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Canonical filtrations and stability of direct images by Frobenius morphisms. (English) Zbl 1201.14030
Summary: We study the stability of direct images by Frobenius morphisms. First, we compute the first Chern classes of direct images of vector bundles by Frobenius morphisms modulo rational equivalence up to torsions. Next, introducing the canonical filtrations, we prove that if $$X$$ is a nonsingular projective minimal surface of general type with semistable $$\varOmega_X^1$$ with respect to the canonical line bundle $$K_X$$, then the direct images of line bundles on $$X$$ by Frobenius morphisms are semistable with respect to $$K_X$$.

##### MSC:
 14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14J29 Surfaces of general type
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