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1. Generalised parabolic Higgs bundles

    1.     In [MR3174731] Bhosle constructs moduli spaces of generalized parabolic Hitchin (GPH) pairs on a nonsingular curve $X$, together with a Hitchin map. In the case where $X$ is the normalisation of an integral projective curve $Y$ compares to the moduli space of GPH pairs on $X$ to the moduli space of Hitchin pairs on $Y$. The case considered by Bhosle is when the structure group is $GL_n(\mathbb{C})$.

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      Problem 1.1.

      Construct an analog of generalised parabolic Hitchin pairs, the corresponding moduli spaces, and the Hitchin maps, for other complex algebraic groups $G_{\mathbb{C}}$. (I.e. Define and study “$G_\mathbb{C}$-principal generalised parabolic Hitchin pairs”.)

        •     If we believe that non-abelian Hodge theory (NAHT) should work in a really robust way for Higgs pairs on nodal curves, there should be comparison theorems.

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          Problem 1.2.

          Compare generalised parabolic Higgs bundles on nodal curves a la Bhosle [MR3174731] to the work of (a) Bezrukovnikov and Kapranov (microlocal sheaves) [MR3530166] and (b) Crawley-Boevey [MR3064416] (or [arXiv:1109.2018]). Also, what does NAHT on the normalization descend to on the nodal curve?

            •     In [MR3291351] Bhosle and Parameswaran define a notion of “strong semistability” for vector bundles on a projective variety, and show that the category of such objects on a projective curve $Y$ forms a neutral Tannakian category – i.e. the category is equivalent to representations of some group $\mathcal{G}_Y$ (the “holonomy group scheme”).

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              Problem 1.3.

              Give a geometric description of the holonomy group scheme.
                  When $Y$ is an integral complex nodal curve many [MR3291351] describe many projections $\mathcal{G}_Y \to \overline{\rho(\pi_1(Y))}$, the Zariski closure of the image of the topological fundamental group under some representation $\rho:\pi_1(Y)\to GL_n(\mathbb{C})$. These maps come from certain neutral Tannakian subcategories associated to rank $n$ degree zero strongly semi stable vector bundles on $Y$.

                  Cite this as: AimPL: Singular geometry and Higgs bundles in string theory, available at http://aimpl.org/singularhiggs.