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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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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) – Torsten Ekedahl Aug 27 '11 at 6:55
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. – roy smith Aug 27 '11 at 17:10
Another interesting reference is the first two sections of chapter 4 in Néron Models by Bosch, Lütkebohmert and Raynaud. – Matthieu Romagny Aug 28 '11 at 7:58
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. – David White Sep 10 '11 at 17:10
up vote 4 down vote accepted

See also

which proves a converse (corr. 2.3)

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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. – Christian Liedtke Aug 27 '11 at 16:54
The paper "essentially finite vector bundles on varieties with trivial vector bundle" (by Biswas, Parameswaran, Subramanian) says more about what happens in positive characteristic. – Martin Brandenburg Mar 9 '12 at 9:13

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