Fully nonlinear elliptic equations for conformal deformations of Chern–Ricci forms

Abstract

We study a general class of fully nonlinear elliptic equations of second order on Hermitian manifolds. We derive a priori estimates and prove some existence results for these equations. In particular, we prove interior estimates for the complex Hessian, which in general do not hold for other types of equations. We apply our results to draw geometric conclusions on conformal deformations of the mixed Chern–Ricci forms, a notion we introduced in the article. Our work is motivated by the close connections of these equations to problems in non-Kähler geometry, and the fact that there have been increasing interests in fully nonlinear pde's beyond the complex Monge–Ampère equation from complex geometry.