a smooth cubic surface in $\mathbf{P}^3$ and $E$ a line on it; moreover take $A=-K\_X$.


Then by Serre duality $$H^1(X, -A-aE)=H^1(X, K\_X + A + aE)=H^1(X, aE).$$


On the other hand $h^0(X, aE)=1$ since $E^2=-1$ and $h^2(X, aE)=h^0(X, K\_X-aE)=0$ since $K\_X$ is not effective. 


Now Riemann-Roch yields $$\chi(X, aE)=\frac{aE(aE-K\_X)}{2}+\chi(\mathcal{O}\_X)=\frac{-a^2+a}{2}+1,$$
hence $$h^1(X, aE)=\frac{a(a-1)}{2}$$ which is not zero for $a \geq 2$.


