# What is known about Higgs bundles with sections?

Let $C$ be a complex curve. Recall that a Higgs bundle on $C$ is a vector bundle $E$ on $C$ equipped with a morphism $E \to E \otimes K_C$. The space of (stable) Higgs bundles is much studied, and is in particular known to be smooth. Moreover there is a "nonabelian Hodge theorem" giving a diffeomorphism between the moduli of Higgs bundles and a certain character variety of $\pi_1(C)$.

What is known about the moduli space of Higgs bundles with a section, i.e., the space parameterizing triples $(E, s \in \mathrm{H}^{0}(E), \phi: E\to E \otimes K_C)$ ? Is it smooth (after imposing some appropriate stability condition)? Is there an analogue of the "nonabelian Hodge theorem"?

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"..and is in particular known to be smooth": I believe this is only true when the rank and degree of the vector bundle are assumed to be coprime. (In this case, stable and semistable mean the same thing.) – user5395 Mar 10 '11 at 18:44
Do you have a reference for the diffeomorphism in the third sentence? – Peter Samuelson Mar 14 '11 at 23:44
It is a homeomorphism, and a good reference is Simpson: Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. No. 79 (1994), 47–129. and Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. No. 80 (1994), 5–79 (1995). – Richard Wentworth Apr 12 '11 at 2:08

For stable $E$ of degree $0$ there are no nonzero holomorphic sections. The interessting things must happen at non-stable bundles. As far as I know, there is a desingulaization procedure for the space of semistable holomorphic bundles $V$ similar to your situation: one takes $E=End V$ together with a holomorphic section of $E.$ Again, in the case of stable bundles there are only multiples of the identity, but for nonstable there are more endomorphisms. For details, see Tyurin#s 'red book' on vector bundles over surfaces (Quantization, Classical and Quantum Field Theory and Theta functions) and the references therein. Maybe you can apply these ideas to your situation.