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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?

(For the complex manifold case this is not true due to the example of Hopf manifolds $\mathbf{C}^n-\{0\}/\{z\to 2z\}$.)

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

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..

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?  (For the complex manifold case this is not true due to the example of Hopf manifolds $\mathbf{C}^n-\{0\}/\{z\to 2z\}$.)

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? 

(For the complex manifold case this is not true due to the example of Hopf manifolds $\mathbf{C}^n-\{0\}/\{z\to 2z\}$.)

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

Suppose $X$ is an algebraic variety with trivial tangent bundle $T_X$ (not only canonical bundle $K_X$), is it true that $X$ is an abelian variety? (For the complex manifold case this is not true due to the example of Hopf manifolds $\mathbf{C}^n-\{0\}/\{z\to 2z\}$.)

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? (For the complex manifold case this is not true due to the example of Hopf manifolds $\mathbf{C}^n-\{0\}/\{z\to 2z\}$.)