# Must an algebraic variety with trivial tangent bundle be an abelian variety?

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..

• 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. Apr 20, 2015 at 22:24
• And no if you don't assume some properness condition.
– ACL
Apr 20, 2015 at 22:29
• ...since any affine algebraic group gives an example. Apr 21, 2015 at 9:31
• 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. Apr 22, 2015 at 3:40
• @oxeimon canonical is the top exterior power. Sep 28, 2016 at 18:55

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.