Chern classes of the Grassmannians and Schubert calculus

Abstract

IN THIS note we continue the theme of [3,4] by using the zeros of a holomorphic vector field to give a geometric demonstration that the ring structure of the cohomology of the Grassman manifolds (the multiplicative aspect of Schubert calculus) is a consequence of general properties of symmetric functions. More precisely, this amounts to showing that certain properties of Schubert cycles such as independence; multiplicative structure etc. are formally deducible from similar properties of a class of symmetric functions called Schur functions provided certain so called meaningless Schubert conditions are ignoied. This principle seems to have originated with Giambelli and was formulated precisely by Lesieur[9]. More recently, Horrocks[6], Lascoux[8], Porteous [ 1 l] and Stanley [ 131 have discussed various aspects of the principle. Recall that if, X is a compact complex manifold and V a holomorphic vector field on X with zeros 2, one may view 2 as the variety defined by the sheaf of ideals i(V)R’ C 6 where i(V):W+fP-’ is the contraction from the sheaf of germs of holomorphic p-forms to (p l)-forms and 6 = R”. The structure sheaf fY2 = Q/i( V)Q’ of 2 is a coherent possibly unreduced sheaf of rings on Z. The main result of [4] is that if X is in addition compact Kaehler and Z finite but nontrivial, then H”(X, I!?~), the ring of functions on Z, has a decreasing filtration F having the property that fiFj C F;+j such that there is an isomorphism of graded rings