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My question is: I am interested in calculating the Betti numbers of a specific polynomial variety (w.r.t. singular cohomology) whose zeros I am looking at over $\mathbb{C}$ (it has integer coefficients). I know the polynomial explicitly.

Please allow me to explain my background. I am new to homological algebra in general. I know what Betti numbers are, and I know how to calculate Betti numbers of simple things, like a circle in $\mathbb{R}^2$ (we know how to triangulate the circle, and it is not too hard to find the simplicial homology that way). So, as you can see, I have a modicum of background in simplicial homology. Basically if someone gives me the triangulation, I know how to proceed.

In my current problem, I know the polynomial explicitly, but it is in very high dimensional space. So I don't know how to proceed. Also, I need the singular Betti numbers (they probably coincide with simplicial, but I am not too sure).

My goals are 1) I want to learn all the relevant background so I can do it myself 2) I want to be able to do this for other polynomials as well. Specifically, I'd greatly appreciate if I can get references from where I can learn all this ground-up in a holistic manner.

Jyothi
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1 Answers1

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It looks like you have access to the equations that define your variety. You could get the de Rham Betti numbers by using the algorithms of Oaku, Takayama, Walther, etc. See for example

  1. oaku,takayama-algorithms for D-modules - restriction, tensor product, localization, and local cohomology groups (J. Pure Appl. Algebra, 2001)
  2. oaku,takayama-an algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation (J. Pure Appl. Algebra, 1999)
  3. scheiblechner-effective de Rham cohomology - the general case
  4. walther-algorithmic determination of the rational cohomology of complex varieties via differential forms (2001)
  5. walther-algorithmic computation of de Rham cohomology of complements of complex affine varieties (1998)
Abhiram Natarajan
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