Localized Donaldson-Thomas theory of surfaces

Understand

Let $S$ be a projective simply connected complex surface and $\mathcal{L}$ be a line bundle on $S$.

• We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of $\mathcal{L}$.
• The moduli space admits a $\mathbb{C}^*$-action induced by scaling the fibers of $\mathcal{L}$.
• We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on $S$.