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 $α\in N[I]$ of positive integers one can consider the space $Q_α$ of algebraic maps of degree $α$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_α$ (Quasimaps), $Q^L_α$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $π: Q^L_α\to Q^D_α$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-MacPherson sheaf $IC_α$ on the Quasimaps' space $Q^D_α$. Namely, we prove that this singular support $SS(IC_α)$ is irreducible. The proof is based on the factorization property of Quasimaps' space and on the detailed analysis of Laumon's resolution $π: Q^L_α\to Q^D_α$.