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Solution to Hessian type equations with prescribed singularity on compact Kahler manifold

Genglong Lin

TL;DR

The paper extends relative pluripotential theory to the Hessian setting on a compact Kähler manifold, developing a robust framework for Hessian equations with prescribed singularities. It introduces relative full mass classes, rooftop envelopes, and an integration-by-parts formalism to study $\omega$-$m$-subharmonic functions and the Hessian operator $H_m$. A key contribution is establishing the relative finite energy class and proving that, under natural integrability conditions on the right-hand side, solutions to $H_m(u)=F(x,u)\omega^n$ exist within a prescribed singularity class and enjoy relative boundedness. Additionally, the authors characterize the finite energy range of the Hessian operator, linking measures representable as Hessian products to energy-control conditions via capacity, thereby bridging variational methods with capacity techniques in the Hessian context.

Abstract

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix an integer $m$ such that $1\leq m\leq n$. We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}. Given a model potential $φ$, we also study degenerate complex Hessian equations of the form $(ω+dd^c \varphi)^m\wedgeω^{n-m}=F(x,\varphi)ω^n$. Under some natrual conditions on $F$, we prove that the solution of this type equation has the same singularity type as $φ$.

Solution to Hessian type equations with prescribed singularity on compact Kahler manifold

TL;DR

The paper extends relative pluripotential theory to the Hessian setting on a compact Kähler manifold, developing a robust framework for Hessian equations with prescribed singularities. It introduces relative full mass classes, rooftop envelopes, and an integration-by-parts formalism to study --subharmonic functions and the Hessian operator . A key contribution is establishing the relative finite energy class and proving that, under natural integrability conditions on the right-hand side, solutions to exist within a prescribed singularity class and enjoy relative boundedness. Additionally, the authors characterize the finite energy range of the Hessian operator, linking measures representable as Hessian products to energy-control conditions via capacity, thereby bridging variational methods with capacity techniques in the Hessian context.

Abstract

Let be a compact Kähler manifold of dimension and fix an integer such that . We reformulate most relative pluripotential results of Darvas-DiNezza-Lu's survey \cite{DNL23} to the Hessian setting. As an application, we use a slightly different method and give an characterization of finite energy range of the Hessian operator, which cannot be directly reformulated by \cite{DNL23}. Given a model potential , we also study degenerate complex Hessian equations of the form . Under some natrual conditions on , we prove that the solution of this type equation has the same singularity type as .
Paper Structure (11 sections, 44 theorems, 95 equations)

This paper contains 11 sections, 44 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relative m-potential theory
  4. Quasi-$m$-sh envelopes
  5. Monotonicity of Hessian product masses
  6. Model potentials and relative full mass classes
  7. Integration by parts
  8. Complex Hessian type equations with prescribed singularity
  9. The relative finite energy class
  10. Relative boundedness of solution
  11. Finite energy range of the Hessian operator

Key Result

Proposition 2.6

Let $U\subset\mathbf{C}^n$ be an open set. Suppose $\{f_j\}_j$ are uniformly bounded quasi-continuous functions which converge in capacity to another quasi-continuous function $f$ on $U$. Let $\{u_1^j\}_j,...,\{u_m^j\}_j$ be uniformly bounded $m$-sh functions on $\Omega$, converging in $m$-capacity where $\beta$ is the standard Kähler form of $\mathbf{C}^n$

Theorems & Definitions (84)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Theorem 3.1
  • ...and 74 more