论文标题
jacobian，prym堆栈和g-higgs配对的频谱数据上的标准图
The Norm map on the compactified Jacobian, the Prym stack and Spectral data for G-Higgs pairs
论文作者
论文摘要
在Filippo viviani的监督下进行的本文致力于研究$ g $ -higgs对的光谱通信，以$ g = sl（r，\ mathbb {c}）$，$ pgl（$ pgl），$ pgl（r，\ mathbb {c}）$，$ sp（$ sp（2r） $ gsp（2r，\ mathbb {c}）$，$ psp（2r，\ mathbb {c}）$，在任何光纤上。 在第一章中，我们介绍了压实的Jacobian堆栈参数化无扭转的Rank-1滑轮，上面是（可能还原，非降低）的射击曲线，我们用压缩的Jacobian来描述Higgs对的光谱对应关系。 在第二章中，我们研究了与纯尺寸1的嵌入嵌入式赛酚方案之间的任何有限的，平坦的形态相关的广义分裂和广义线捆绑包的概念。在我们在磁场上处理投影性曲线的情况下，我们研究压缩的jacobian之间的标准图和不向图像。 在第三章中，我们介绍了与平稳的代码域之间的任何有限的，平坦的形态相关的prym堆栈。如果域用局部平面奇异性降低，并且该场是代数封闭的，我们表明通常的PRYM方案包含在Prym堆栈中，作为开放且致密的子集。 在第四章中，我们研究了$ g $ -Hitchin纤维化的纤维，以$ g $在古典的谎言组中变化。特别是，在$ g = sl（r，\ mathbb {c}）$，$ pgl（r，\ mathbb {c}）$的情况下，任何纤维的描述都涉及形态学从光谱曲线到基本曲线所引起的标准图所引起的固定图的二叠纪光纤。如果是$ g = sp（2r，\ mathbb {c}）$，$ gsp（2r，\ mathbb {c}）$，$ psp（2r，\ mathbb {c}）$，对任何光纤的描述涉及在合格的jacobian compactified jacobian的均衡器的均衡堆栈。
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.
