Arithmetic Intersection Theory on Flag Varieties
Abstract
Let F be the complete flag variety over Spec(Z) with the tautological filtration 0 ⊂ E1 ⊂ E2 ⊂ ... ⊂ En=E of the trivial bundle E over F. The trivial hermitian metric on E() induces metrics on the quotient line bundles Li(). Let c1(Li) be the first Chern class of Li in the arithmetic Chow ring CH(F) and xi = -c1(Li). Let h(X1,...,Xn) be a polynomial with integral coefficients in the ideal <e1,...,en> generated by the elementary symmetric polynomials ei. We give an effective algorithm for computing the arithmetic intersection h(x1,...,xn) in CH(F), as the class of a SU(n)-invariant differential form on F(). In particular we show that all the arithmetic Chern numbers one obtains are rational numbers. The results are true for partial flag varieties and generalize those of Maillot for grassmannians. An `arithmetic Schubert calculus' is established for an `invariant arithmetic Chow ring' which specializes to the Arakelov Chow ring in the grassmannian case.
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