L2 theory on pseudoconcave domains in CP^n

Determine whether, for a bounded pseudoconvex domain Ω ⊂ CP^n with smooth boundary and its pseudoconcave complement Ω^+ = CP^n \overline{Ω}, the L^2 Dolbeault cohomology groups H^{p,q}_{L^2}(Ω^+) vanish for all degrees with p ≠ q and q < n − 1.

Background

The paper develops L^2 and Sobolev theory for the ∂̄-operator on domains in complex projective space CP^n, proving several solvability and regularity results on pseudoconvex and pseudoconcave settings. While weighted L^2 methods and Sobolev estimates are established in various configurations, a complete L^2 theory on pseudoconcave domains has not been settled.

The authors explicitly highlight that the L^2 theory for pseudoconcave domains, even with smooth boundaries, remains unresolved and point to a specific problem (Problem L2 pseudoconcave) that asks for the vanishing of L^2 Dolbeault cohomology for the complement Ω^+ of a pseudoconvex domain Ω.