Department of Mathematics
Colloquium
by
T.R. Ramadas

The talk is based on joint work with V.B. Mehta. ( V.B. Mehta and T.R. Ramadas "Moduli of vector bundles, Frobenius splitting and invariant theory" Ann. of Math. 144 (1996), 269-313). The notion of Frobenius Splitting, which applies to algebraic varieties over fields of positive characteristic, is due to Mehta and Ramanathan. Varieties with ample anti-canonical bundles are often F-split. When they are, very pleasant consequences follow, notably vanishing theorems. We prove that the moduli spaces of rank two (parabolic) bundles on the generic curve (over an algebraically closed field of characteristic >2) are F-split. The Verlinde Formula is a corollary. There are results for the local structure of the moduli spaces as well, but that is another story. (V.B. Mehta and T.R. Ramadas "Frobenius Splitting and Invariant Theory", Trasformation Groups, 2 (1997) 1-13).

References:

1. Mehta, V. B. and Ramanathan, A. :Frobenius splitting and cohomology vanishing for Schubert varieties. Ann. of Math. (2) 122(1985), no. 1, 27-40.

2. van der Kallen, Wilberd: Lectures on Frobenius splittings and B-modules. Notes by S. P. Inamdar. Published for the Tata Institute of Fundamental Research, Bombay; by Springer-Verlag, Berlin, 1993. x+98 pp. ISBN: 3-540-56672-4

3. Sorger, Christoph La formule de Verlinde. (French) [The Verlinde formula] Sminaire Bourbaki, Vol. 1994/95. Astrisque No. 237 (1996), Exp. No. 794, 3, 87-114.