Twistor space for local systems on an open curve (Q6573210)

The moduli space of solutions to the \(\mathrm{SU}(2)\)-self-duality equations over a smooth compact Riemann surface \(\Sigma\) was constructed by Hitchin as a hyperkähler manifold. Its twistor space, in terms of Penrose theory, was interpreted by Deligne as a space obtained by the glueing of the Hodge moduli spaces of \(\lambda\)-connections over \(\Sigma\) and over \(\bar{\Sigma}\) via the monodromy representation. The construction of a Deligne-Hitchin moduli space of \(\lambda\)-connections over smooth quasi-projective curves is the main concern of this article.\N\NLet \(X\) be a smooth quasi-projective curve with basepoint \(x \in X\). In his preceding work [Proc. Indian Acad. Sci., Math. Sci. 132, No. 2, Paper No. 54, 26 p. (2022; Zbl 1535.14030)], the author constructed a Deligne-Hitchin moduli space for parabolic bundles of rank 2 as a quotient by the Hecke gauge groupoid allowing permutation of orderings of the filtration steps at points where the eigenvalues are distinct. Since there are only two eigenvalues to consider in this case, one can freely change the order of the eigenspaces in the parabolic filtration.\N\NIn the present article, the author extends the ideas considered in the rank 2 case to any rank. The first main step in this course involves the appropriate definition of the Hecke gauge groupoid. For higher rank parabolic bundles when there are more than two eigenvalues, the idea in the definition is to say that two eigenspaces may be permuted in the ordering of the filtrations, whenever they have distinct residual eigenvalues. For certain genericity hypotheses, a tame harmonic bundle over \(X\) yields a well-defined preferred section\N\[\N\rho: \mathbb{P}^1 \to \left( \tilde{\mathcal{M}}_{\text{DH}}(X,x), \mathcal{G}_{\text{DH}}\right)\N\]\Nand another major step is to construct a separated analytic neighborhood of the preferred section \(\rho\) within the non-separated moduli space groupoid. Provided this separated and smooth local neighborhood, as well as the mixed twistor structure on the normal bundle, then the mixed twistor structure on the formal completion follows.