Essentially finite vector bundles on varieties with trivial tangent bundle

Biswas, Indranil; Parameswaran, A. J.; Subramanian, S.

doi:10.1090/S0002-9939-2011-10804-9

Essentially finite vector bundles on varieties with trivial tangent bundle
HTML articles powered by AMS MathViewer

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