# Poincare Polynomials of Some Moduli Varieties

## Abstract

For smooth projective varieties V/ℂ, the Weil conjectures as established by Deligne, tell us that the number of rational points of the corresponding variety V over F_{q}n, for all n, determine the Betti numbers of V (for precise details cf., §5). This theme has been taken up by Harder and Narasimhan in [H-N] and by Desale and Ramanan in [D-R] to compute the Poincaré polynomial of the moduli space M(n,d) of semi-stable vector bundles of rank n and degree d, where n and d are coprime. More recently, Atiyah and Bott [A-B] following a geometric approach compute the Poincaré polynomial of the moduli space M(n,d) when (n,d) = l, and also show that there is no torsion in the cohomology in this case. Let N be the smooth compactification of M(2,0)^{s}O_{x} (the stable bundles with detE ≅ O_{x}) constructed in [S]. In [B-S], an approach modelled on [A-B] was studied and this gave only partial success in the computation of the cohomology of N.

## Keywords

Modulus Space Vector Bundle Line Bundle Rational Point Isomorphism Class## Preview

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