Cohomology of the normal bundle of a divisor

Question

Let $X$ be a smooth projective variety over the field of complex numbers of dimension $n$ and $D\subset X$ a smooth divisor. Consider the normal bundle $O_X(D)|_D$ of $D$ in $X$. Is there any geometric interpretation of $H^{n-1}(D,O_X(D)|_D)$? In particular is there any relation of the dimension of this vector space with the geometric genus of $D$?