Let $X$ be a smooth, projective variety, $V\_1, V\_2$ smooth, closed subvarieties of the same dimension and $E$ a locally free sheaf on $X$. There exist natural morphisms $$r\_1: H^i\_{V\_1}(E) \to H^i(E) \mbox{ and } r\_2: H^i\_{V\_2}(E) \to H^i(E). $$
Suppose that $V\_1 \cap V\_2=\emptyset$. The question is, for which values of $i$ can we say that $\mbox{Im}(r\_1) \cap \mbox{Im}(r\_2)=0$?



Cohomology with compact support
For $i=0$ this holds essentially by the definition of $H^i\_V$.


For $i>0$ it generally fails. Let $X$ be a curve and $V\_i$ arbitrary, different points. Then the complements $X\setminus V\_i$ are affine and hence both $r\_1$ and $r\_2$ are surjective.


