Minimal slopes and bubbling for complex Hessian equations

TL;DR

This paper investigates the existence and uniqueness of canonical solutions with singularities for complex Hessian equations, especially in unstable cases, using geometric flows and analyzing bubbling phenomena on Kahler surfaces.

Contribution

It introduces a new approach to find canonical solutions in unstable cases of complex Hessian equations, including the J-equation and deformed Hermitian Yang-Mills equation, via limits of geometric flows.

Findings

Existence of unique canonical solutions on Kahler surfaces in unstable cases.

Canonical solutions can be obtained as limits of geometric flows such as the J-flow.

Bubbling phenomena lead to singular algebraic spaces as Gromov-Hausdorff limits.

Abstract

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.