The Laumon's resolution of Drinfeld's compactification is small

TL;DR

This paper proves Drinfeld's conjecture that the natural map from a smooth compactification to a singular one is a small resolution, and analyzes the Hodge structure of intersection cohomology sheaves in the context of algebraic maps from curves to flag varieties.

Contribution

It confirms Drinfeld's conjecture that the map from the smooth quasiflag compactification to the quasimaps compactification is a small resolution, and computes the Hodge structures of IC sheaves.

Findings

The map $ abla: ext{Quasiflag} o ext{Quasimaps}$ is a small resolution.

The Hodge structure on IC sheaf stalks is pure Tate.

The generating function for IC stalks matches Lusztig's $q$-analogue of Kostant's partition function.

Abstract

Let $C$ be a smooth projective curve of genus 0. Let $\FF$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $\alpha$ of positive integers one can consider the space $\MM\alpha$ of algebraic maps of degree $\alpha$ from $C$ to $\FF$. This space has drawn much attention recently in connection with Quantum Cohomology. The space $\MM\alpha$ is smooth but not compact. The problem of compactification of $\MM\alpha$ proved very important. One compactification $\MML\alpha$ (the space of {\em quasiflags}), was constructed in \cite{L}. However, historically the first and most economical compactification $\MMD\alpha$ (the space of {\em quasimaps}) was constructed by Drinfeld (early 80-s, unpublished). The latter compactification is singular, while the former one is smooth. Drinfeld has conjectured that the natural map $\pi:\MML\alpha\to\MMD\alpha$ is a small resolution of singularities. In the present note we prove this conjecture. As a byproduct, we compute the stalks of $IC$ sheaf on $\MMD\alpha$ and, moreover, the Hodge structure in these stalks. Namely, the Hodge structure is a pure Tate one, and the generating function for the $IC$ stalks is just the Lusztig's $q$-analogue of Kostant's partition function (see \cite{Lu}).