Does there exist a complex manifold in $n$ complex dimensions which is compact and is also parallellizable? That is, there exists $n$ holomorphic sections who are a basis for the holomorphic tangent space at every point?
Edit: The 2-torus is a complex lie group and thus parallellizable and is obviously compact.
Well, you have $n$-dimensional tori $\mathbb{C}^n/\Lambda$.
More generally, Wang completely classified such manifolds in Complex Parallelisable Manifolds. In particular, he proved the following:
Theorem: A compact connected complex manifold $M$ has holomorphically trivial tangent bundle if and only if it it is biholomorphic to a coset space $G/D$ where $G$ is a complex Lie group with discrete subgroup $D$.
If one further requires $M$ to be Kähler, then tori are the only examples.
Note, there are examples of complex manifolds which are real parallelisable but not complex parallelisable. For example, $\mathbb{CP}^1\times(\mathbb{C}/\Lambda)$.