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Ramanujam's vanishing theorem states for $X$ a Kähler surface and (1)$D$ a 1-connected divisor, (2)$h^0(O_X(nD))\ge 2$ for some $n\ge 1$, (3) $|nD|$ is not an irrational pencil, we have $h^1(O_X(-D))=0$.

In the proof given in BPV, it used a topological lemma which is based on Hodge decomposition of a Kähler manifold.

I was wondering if the theorem can be generalized to singular varieties. So I want to know if there is an algebraic proof which avoids hodge decomposition? Or is there already such a generalization for singluar variety?

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Perhaps this helps: Kyungho Oh, Vanishing Theorems for Singular Varieties, (Proceedings of the American Mathematical Society).

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