6 problems

Let $S$ be a noncompact smooth curve with smooth compactification $\overline S$, let $j:S\hookrightarrow\overline S$ be the inclusion, and let $L_{\rho}$ be the local system associ…

Let $X$ be a compact Riemann surface, let $P\subset X$ be a finite set, and let $(V,\nabla_V)$ be a semisimple holomorphic flat bundle on $X\setminus P$. Let $(H,D_V,h)$ be the tam…

Let $X$ be a compact Riemann surface with a finite set of punctures, and let $(H,D_V,h)$ be a flat holomorphic vector bundle with a tame harmonic metric on $X$ minus those puncture…

Let $\alpha_S$ be the canonical homeomorphism between the relative de Rham and Higgs moduli spaces, and assume that it is an isomorphism of differentiable fiber bundles. Denote the…

Conjecture 2. The $L^2$ Dolbeault cohomology is isomorphic to the $L^2$ Higgs cohomology; equivalently, the complexes are quasi-isomorphic:

Conjecture 1. The intersection cohomology is isomorphic to the $L^2$ de Rham cohomology