vanishing theorem for Nakano k-positive vector bundles ?

Question

Hi,

The Nakano vanishing theorem for vector bundles says apparently the following: Let $X$ be a compact kähler manifold of dimension $n$, and $E$ an hermitian holomorphic vector bundle. If $E$ if Nakano-positive, then $H^{n,q}(X,E)=0$ for $q \geq 1$.

In a paper by Siu (Complex-analycity of harmonic maps..., JDG 1982), there is a definition of Nakano $k$-positivity (and classical Nakano positivity is equivalent to $1$-positivity) : a bundle is Nakano $k$-positive if $\Lambda^k E$ is Nakano positive. Then he refers to "classical" vanishing theorems for $k$-positive bundles, but I didn't find the statements.

Is the following statement one of these results:

If $E$ is Nakano $k$-positive, then $H^{n,q}(X,E)=0$ for $q \geq k$ ?

Are there references I can read for these vanishing theorems ? Thanks.

Answer 1

I recommend you to read Bernard Shiffman and Andrew John Sammese, Vanishing Theorems on Complex manifolds. In Chapter III, they dealt the k-positivity for line bundles and in Chapter VI they treated the vector bundle case.

Best regards

Shaban Khidr