TITLE: Must an algebraic variety with trivial tangent bundle be an abelian variety?
QUESTION [21 upvotes]: 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..

REPLY [28 votes]: 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.