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Monge-Ampere type equations on compact Hermitian manifolds with bounded mass property

Xuan Li

TL;DR

This work advances complex Monge–Ampère theory on compact Hermitian manifolds under the bounded mass property by developing a Hermitian relative pluripotential framework. It introduces rooftop envelopes and relative full-mass classes to extend non-pluripolar MA analysis beyond closed Kähler forms, and proves existence of solutions to MA equations with prescribed singularities and non-pluripolar RHS. A key novelty is the Hermitian extension of the Darvas–Di Nezza–Lu distance on model singularities, enabling stability results as singularities vary. The paper also provides energy estimates, a supersolution-based construction, and a range characterization for Monge–Ampère operators, highlighting convergence in capacity and robustness of the Hermitian theory with practical implications for singular metric problems.

Abstract

In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the existence of solutions to complex Monge-Ampere type equations with prescribed singularities, allowing for non-pluripolar measures on the right-hand side. We also obtain stability results when singularity types vary, by extending the Darvas-Di Nezza-Lu distance to the Hermitian context.

Monge-Ampere type equations on compact Hermitian manifolds with bounded mass property

TL;DR

This work advances complex Monge–Ampère theory on compact Hermitian manifolds under the bounded mass property by developing a Hermitian relative pluripotential framework. It introduces rooftop envelopes and relative full-mass classes to extend non-pluripolar MA analysis beyond closed Kähler forms, and proves existence of solutions to MA equations with prescribed singularities and non-pluripolar RHS. A key novelty is the Hermitian extension of the Darvas–Di Nezza–Lu distance on model singularities, enabling stability results as singularities vary. The paper also provides energy estimates, a supersolution-based construction, and a range characterization for Monge–Ampère operators, highlighting convergence in capacity and robustness of the Hermitian theory with practical implications for singular metric problems.

Abstract

In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the existence of solutions to complex Monge-Ampere type equations with prescribed singularities, allowing for non-pluripolar measures on the right-hand side. We also obtain stability results when singularity types vary, by extending the Darvas-Di Nezza-Lu distance to the Hermitian context.
Paper Structure (14 sections, 50 theorems, 199 equations)

This paper contains 14 sections, 50 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Non-pluripolar products and quasi-psh envelopes
  4. Upper and lower Monge-Ampère volumes
  5. The Relative Full Mass Classes
  6. Existence of the Rooftop Envelope
  7. Several useful criteria
  8. A semi-metric on the space of model singularity types
  9. Monge-Ampère Equations in the Full Mass Class
  10. Energy estimates
  11. Existence of solutions
  12. Monge-Ampère Equations with Prescribed Singularity Type
  13. Solving complex Monge-Ampère type equations
  14. Characterization of the range of the Monge-Ampère operators

Key Result

Theorem 1.1

Assume $\overline{\mathop{\mathrm{vol}}}(\omega_X)<+\infty$ and $\underline{\mathop{\mathrm{vol}}}(\omega_X)>0$. Let $\theta$ be a big form and $\phi$ a $\theta$-psh model potential. Let $\mu$ be a positive Radon measure which does not charge pluripolar sets. Then, (i) there exists a function $\varp (ii) for any $\lambda > 0$, there exists a unique $\varphi \in \mathcal{E}(X, \theta,\phi)$ such th

Theorems & Definitions (91)

  • Theorem 1.1: (Theorem \ref{['MA_prescribed']} and Theorem \ref{['MA_pre_0']})
  • Theorem 1.2: (Theorem \ref{['infinity_energy']})
  • Theorem 1.3: (Theorem \ref{['1-ener_thm']})
  • Proposition 1.1: (Proposition \ref{['general_asl']})
  • Theorem 1.4: (Theorem \ref{['thm_sum']})
  • Corollary 1.1: (Corollary \ref{['cor_6.1']})
  • Theorem 1.5: (Theorem \ref{['cauchy']})
  • Theorem 1.6: (Theorem \ref{['model_capacity']})
  • Corollary 1.2: (Corollary \ref{['cor_6.2']})
  • Definition 2.1
  • ...and 81 more