
## 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### 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.

#### 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.