Extensão

Higgs bundles: a geometric-categorical perspective.

Palestrante: Cristhian Paúl Neyra Salvador
Programa de Pós-Graduação em Matemática da UFSCar

Data: 23/09
Horário: das 08:25 às 08:50 horas
Local: Auditório do DM

Resumo: Let X be a smooth projective curve over an algebraically closed field of characteristic zero, let D be an effective divisor, and set L=K_X(D). An L-twisted Higgs bundle is a pair (E,φ) with φ:E→E⊗L. The aim of this talk is to explain how some of the basic geometric constructions attached to Higgs bundles admit a natural categorical formulation, and how this viewpoint organizes the passage from individual Higgs fields to families and moduli problems.

By adjunction, a Higgs field determines an action L^-1⊗E→E, hence a module structure over the symmetric algebra Sym_Ο_XL^-1. Since Tot(L)=Spec_X_X(Sym_Ο_XL^-1)), the standard equivalence for relative affine schemes identifies such modules with quasi-coherent sheaves on Tot(L). The characteristic polynomial of φ, together with the Cayley--Hamilton theorem, then forces the corresponding sheaf to be supported on the associated spectral curve. This recovers the spectral description of Higgs bundles and explains, in the smooth case, the appearance of Picard varieties; for singular spectral curves, rank-one torsion-free sheaves and suitable compactified Jacobians enter naturally.

Passing to families leads to a groupoid-valued moduli problem. The Hitchin map is interpreted as a morphism to the affine base B_L=⊕_i=1^rH⁰(X,Lⁱ), while the universal spectral curve over B_L packages the fiberwise constructions into a relative one. We then consider the effect of the divisor D: restriction defines a morphism ρ_D:B_L→B_D, with B_D:=⊕_i=1^rH⁰(X,Lⁱ|_D), and fixing boundary spectral data β is expressed by the fiber product B_L,β=B_L×BDβ. Thus the boundary condition is realized categorically as a base change of the Hitchin system. When D is nonreduced, the same restriction retains infinitesimal jet data, relating the boundary condition to deformations that preserve the prescribed spectral behavior.