  The existence of smooth solutions to a broad class of complex Hessian
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
prove that there always exist unique canonical solutions to these two equations
on Kahler surfaces in the unstable cases. Such canonical solutions with
singularities are also shown to be the limits of the corresponding J-flow and
the cotangent flow on certain projective bundles. We further present the
bubbling phenomena for the J-equation by constructing minimizing sequences of
the moment-map energy functionals, whose Gromov-Hausdorff limits are singular
algebraic spaces.
