TITLE: What are examples of parallelizable complex projective varieties?
QUESTION [13 upvotes]: A smooth complex projective variety is the zero-locus, inside some $\mathbb{CP}^n$, of some family of homogeneous polynomials in $n+1$ variables satisfying a certain number of conditions that I won't spell out. It is in particular a differentiable manifold. A parallelizable manifold is a (differentiable) manifold with a trivial tangent bundle, i.e. $TM \cong M \times \mathbb{R}^n$ (equivalently, a manifold of dimension $n$ is parallelizable if it admits $n$ vector fields that are everywhere linearly independent).
Being a projective variety is an algebro-geometric condition, whereas being parallelizable is more of a algebro-topological condition. I'd like to know how the two interact. For example, according to Wikipedia, some complex tori are projective. But like all Lie groups, a complex torus is parallelizable.

What are other examples of smooth complex projective varieties that are parallelizable?

REPLY [3 votes]: Here is a small, rather unsatisfying collection of examples arising from the following result:

A (non-trivial) product of spheres is parallelizable if and only if at least one of the spheres is odd-dimensional. 

One can ask whether any such products can be given a complex structure which makes them projective. First of all, such a manifold would be Kähler and hence symplectic. By considering the cohomology ring of such a product, we can see that the only possibilities are $(S^1)^{2m}\times(S^2)^k$ with $m > 0$. Such spaces have many projective complex structures. For $n = 0$, we obtain the projective tori that you already mentioned, together with their products. For $n > 0$, we get the product of a non-zero number of algebraic tori with $n$ copies of $\mathbb{CP}^1$.