Poincare Polynomials of Some Moduli Varieties
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 Fqn, 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)sOx (the stable bundles with detE ≅ Ox) 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.
KeywordsModulus Space Vector Bundle Line Bundle Rational Point Isomorphism Class
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- [B-l]V. BALAJI, Cohomology of certain moduli spaces of vector bundles Proc. Ind. Acad. Sci. (Math. Sci) Vol 98 (1988), 1–24.Google Scholar
- [B-S]V. BALAJI AND C.S. SESHADRI, Cohomology of a moduli space of vector bundles, “The Grothendieck Festschrift” Volume I, (1990), 87–120Google Scholar
- [Bi]E. BIFET, Sur les points fixes du schema Quot r0x/x/k Sous l’action du tore (Gr m, C.R. Acad. Paris, t. 309, Sèrie I, (1982), 609–612.Google Scholar
- [K-l]F.C. KIRWAN, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31, Princeton. Univ. Press (1984).Google Scholar
- [K-2]F.C. KIRWAN, On the homology of compact ifications of moduli spaces of vector bundles over a Riemann surface, Proc. Lond. Math. Soc. 53, (1986), 237–267.Google Scholar
- [S]C.S. SESHADRI, Desingularisation of moduli varieties of vector bundles on curves, International Symposium on Algebraic Geometry, Kyoto, (1977), 155–184, Kinokunia (Tokyo).Google Scholar
- [Se]J.-P. Serre, Algebraic Groups and Class Fields, Graduate Texts in Mathematics 117, Springer-Verlag.Google Scholar