On Kodaira vanishing for singular varieties

Abstract:

If $X$ is a complex projective variety with an ample line bundle $L$, we show that ${H^i}(X,{L^{ - 1}}) = 0$ for any $i{\text { < codim[Sing(}}X{\text {)]}}$, 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