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9. Other problems

This is the section for problems which (a) were not multi-part and (b) I had fewer notes on.

    1. Simplify the fibres of the Hodge bundle.


      Add remark

      Problem 9.1.

      Seek notion of genericity for complex structures simplifying the stratification of the singular locus of the Hitchin fibration (for $SL_2(\mathbb{C})$).

        • Relation to Hilbert schemes of points.


          Add remark

          Problem 9.2.

          Better understand the relationship between the moduli of Higgs bundles on $C$ and the Hilbert scheme of points $\text{Hilb}^{\text{pt}}(T^*C)$.

            • Compactifications of the moduli of Higgs bundles.


              Add remark

              Problem 9.3.

              Compare compactifications of the moduli space of Higgs bundles for $GL_n(\mathbb{C})$ of Hausel, Schmitt, and MSWW-F.

                • Deformations from Hochschild homology.


                  Add remark

                  Problem 9.4.

                  Give a geometric realisation of objects given by deforming a Hitchin pair by a non-zero class in $HH_0(C)$ (c.f. Keller).

                    • Branes in the moduli of Higgs bundles.


                      Add remark

                      Problem 9.5.

                      Review techniques for constructing Lagrangians in hyperkahler spaces, see which can be applied to the moduli space of Higgs bundles, and explicitly translate/apply those techniques.

                        • $tt^*$-equations and Higgs bundles.


                          Add remark

                          Problem 9.6.

                          Make explicit the relationship between the $tt^*$-equations and Higgs bundles in a specific example.

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