Harmonic metrics of generically regular semisimple Higgs bundles on noncompact Riemann surfaces (Q6084970)

The aim of this paper is to study Higgs bundles equipped with a nondegenerate symmetric pairing under the assumption that the Higgs field is generically regular semisimple. The authors prove that on any Riemann surface a generically regular semisimple Higgs bundle equipped with a nondegenerate symmetric pairing always has a harmonic metric compatible with the pairing. They study the classification of such compatible harmonic metrics in the case where the Riemann surface is the complement of a finite set \(D\) in a compact Riemann surface. In particular, they prove the uniqueness of a compatible harmonic metric if the Higgs bundle is wild and regular semisimple at each point of \(D\). The paper is organized as follows: Section 1 is an introduction to the subject and gives a statement of the results. Section 2 is devoted to harmonic bundles with a real structure. Section 3 deals with good filtered Higgs bundles with symmetric pairing. Section 4 treats the generically regular semisimple case. Appendix A is on compatibility of hermitian metric and skew-symmetric pairing, and Appendix B presents the classification of regular filtered extensions in an easy case.