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On Kodaira Vanishing for Singular Varieties

Donu Arapura and David B. Jaffe
Proceedings of the American Mathematical Society
Vol. 105, No. 4 (Apr., 1989), pp. 911-916
DOI: 10.2307/2047052
Stable URL: http://www.jstor.org/stable/2047052
Page Count: 6
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On Kodaira Vanishing for Singular Varieties
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Abstract

If $X$ is a complex projective variety with an ample line bundle $\mathscr{L}$, we show that $H^i(X, \mathscr{L}^{-1}) = 0$ for any $i < \operatorname{codim}\lbrack \operatorname{Sing}(X) \rbrack$, provided that $X$ satisfies Serre's condition $S_{i + 1}$. We also give examples to show that these results are sharp. Finally, we prove a vanishing theorem (for $H^1$) for seminormal varieties.

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