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Harvard-M.I.T. Algebraic Geometry Seminar}

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ARITHMETIC SCHUBERT CALCULUS AND APPLICATIONS
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{\bf 
HARRY TAMVAKIS
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Brandeis University

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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.


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April 16, 2002

3:00 p.m.

Harvard Room 507
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