Stratifications on the moduli space of Higgs bundles

Abstract

The moduli space of Higgs bundles has two stratifications. The Bia lynicki-Birula stratification comes from the action of the non-zero complex numbers by multiplication on the Higgs field, and the Shatz stratification arises from the Harder-Narasimhan type of the vector bundle underlying a Higgs bundle. While these two stratifications coincide in the case of rank two Higgs bundles, this is not the case in higher rank. In this paper we analyze the relation between the two stratifications for the moduli space of rank three Higgs bundles.Let X be a closed Riemann surface of genus g and let K = K X = T * X be the canonical line bundle of X. Definition 2.1. A Higgs bundle over X is a pair (E, Φ) where the underlying vector bundle E → X is a holomorphic vector bundle and the Higgs field Φ : E → E ⊗ K is a holomorphic endomorphism of E twisted by K.

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

“…Hausel [16] estimates the homotopy groups π n (M k (2, 1)) using two main tools: first the coincidence mentioned before between the Bia lynicki-Birula stratification and the Shatz stratification; and second, the well-behaved embeddings M k (2, 1) ֒→ M k+1 (2,1). These inclusions are also well-behaved in general for GCD(r, d) = 1; nevertheless, those two stratifications above mentioned do not coincide in general (see for instance [11]).…”

A Brief Survey of Higgs Bundles

“…Recall that here, we only announce the results, no formal proofs are given. The reader can find the detailed proofs in [5]. First, we will present the bounds on Harder-Narasimhan types for rank three Higgs bundles.…”

“…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].…”

“…. Strictly speaking, we should use the saturated bundle of the sheaf I = φ 21 (E 1 ) ⊗ K −1 as in [5]. We abuse of notation here.…”

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.…”