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This entry was posted on Sunday, June 10th, 2007 at 4:23 am and is filed under General. You can follow any responses to this entry through the RSS 2.0 feed.
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What an honor to provide the first comment! I had lunch with Arun yesterday and he told me a bit about what you and Hausel were doing — it sounds really great. So here’s something I don’t understand on first glance. In general if you have affine varieties lying around and you can count the points over finite fields, you have some hope of computing the E-polynomial. But typically this doesn’t allow you to see the literal mixed Hodge polynomial (or Poincare polynomial), right? Which I guess is why the formula for the mixed Hodge polynomial in your paper is conjectural. But you do get it for n = 2. So in that case, do you have some kind of purity trick that allows you to deduce the mixed Hodge from the E? Or do you know something about the Hodge structure there by means having nothing to do with counting points over finite fields?

Hi Jordan. The reason we can calculate the mixed Hodge polynomial for n=2 is because in that case we know the cohomology ring explicitly by generators and relations. One can also determine the weights on the generators. Then we only have to find for the cohomology ring a monomial basis in the generators, evaluate the weight and degree of the monomials (using the fact that the MHS is compatible with cup product in cohomology) and add it up to get the mixed Hodge polynomial. One attractive feature of this calculation, that by purely cohomological calculations we can get the E-polynomial of the GL_2 character variety, which reflects the structure of the character table of GL_2(F_q).

What an honor to provide the first comment! I had lunch with Arun yesterday and he told me a bit about what you and Hausel were doing — it sounds really great. So here’s something I don’t understand on first glance. In general if you have affine varieties lying around and you can count the points over finite fields, you have some hope of computing the E-polynomial. But typically this doesn’t allow you to see the literal mixed Hodge polynomial (or Poincare polynomial), right? Which I guess is why the formula for the mixed Hodge polynomial in your paper is conjectural. But you do get it for n = 2. So in that case, do you have some kind of purity trick that allows you to deduce the mixed Hodge from the E? Or do you know something about the Hodge structure there by means having nothing to do with counting points over finite fields?

Hi Jordan. The reason we can calculate the mixed Hodge polynomial for n=2 is because in that case we know the cohomology ring explicitly by generators and relations. One can also determine the weights on the generators. Then we only have to find for the cohomology ring a monomial basis in the generators, evaluate the weight and degree of the monomials (using the fact that the MHS is compatible with cup product in cohomology) and add it up to get the mixed Hodge polynomial. One attractive feature of this calculation, that by purely cohomological calculations we can get the E-polynomial of the GL_2 character variety, which reflects the structure of the character table of GL_2(F_q).