Global Intersection Cohomology of Quasimaps' Spaces

1997/02/14 by Michael Finkelberg, Alexander Kuznetsov, Finkelberg, Michael +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9702010

21 pages, AmsLatex 1.1

arxiv created 1997/02/14 · arxiv updated 2016/08/30

Abstract

Let C be a smooth projective curve of genus 0. Let \CB be the variety of complete flags in an n-dimensional vector space V. Given an (n-1)-tuple α∈\BN[I] of positive integers one can consider the space \CQ_α of algebraic maps of degree α from C to \CB. This space admits some remarkable compactifications \CQ^D_α (Quasimaps), \CQ^L_α (Quasiflags), \CQ^K_α (Stable Maps) of \CQ_α constructed by Drinfeld, Laumon and Kontsevich respectively. It has been proved that the natural map π: \CQ^L_α→ \CQ^D_α is a small resolution of singularities. The aim of the present note is to study the cohomology H^\bullet(\CQ^L_α,\BQ) of Laumon's spaces or, equivalently, the Intersection Cohomology H^\bullet(\CQ^L_α,IC) of Drinfeld's Quasimaps' spaces. We calculate the generating function P_G(t) (``Poincaré polynomial'') of the direct sum ⊕_α∈\BN[I]H^\bullet(\CQ^D_α,IC) and construct a natural action of the Lie algebra \fraksl_n on this direct sum by some middle-dimensional correspondences between Quasiflags' spaces. We conjecture that this module is isomorphic to distributions on nilpotent cone supported at nilpotent subalgebra.

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