A global analogue of the Springer resolution for $SL_2$

TL;DR

This paper constructs a Springer-like resolution for the global nilpotent cone associated with SL_2, advancing understanding of its geometry and laying groundwork for broader applications in the Langlands programs.

Contribution

It introduces a global analogue of the Springer resolution for SL_2, providing new insights into the structure of the global nilpotent cone and its irreducible components.

Findings

Proved the global nilpotent cone is equidimensional.

Provided enumeration of irreducible components.

Constructed a resolution of singularities for the cone.

Abstract

The global nilpotent cone N is a singular stack associated to the choice of an algebraic group G, a smooth projective curve X, and a line bundle L on X, which is of fundamental importance to the Geometric Langlands Program, and which is of emerging importance to the Classical Langlands Program. In analogy with the ordinary Springer resolution, we construct and study a resolution of singularities of N in the special case where G=SL_2. As an immediate application, we prove that N is equidimensional and also provide an enumeration of its irreducible components. We hope this is the first step in constructing a global Springer resolution for an arbitrary reductive group.