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.. 



Must an algebraic variety with trivial tangent bundle be an abelian variety?
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](http://www.ams.org/journals/proc/1954-005-05/S0002-9939-1954-0074064-3/home.html)*, Proc. Amer. Math. Soc. **5** (1954), 771–776. 


