Direct images of bundles under Frobenius morphisms

Abstract

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with $\mathrm{char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of $F_{\ast }W$ is bounded by instability of $W\otimes {\mathrm{T}}^{\ell }({\Omega }_X^1)$ ($0\le \ell \le n(p-1)$)(Corollary \ref{cor3.8}). When $X$ is a smooth projective curve of genus $g\ge 2$, it implies $F_{\ast }W$ being stable whenever $W$ is stable.

the final version to appear in Invent. math. (2008)

References in corpus (1)

• Stability of direct images under Frobenius morphism

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