General Dolbeault-type theorem for higher degrees

Prove a general Dolbeault-type theorem establishing an isomorphism H^{0,p}_{\bar D}(Q) \cong \check H^p(Q) for degrees beyond p = 1, thereby extending the p = 1 identification between \bar D-cohomology and Čech cohomology to all higher degrees in the heterotic SU(3) setting.

Background

The paper constructs a Čech-theoretic framework adapted to the non-standard operator \bar D and proves a Dolbeault-type theorem identifying H^{0,1}_{\bar D}(Q) with \check H^1(Q).

Extending this isomorphism to higher degrees would provide a unified algebraic-topological tool for computing \bar D-cohomology and thereby analyzing deformations and obstructions at all orders.