Theses
THESIS DEFENSE -- Classification of very stable Higgs bundles
Speaker: Miguel González (ICMAT-UCM)Advisor/s: Oscar García-Prada (ICMAT-CSIC), Tamás Hausel (Institute of Science and Technology Austria, ISTA)Date: Thursday, 22 October 2026 - 16:00Place: Aula Miguel de Guzmán, Facultad de Matemáticas, Universidad Complutense de Madrid
Abstract:
Very stable Higgs bundles were introduced by Hausel and Hitchin with the motivation of identifying well-behaved complex Lagrangian subvarieties of the moduli space of Higgs bundles. Their upward flows constitute affine, irreducible, closed complex Lagrangian subvarieties which are invariant by the natural scaling action, providing simple testing grounds for mirror symmetry predictions. Moreover, they correspond to the zero-dimensional attractor spaces of fixed points in the nilpotent cone of the moduli space, playing a key role in understanding its topology. We present several results centred around the classification of these objects in the moduli space of G-Higgs bundles for any connected semisimple complex algebraic group G. On one hand, we classify very stable Higgs bundles with generically regular Higgs field, generalising the result for G=GL_n(C) of Hausel and Hitchin and relating the problem to the combinatorics of a torus action on the affine Grassmannian of G. The resulting techniques can be extended to solve a similar problem in the moduli space of strongly parabolic G-Higgs bundles, incorporating more general affine flag varieties. On the other hand, for the remaining Higgs bundles with non-regular Higgs fields, we apply recently developed techniques involving a topological invariant of the fixed point locus of the scaling action, the Toledo invariant, to rule out many of its components from containing very stable Higgs bundles. We also provide some additional results about the closed Lagrangians we identify, such as the computation of their virtual equivariant multiplicity or, in selected simple cases, the description of the Hitchin system over them and the computation of their dual brane.
