Abstract：

The existence of smooth solutions to a broad class of complex Hessian type equations is related to nonlinear Nakai type criteria on intersection numbers on Kahler manifolds. Such a Nakai criteria can be interpreted as a slope stability condition analogous to the slope stability for Hermitian vector bundles over Kahler manifolds. In the present work, we initiate a program to find canonical solutions to such equations in the unstable case when the Nakai criteria fails. Conjecturally such solutions should arise as limits of natural parabolic flows and should be minimisers of the corresponding moment-map energy functionals. We implement our approach for the J-equation and the deformed Hermitian Yang-Mills equation on surfaces and some examples with symmetry. We further present the bubbling phenomena for the J-equation by constructing minimizing sequences of the moment-map energy functionals.