The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Abstract

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha \in N[I]$ of positive integers one can consider the space $Q_{\alpha }$ of algebraic maps of degree $\alpha$ from $C$ to $B$. This space admits some remarkable compactifications $Q_{\alpha }^D$ (Quasimaps), $Q_{\alpha }^L$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $\pi :Q_{\alpha }^L\to Q_{\alpha }^D$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $I{C}_{\alpha }$ on the Quasimaps' space $Q_{\alpha }^D$. Namely, we prove that this singular support $SS(I{C}_{\alpha })$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $\pi :Q_{\alpha }^L\to Q_{\alpha }^D$.

8 pages, AmsLatex 1.1

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