Endomorphisms of projective bundles over a certain class of varieties
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Let $B$ be a simply-connected projective variety such that the first cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector bundle over $B$ and $X=\p(E)$. It is easily seen that a power of any endomorphism of $X$ takes fibers to fibers. We prove that if $X$ admits an endomorphism which is of degree greater than one on the fibers, then $E$ splits into a direct sum of line bundles.
Abstract
Let $B$ be a simply-connected projective variety such that the first cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector bundle over $B$ and $X=\p(E)$. It is easily seen that a power of any endomorphism of $X$ takes fibers to fibers. We prove that if $X$ admits an endomorphism which is of degree greater than one on the fibers, then $E$ splits into a direct sum of line bundles.
참고문헌 (10)
- E. Amerik, "On endomorphisms of projective bundles", Manuscripta Math, 111(1), pp.17~28 (2003)
- E. Amerik, "Exceptional points of an endomorphism of the projective plane", Math. Z, 249(4), pp.741~754 (2005)
- E. Amerik, "A boundedness theorem for morphisms between threefolds", Ann. Inst. Fourier (Grenoble), 49(2), pp.405~415 (1999)
- A. Beauville, "Endomorphisms of hypersurfaces and other manifolds", Internat. Math. Res. Notices, 2001(1), pp.53~58 (2001)
- D. Birkes, "Orbits of linear algebraic groups", Ann. of Math, 93, pp.459~475 (1971)
- J.-Y. Briend, "Deux caracterisations de la mesure d'equilibre d'un endo-morphisme de Pk(C)", Erratum, Publ. Math. IHES, 109, pp.295~296 (2009)
- S. Druel, "Structures de contact sur les variétés toriques", Math. Ann, 313(3), pp.429~435 (1999)
- S. Kobayashi, "Meromorphic mappings onto compact complex spaces of general type", Invent. Math, 31(1), pp.7~16 (1975)
- N. Nakayama, "Ruled surfaces with non-trivial surjective endomorphisms", Kyushu J. Math, 56(2), pp.433~446 (2002)
- N. Nakayama, "Building blocks of étale endomorphisms of complex projective manifolds", Proc. Lond. Math. Soc, 99(3), pp.725~756 (2009)