Hodge numbers of 2n-dimensional torus

In summary, the conversation is about finding the Hodge numbers for a 2n-dimensional torus and one person suggests using the span of wedge products of complex numbers on the torus, while another person explains that the tangent bundle of a complex torus is trivial and can be used to compute the Hodge numbers from first principles.
  • #1
Paul Cook
2
0
Hi,

A small but exceptionally annoying algebraic topology question:

I'm trying to find the Hodge numbers (from the Hodge-de Rham cohomology) for a 2n-dimensional torus (that is, n complex dimensions).

Anyone have any ideas? It's a rather technical question, but I don't really want to deduce it from first principles.

Thanks!
 
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  • #2
well, off the top of my head, i would say, isn't [itex]H^{p,q}[/itex] spanned by [itex]dz_{i_1}\wedge\dotsb\wedge dz_{i_p}\wedge d\overline{z}_{j_1}\wedge\dotsb\wedge d\overline{z}_{j_q}[/itex] on the torus?

then we would have

[tex]b^{p,q}=\binom{n}{p}\binom{n}{q}[/tex]
 
Last edited:
  • #3
I believe lethe is right. the thing that makes this case so easy is that the tangent bundle of a complex torus is trivial, so you just need to look only at the tangent space at the origin.

I.e. the wedge product of the trivial bundle is the trivial bundle whose fibres are the wedge product of the given vector space. Thus lethe has just computed the pth wedge of C^n times the qth wedge of C^n, (or their duals).

The dimension of this last space is the number of sections of the (dual) hodge (p,q) bundle, because that bundle is trivial, and that is the definition of the hodge number h^(p,q).

So this is actually computed from first principles.
 

1. What are Hodge numbers of 2n-dimensional torus?

Hodge numbers of 2n-dimensional torus refer to a set of integers that describe the topological properties of a 2n-dimensional torus. They are used in algebraic geometry and complex analysis to classify and study these mathematical structures.

2. How are Hodge numbers calculated?

Hodge numbers are typically calculated using the Hodge diamond, which is a geometric representation of the Hodge numbers. The numbers in the diamond are calculated using the Betti numbers and the dimension of the torus.

3. What is the significance of Hodge numbers in mathematics?

Hodge numbers are important in mathematics because they provide a way to classify and study 2n-dimensional tori. They also have applications in other areas of mathematics, such as algebraic topology and differential geometry.

4. What information do Hodge numbers provide about a 2n-dimensional torus?

Hodge numbers provide information about the topology and geometric structure of a 2n-dimensional torus. They can also give insight into the algebraic and differential properties of these mathematical objects.

5. Can Hodge numbers be used to classify other mathematical structures?

Yes, Hodge numbers can be used to classify other complex algebraic varieties, such as projective varieties and Calabi-Yau manifolds. They are also used in the study of higher-dimensional algebraic geometry and mirror symmetry.

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