Canonical representatives of complexified Kähler classes

Motivated by constructions appearing in mirror symmetry, we consider the problem of finding canonical representatives for a complexified Kähler class on a compact complex manifold. These are complex cohomology classes whose imaginary part is a Kähler class, while the real part is an arbitrary real (1,1)-class. As is often the case in complex geometry, one way to fix a representative of such a class is to impose an elliptic PDE. In this talk, I will explain why a natural choice of PDE is a coupling of the deformed Hermitian Yang-Mills equation and the constant scalar curvature equation. We will then see how to prove the existence of solutions in some special cases and talk about some obstructions to the existence of solutions. Based on arXiv:2209.14157, joint work with Jacopo Stoppa.