Degenerate complex Hessian equations on compact Kahler manifolds

Abstract

Let (X, ω) be a compact Kähler manifold of dimension n and fix 1 ≤ m ≤ n. We prove that the total mass of the complex Hessian measure of ω-m-subharmonic functions is non-decreasing with respect to the singularity type. We then solve complex Hessian equations with prescribed singularity, and prove a Hodge index type inequality for positive currents.

Mixed Hessian inequalities and uniqueness in the class $$\mathcal {E}(X,\omega ,m)$$ E ( X , ω , m )

“…More precisely, it is not clear whether the convolution of a m-sh functions (when ω is not flat) with a smooth kernel is m-sh. Nevertheless any m-sh function can be approximated by smooth m-sh functions (see [20,21]). …”

“…It follows from [20] that for any u ∈ SH m (X, ω) there exists a decreasing sequence of smooth ω-m-sh functions on X which converges to u. Following the classical pluripotential method of Bedford and Taylor [2] one can then define the complex Hessian operator for any bounded ω-m-sh function:…”

“…Theorem 2.7 [20] Let μ be a non-negative Radon measure on X vanishing on all m-polar sets. Assume that μ(X ) = X ω n .…”

“…The variational method used in [20] (which originated from [4]) can be applied in the same way to get the following result:…”

“…This finally terminated the program for proving an analogue of the Calabi-Yau theorem in the Hessian setting. Afterwards a fruitful potential theory was established culminating in the solution of the smoothing problem for m-subharmonic functions (see [20,21]). …”