Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ENDOMORPHISMS OF PROJECTIVE BUNDLES OVER A CERTAIN CLASS OF VARIETIES

  • Amerik, Ekaterina (National Research University Higher School of Economics Laboratory of Algebraic Geometry and Applications) ;
  • Kuznetsova, Alexandra (National Research University Higher School of Economics Laboratory of Algebraic Geometry and Applications)
  • Received : 2016.09.13
  • Accepted : 2017.04.05
  • Published : 2017.09.30
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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 $X={\mathbb{P}}(E)$. 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.

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References

  1. E. Amerik, On endomorphisms of projective bundles, Manuscripta Math. 111 (2003), no. 1, 17-28. https://doi.org/10.1007/s00229-002-0347-z
  2. E. Amerik and F. Campana, Exceptional points of an endomorphism of the projective plane, Math. Z. 249 (2005), no. 4, 741-754. https://doi.org/10.1007/s00209-004-0727-z
  3. E. Amerik, M. Rovinsky, and A. Van de Ven, A boundedness theorem for morphisms between threefolds, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 405-415. https://doi.org/10.5802/aif.1679
  4. A. Beauville, Endomorphisms of hypersurfaces and other manifolds, Internat. Math. Res. Notices 2001 (2001), no. 1, 53-58. https://doi.org/10.1155/S1073792801000034
  5. D. Birkes, Orbits of linear algebraic groups, Ann. of Math. 93 (1971), 459-475. https://doi.org/10.2307/1970884
  6. J.-Y. Briend and J. Duval, Deux caracterisations de la mesure d'equilibre d'un endomorphisme de ${\mathbb{P}}^k$(C), Publ. Math. IHES 93 (2001), 145-159; Erratum, Publ. Math. IHES 109 (2009), 295-296. https://doi.org/10.1007/s10240-001-8190-4
  7. S. Druel, Structures de contact sur les varietes toriques, Math. Ann. 313 (1999), no. 3, 429-435. https://doi.org/10.1007/s002080050268
  8. S. Kobayashi and T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7-16. https://doi.org/10.1007/BF01389863
  9. N. Nakayama, Ruled surfaces with non-trivial surjective endomorphisms, Kyushu J. Math. 56 (2002), no. 2, 433-446. https://doi.org/10.2206/kyushujm.56.433
  10. N. Nakayama and D.-Q. Zhang, Building blocks of etale endomorphisms of complex projective manifolds, Proc. Lond. Math. Soc. (3) 99 (2009), no. 3, 725-756. https://doi.org/10.1112/plms/pdp015