What are examples of parallelizable complex projective varieties?

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?

Solution 1:

At Najib Idrissi's request, here is an answer to a different question:

Theorem: the only Kaehler manifolds which are holomorphically parallelisable are complex tori.

Reference:

Wang, Hsien-Chung. Complex parallisable manifolds. Proc. Amer. Math. Soc. 5 (1954), 771–776.

Edit: OK, here's a way to get a family of examples containing those given by Michael Albanese.

Proposition: If $M$ is a parallelisable (real) manifold of dimension $d \geq 1$, and $C$ is any orientable (real) surface, then $M \times C$ is parallelisable.

Proof: $C$ embeds in $\mathbf R^3$, so (e.g. thinking about an outward unit normal vector) we see that the tangent bundle of $C$ is trivialised by adding one copy of the trivial bundle over $C$: $$TC \oplus \mathbf R_C = \mathbf R^3_C.$$ Now the tangent bundle of $M \times C$ is $\pi_1^* TM \oplus \pi_2^* TC$, which by hypothesis is $$\pi_1^* \mathbf R_M^d \oplus \pi_2^* TC = \mathbf R_{M\times C}^{d-1} \oplus \pi_2^* (T_C \oplus \mathbf R_C) = \mathbf R_{M\times C}^{d+2}. \quad \square$$

Corollary: Any complex manifold of the form $A \times C_1 \times \cdots \times C_n$ where $A$ is a complex torus of positive dimension and the $C_i$ are Riemann surfaces is parallelisable.

Proof of Corollary: As the OP remarked, $A$ is parallelisable. Now apply the Proposition $n$ times. $\square$

Solution 2:

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 $\mathbb{CP}^1$.