Fully nonlinear parabolic equations of real forms on Hermitian manifolds

Over many decades fully nonlinear PDEs, and the complex Monge-Ampère equation in particular played a central role in the study of complex manifolds. Most previous works focused on problems that can be expressed through equations involving real ( 1 , 1 ) (1,1) forms. As many important questions, especially those linked to higher cohomology classes, in algebraic and complex geometry involve real ( p , p ) (p, p) forms for p > 1 p > 1 , there is a strong need to develop PDE techniques to study these questions. In this paper we consider a fully nonlinear equation for ( p , p ) (p, p) forms on compact Hermitian manifolds. We establish the existence of classical solutions for a large class of these equations by a parabolic approach, proving the long-time existence and convergence of solutions to the elliptic case.