Essentially finite vector bundles on varieties with trivial tangent bundle
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- by Indranil Biswas, A. J. Parameswaran and S. Subramanian PDF
- Proc. Amer. Math. Soc. 139 (2011), 3821-3829 Request permission
Abstract:
Let $X$ be a smooth projective variety, defined over an algebraically closed field of positive characteristic, such that the tangent bundle $TX$ is trivial. Let $F_X : X \longrightarrow X$ be the absolute Frobenius morphism of $X$. We prove that for any $n \geq 1$, the $n$–fold composition $F^n_X$ is a torsor over $X$ for a finite group–scheme that depends on $n$. For any vector bundle $E \longrightarrow X$, we show that the direct image $(F^n_X)_*E$ is essentially finite (respectively, $F$–trivial) if and only if $E$ is essentially finite (respectively, $F$–trivial).References
- Indranil Biswas and Yogish I. Holla, Comparison of fundamental group schemes of a projective variety and an ample hypersurface, J. Algebraic Geom. 16 (2007), no. 3, 547–597. MR 2306280, DOI 10.1090/S1056-3911-07-00449-3
- I. Biswas and J. P. dos Santos, Vector bundles trivialized by proper morphisms and the fundamental group scheme, Jour. Inst. Math. Jussieu (to appear), DOI:10.1017/S1474748010000071.
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- Jun-ichi Igusa, On some problems in abstract algebraic geometry, Proc. Nat. Acad. Sci. U.S.A. 41 (1955), 964–967. MR 74085, DOI 10.1073/pnas.41.11.964
- George R. Kempf, Varieties with trivial tangent bundles, Topics in algebraic geometry (Guanajuato, 1989) Aportaciones Mat. Notas Investigación, vol. 5, Soc. Mat. Mexicana, México, 1992, pp. 109–111. MR 1308335
- V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent bundle, Compositio Math. 64 (1987), no. 2, 191–212. With an appendix by Srinivas and M. V. Nori. MR 916481
- V. B. Mehta and S. Subramanian, Some remarks on the local fundamental group scheme, Proc. Indian Acad. Sci. Math. Sci. 118 (2008), no. 2, 207–211. MR 2423233, DOI 10.1007/s12044-008-0013-9
- David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, London, 1970. MR 0282985
- Madhav V. Nori, On the representations of the fundamental group, Compositio Math. 33 (1976), no. 1, 29–41. MR 417179
- Madhav V. Nori, The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122. MR 682517, DOI 10.1007/BF02967978
Additional Information
- Indranil Biswas
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
- MR Author ID: 340073
- Email: indranil@math.tifr.res.in
- A. J. Parameswaran
- Affiliation: Kerala School of Mathematics, Kunnamangalam (PO), Kozhikode, Kerala 673571, India
- Email: param_aj@yahoo.com
- S. Subramanian
- Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
- Email: subramnn@math.tifr.res.in
- Received by editor(s): March 22, 2010
- Received by editor(s) in revised form: September 15, 2010, and September 16, 2010
- Published electronically: March 15, 2011
- Communicated by: Lev Borisov
- © Copyright 2011 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 3821-3829
- MSC (2010): Primary 14L15, 14F05
- DOI: https://doi.org/10.1090/S0002-9939-2011-10804-9
- MathSciNet review: 2823029