We review recent results for heterotic moduli and the Hull–Strominger system. In particular, we discuss mathematical properties of the recently derived deformation operator \(\overline{D}\) associated to the deformation complex of heterotic SU(3) solutions. We review results on Serre duality, showing that the operator has a vanishing index, and discuss a notion of Čech cohomology and a particular instance of a Dolbeault theorem for \(\overline{D}\) . Specifically, the cohomology parametrising infinitesimal deformations is isomorphic to the first Čech cohomology of an associated cochain complex. This will be useful for future research, as it provides a more algebraic handle on the heterotic moduli problem, which is useful for understanding notions of stability, geometric invariants, and enumerative geometry for the Hull–Strominger system.