Poincaré duality for $$L^p$$ cohomology on subanalytic singular spaces

We investigate the problem of Poincaré duality for $$L^p$$ L p differential forms on bounded subanalytic submanifolds of $$\mathbb {R}^n$$ R n (not necessarily compact). We show that, when p is sufficiently close to 1 then the $$L^p$$ L p cohomology of such a submanifold is isomorphic to its singular homology. In the case where p is large, we show that $$L^p$$ L p cohomology is dual to intersection homology. As a consequence, we can deduce that the $$L^p$$ L p cohomology is Poincaré dual to $$L^q$$ L q cohomology, if p and q are Hölder conjugate to each other and p is sufficiently large.

Semi-abelian Spectral Data for Singular Fibres of the 𝖲𝖫(2,ℂ)-Hitchin System

We describe spectral data for singular fibres of the $\textsf{SL}(2,{\mathbb{C}})$-Hitchin fibration with irreducible and reduced spectral curve. Using Hecke transformations, we give a stratification of these singular spaces by fibre bundles over Prym varieties. By analysing the parameter spaces of Hecke transformations, this describes the singular Hitchin fibres as compactifications of abelian group bundles over abelian torsors. We prove that a large class of singular fibres are themselves fibre bundles over Prym varieties. As applications, we study irreducible components of singular Hitchin fibres and give a description of $\textsf{SL}(2,{\mathbb{R}})$-Higgs bundles in terms of these semi-abelian spectral data.