Higgs Triples and Ruled Surfaces

Aiming to understand complexes of coherent sheaves on algebraic Poisson surfaces and the associated deformation quantizations and moduli problems, we begin our study by examining the case of ruled surfaces over a smooth projective curve $X$, namely the Poisson surface will be $S=\mathbb{P}(\mathcal{O}\oplus\omega)$, where $\omega$ is the canonical line bundle of $X$. Fixing a vector bundle $F\to X$, after revisiting the background technology of \textsl{spectral data and Higgs bundles} we aim to encode $(D,\,F)$-framed sheaves on $S$ as a form of \textsl{extended Higgs data} [Chapter 3], i.e. Higgs triples, as introduced by A. Minets , and $F$-prolonged Higgs bundles. We present our first main result, demonstrating the correspondence between pure $F$-prolonged Higgs bundles on $X$ and $(D,\,F)$-framed torsion free sheaves on $S$, globally generated along the fibers of the natural projection. Moreover, exploring the close relation between the two types of extended Higgs data, we aim to place them in the context of perverse coherent sheaves on $X$ and examine the stability of the Higgs data as a polynomial stability in the sense of Bayer \cite{bayer}. So, using the polynomial stability given by the dual to the large volume perversity, we recover the notion of stability for Higgs triples as introduced by Minets, but also derive a stability condition for (pure) $F$-prolonged Higgs bundles, so the stable objects correspond to Huybrechts-Lehn stable $(D,\, F)$-framed torsion free sheaves