Intereting Posts

how can we show $\frac{\pi^2}{8} = 1 + \frac1{3^2} +\frac1{5^2} + \frac1{7^2} + …$?
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Every metric space can be isometrically embedded in a Banach space, so that it's a linearly independent set
How many Hamiltonian cycles are there in a complete graph $K_n$ ($n\geq 3$) Why?
$f(x)=1/q$ for $x=p/q$ is integrable
Relation between blowing up at a point and at a variety
Left Adjoint of a Representable Functor
Recursion tree T(n) = T(n/3) + T(2n/3) + cn
how to prove this combinatorial identity I accidentally find?
Solving Special Function Equations Using Lie Symmetries
Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules
Who has the upper hand in a generalized game of Risk?
How to prove that equality is an equivalence relation?
How to show that every Boolean ring is commutative?

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?

- Presheaf which is not a sheaf — holomorphic functions which admit a holomorphic square root
- Are minimal prime ideals in a graded ring graded?
- Fundamental theorem of Algebra using fundamental groups.
- Algebraic Curves and Second Order Differential Equations
- On a certain morphism of schemes from affine space to projective space.
- On limits, schemes and Spec functor

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- Why study Algebraic Geometry?

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

Theorem:the only Kaehler manifolds which areholomorphicallyparallelisable 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. $\square$

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 $\mathbb{CP}^1$.

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