The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs

TL;DR

This thesis explores the spectral correspondence for G-Higgs pairs, introducing compactified Jacobian and Prym stacks, and analyzing fibers of the G-Hitchin fibration for classical Lie groups.

Contribution

It develops a comprehensive framework for spectral correspondence using stacks, extending classical concepts to reducible and non-reduced curves, and describes fibers of the G-Hitchin fibration in stack-theoretic terms.

Findings

Compactified Jacobian stacks parametrizing torsion-free sheaves are described.

Prym stacks are introduced and relate to classical Prym schemes.

Fibers of the G-Hitchin fibration are characterized via stacky Norm maps and equalizer stacks.

Abstract

This thesis, done under the supervision of Filippo Viviani, is devoted to the study of the spectral correspondence for $G$-Higgs pairs, in the case of $G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, $Sp(2r,\mathbb{C})$, $GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, over any fiber. In the first chapter we introduce the compactified Jacobian stack parametrizing torsion-free rank-1 sheaves over a (possibly reducible, non reduced) projective curve and we describe the spectral correspondence for Higgs pairs in terms of the compactified Jacobian. In the second chapter, we study the notion of direct and inverse image for generalized divisors and generalized line bundles associated to any finite, flat morphism between embeddable noetherian schemes of pure dimension 1. In the case when we deal with projective curves over a field, we study the Norm and inverse image maps between compactified Jacobians. In the third chapter we introduce the Prym stack associated to any finite, flat morphism between projective curves over a field, with smooth codomain. If the domain is reduced with locally planar singularities and the field is algebraically closed, we show that the usual Prym scheme is contained in the Prym stack as an open and dense subset. In the fourth chapter, we study the fibers of the $G$-Hitchin fibration, for $G$ varying among classical Lie groups. In particular, in the case of $G=SL(r,\mathbb{C})$, $PGL(r, \mathbb{C})$, the description of any fiber involves the stacky fibers of the Norm map induced by the morphism from the spectral curve to the base curve. In the case of $G=Sp(2r,\mathbb{C})$, $GSp(2r,\mathbb{C})$, $PSp(2r,\mathbb{C})$, the description of any fiber involves the equalizer stack of two maps defined on the compactified Jacobian of the spectral curve.