Stabilization of the Homotopy Groups of the Moduli Spaces of k-Higgs Bundles

Abstract

The work of Hausel proves that the Bia lynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli spaces of k-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of M k (2, d), and generalize it to M k (3, d), the moduli spaces of k-Higgs bundles of degree d, and ranks two and three respectively, over a compact Riemann surface X, using the results from the works of Hausel and Thaddeus, among other tools.

A Brief Survey of Higgs Bundles

“…Recall that here, we only announce the result, no proof is given. The reader can find the full detailed proof in [20].…”

“…We are currently working on interesting results related with the homotopy of the moduli space of Higgs bundles, in terms of stratifications in the nilpotent cone [6], and in terms of Morse Theory and Variations of Hodge Structures [21,20]. We encourage the reader to explore those results in the references above mentioned.…”

“…Research supported by Universidad de Costa Rica through Escuela de Matemática and through CIMM (Centro de Investigaciones Matemáticas y Metamatemáticas), Project 820-B8-224. This work is partly based on the Ph.D. Project [19] called "Homotopy Groups of the Moduli Space of Higgs Bundles", supported by FEDER through Programa Operacional Factores de Competitividade-COMPETE, and also supported by FCT (Fundação para a Ciência e a Tecnologia) through the projects PTDC/MAT-GEO/0675/2012 and PEst-C/MAT/UI0144/2013 with grant reference SFRH/BD/51174/2010.…”

“…The paper is organized as follows: in Section 2 we recall some preliminary facts about differential geometry, specifically about smooth vector bundles and holomorphic vector bundles; in Section 3, we present the gauge group action over the set of connections and the induced action over the holomorphic structures; in Section 4, we define the moduli space of Higgs bundles, and we present the two principal Hitchin constructions for rank two: the differential geometry flavor, and the algebraic geometry flavor in 4.2; in 4.3 we present Simpsons contribution to higher rank; in Section 5, we recall some basic facts about stratifications of the moduli space of Higgs bundles: Shatz stratification in 5.1 and Białynicki-Birula stratification in 5.2; in Section 6, we present our most recent results, those related to stratifications published in [5], and those related to homotopy published in [20].…”

Stratifications on the nilpotent cone of the moduli space of Hitchin pairs

“…In our earlier work [6,20] (see also [19,21]) we investigated the limit as z → 0 of any Higgs bundle and its relation to the Harder-Narasimhan filtration of the underlying vector bundle, in order to better understand the relation between the Bia lynicki-Birula and Shatz stratifications of the moduli space (the latter being defined by the Harder-Narasimhan type). The case of rank two had already considered by Hitchin [10], who observed that in this case the two stratifications coincide.…”