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Suppose $X$ is a proper algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety?

I think the holomorphic tangent bundle of a Hopf surface will not be trivial, according to the comment below..

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    $\begingroup$ Yes, assuming the variety is a complex projective manifold. (Prove that the assumption implies that the Albanese map $X \to \mathbb{Alb}(X)$ is etale.) No in positive characteristic. For examples, see Mehta and Srinivas, "Varieties in positive characteristic with trivial tangent bundle," Compositio Math., 1987. $\endgroup$
    – Vesselin Dimitrov
    Commented Apr 20, 2015 at 22:24
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    $\begingroup$ And no if you don't assume some properness condition. $\endgroup$
    – ACL
    Commented Apr 20, 2015 at 22:29
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    $\begingroup$ ...since any affine algebraic group gives an example. $\endgroup$
    – Allen Knutson
    Commented Apr 21, 2015 at 9:31
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    $\begingroup$ Does the Hopf manifold really have trivial tangent bundle? The sections $zd/dz, wd/dz, zd/dw, wd/dw$ on $\mathbb C^2$ are scale-invariant and so descend to the Hopf surface. But the structure sheaf of the Hopf surface has a 1-dimensional space of sections, so a trivial rank two vector bundle should have a $2$-dimensional space of sections. But there is no linear relation among these. $\endgroup$
    – Will Sawin
    Commented Apr 22, 2015 at 3:40
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    $\begingroup$ @oxeimon canonical is the top exterior power. $\endgroup$
    – S. Carnahan ♦
    Commented Sep 28, 2016 at 18:55

1 Answer 1

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More generally, in the complex case the following result holds.

Theorem. Let $X$ be a compact Kähler manifold which is complex parallelisable, i.e. such that $T_X$ is holomorphically trivial. Then $X$ is a complex torus. In particular, if $X$ is algebraic then $X$ is an abelian variety.

For the proof, see

H. C. Wang: Complex parallisable manifolds, Proc. Amer. Math. Soc. 5 (1954), 771–776.

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