Direct images of bundles under Frobenius morphisms

Details Direct images of bundles under Frobenius morphisms Direct images of bundles under Frobenius morphisms

2006-11-13

arxiv.org/abs/math/0611360v2

Mathematics

Algebraic Geometry

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

Scientific paper

10.1007/s00222-008-0125-y

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

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