Stratifications on the moduli space of Higgs bundles (Q1687819)

Let \(X\) be a compact Riemann surface of genus at least \(2\). A moduli space \({\mathcal M}(r, d)\) for equivalence classes of semistable Higgs bundles of rank \(r\) and degree \(d\) over \(X\) was constructed by \textit{N. Nitsure} [Proc. Lond. Math. Soc. (3) 62, No. 2, 275--300 (1991; Zbl 0733.14005)]. The authors consider two natural stratifications on \({\mathcal M}(r, d)\): {\parindent=0.7cm\begin{itemize}\item[--] The \textsl{Shatz stratification} is defined by the Harder-Narasimhan types of the underlying vector bundles \(E\) of the semistable Higgs bundles \((E, \Phi) \in {\mathcal M}(r, d)\). \item[--] The \textsl{Białynicki-Birula stratification} is defined by the \textsl{Hodge bundles} \(\lim_{z \to 0} ( E, z \Phi )\) associated to \((E , \Phi) \in {\mathcal M}(r, d)\). These are fixed points of the natural \({\mathbb C}^*\)-action on \({\mathcal M}(r, d)\) (cf. [\textit{C. T. Simpson}, Publ. Math., Inst. Hautes Étud. Sci. 75, 5--95 (1992; Zbl 0814.32003)]). \end{itemize}} The authors show that for any \(r\), the open dense locus \[ \{ ( E, \Phi) : E \text{ a semistable vector bundle} \} \] is a common stratum of both the Shatz and Białynicki-Birula stratifications. They give another proof of a result of \textit{T. Hausel} [Geometry of Higgs bundles. Cambridge, UK: Cambridge University (PhD thesis) (1998)] that the stratifications coincide fully for \(r = 2\). They then analyse the rank \(3\) case in detail. Here the two stratifications no longer coincide: For certain Harder-Narasimhan types, the Białynicki-Birula stratum of a rank 3 Higgs bundle \(( E, \Phi )\) depends on both \(E\) and \(\Phi\). In the introduction, the authors indicate the relevance of their results to several questions on moduli of Higgs bundles. The proof of the main result includes an ingenious argument using one-parameter families of gauge transformations to obtain desirable representatives of the moduli points of \(\lim_{z \to 0} ( E, z \Phi )\) where \(E\) is an unstable vector bundle.