I am looking for the reference where I can find the proof of the following:
If $A$ is an abelian variety then its tangent bundle is trivial.
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6$\begingroup$ This of course is a general fact for algebraic groups and there are references for that. However, a reference for abelian varieties is given by David Mumford: Abelian varieties, Ch 4 (iii) $\endgroup$– Torsten EkedahlCommented Aug 27, 2011 at 6:55
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$\begingroup$ i suggest you first try as an exercise to think of an interesting map from AxV to TA where A is an abelian variety and V the tangent space at the origin, and TA is the tangent bundle. $\endgroup$– roy smithCommented Aug 27, 2011 at 17:10
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2$\begingroup$ Another interesting reference is the first two sections of chapter 4 in Néron Models by Bosch, Lütkebohmert and Raynaud. $\endgroup$– Matthieu RomagnyCommented Aug 28, 2011 at 7:58
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$\begingroup$ This question just got bumped to the front-page by MathOverflow. @yuvi, can you accept the answer below so it'll get registered as solved by the software. Since this was already on the front-page I retagged as textbook-recommendation since this fact you seek is discussed in many textbooks and effectively you're asking which is best. For my money, Torsten's answer is the most useful such text. $\endgroup$– David WhiteCommented Sep 10, 2011 at 17:10
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1 Answer
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See also
http://www-fourier.ujf-grenoble.fr/~mbrion/notes_bremen.ps
which proves a converse (corr. 2.3)
answered Aug 27, 2011 at 14:09
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6$\begingroup$ However, the converse is false in positive characteristic: Igusa has given an example of a variety with trivial tangent bundle in characteristic $2$ that is not an Abelian variety: it arises as $(E\times E')/({\mathbb{Z}/2\mathbb{Z})$, where $E$ and $E'$ are elliptic curves, and the group $\mathbb{Z}/2\mathbb{Z}$ acts via sign involution on one factor and via translation of a $2$-torsion point on the other factor. $\endgroup$ Commented Aug 27, 2011 at 16:54
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$\begingroup$ The paper "essentially finite vector bundles on varieties with trivial vector bundle" (by Biswas, Parameswaran, Subramanian) says more about what happens in positive characteristic. $\endgroup$ Commented Mar 9, 2012 at 9:13