Harvard-MIT Algebraic Geometry Seminar
		     April 16, 2002 at 3:00 p.m.
			   Harvard Room 507



	    Arithmetic Schubert calculus and applications



		      Harry Tamvakis (Brandeis)



Abstract:

Let G be the Grassmannian which parametrizes m-dimensional subspaces of
affine (m+n)-space for any base field, considered as a scheme over the
ring of rational integers. We study the structure of the Arakelov Chow
ring of G, and describe the resulting `arithmetic Schubert calculus'.
The applications include: (i) an explicit formula for the action of the
Hodge star operator on the Schubert forms on G(C), (ii) a computation of
the Faltings height of G under its Pl"ucker embedding in projective
space, and (iii) a proof of the arithmetic hard Lefschetz conjecture and
Hodge index inequalities when m=1 or 2. The latter are related to good
estimates for a class of special functions called Racah polynomials.