arXiv subjects

Chinh H. Lu

Publications and source records attributed to Chinh H. Lu.

At least 19 recordsLinked to original sources

arXiv2026

Kähler-Ricci Flow: from divisors to cusps

We study the geometric regularization of positive closed currents by the Kähler-Ricci flow on compact Kähler manifolds. In a previous work of ours, it was shown that the Kähler-Ricci flow immediately smoothes out such a current when it has zero Lelong numbers. We study here the case when $T_0$ has divisorial singularities, showing that the flow gradually replaces the latter by Poincaré type ones, providing an approximation of $T_0$ by complete Kähler metrics with bounded curvature in a Zariski open set.

math.DG↗

arXiv2025

Monge-Ampère equations with prescribed singularities on compact Hermitian manifolds

Given a compact complex manifold $X$, we study the existence and the uniqueness of weak solutions to degenerate Monge-Ampère equations on $X$ with prescribed singularities when the reference form is semipositive and big, while the right hand side is a non-pluripolar positive Radon measure. This generalizes our previous work to more general hermitian manifolds and also to the case of solutions with prescribed singularities.

math.CV↗

arXiv2025

Singular Calabi-Yau metrics

These are notes of lectures given by the first named author during the CIME Summer school Calabi-Yau varieties. We survey known results concerning the complex Monge-Ampère equations in Hermitian contexts obtained by many authors during the last fifteen years.

math.CV↗

arXiv2025

A new approach to the Monge-Ampère eigenvalue problem

We study the eigenvalue problem for the complex Monge-Ampère operator in bounded hyperconvex domains in $\C^n$, where the right-hand side is a non-pluripolar positive Borel measure. We establish the uniqueness of eigenfunctions in the finite energy class introduced by Cegrell, up to positive multiplicative constants, and provide a Rayleigh quotient type formula for computing the eigenvalue. Under a natural continuity assumption on the measure, we further show that both the eigenvalue and eigenfunctions can be obtained via an iterative procedure starting from any negative finite energy function. Our approach relies on the fine properties of plurisubharmonic envelopes, which allow a partial sublinearization of the nonlinear problem. As far as we know, this method is new, even in the linear case, and not only yields new results but also significantly simplifies existing arguments in the literature. Moreover, it extends naturally to the setting of complex Hessian operators. Finally, by translating our results from the complex Monge-Ampère setting via a logarithmic transformation, we also obtain several interesting analogues for the real Monge-Ampère operator.

math.CV↗

arXiv2025

Uniform estimates: from Yau to Kolodziej

In this note we provide a new and efficient approach to uniform estimates for solutions to complex Monge-Ampere equations, as well as for solutions to geometric PDE's that satisfy a determinantal majorization.

math.DG↗

arXiv2025

On uniqueness of solutions to complex Monge-Ampère mean field equations

We establish the uniqueness of solutions to complex Monge-Ampère mean field equations when the temperature parameter is small. In the local setting of bounded hyperconvex domains, our result partially confirms a conjecture by Berman and Berndtsson. Our approach also extends to the global context of compact complex manifolds.

math.CV↗

arXiv2024

Volumes of Bott-Chern classes

We study the volumes of transcendental and possibly non-closed Bott-Chern $(1,1)$-classes on an arbitrary compact complex manifold $X$. We show that the latter belongs to the class $\mathcal{C}$ of Fujiki if and only if it has the $\textit{bounded mass property}$ -- i.e., its Monge-Ampère volumes have a uniform upper-bound -- and there exists a closed Bott-Chern class with positive volume. This yields a positive answer to a conjecture of Demailly-Păun-Boucksom. To this end we extend to the hermitian context the notion of non-pluripolar products of currents, allowing for the latter to be merely ${\it quasi}$-${\it closed}$ and ${\it quasi}$-${\it positive}$. We establish a quasi-monotonicity property of Monge-Ampère masses, and moreover show the existence of solutions to degenerate complex Monge-Ampère equations in big classes, together with uniform a priori estimates. This extends to the hermitian context fundamental results of Boucksom-Eyssidieux-Guedj-Zeriahi.

math.DG↗

arXiv2024

Kiselman Minimum Principle and Rooftop Envelopes in Complex Hessian Equations

We initiate the study of $m$-subharmonic functions with respect to a semipositive $(1,1)$-form in Euclidean domains, providing a significant element in understanding geodesics within the context of complex Hessian equations. Based on the foundational Perron envelope construction, we prove a decomposition of $m$-subharmonic solutions, and a general comparison principle that effectively manages singular Hessian measures. Additionally, we establish a rooftop equality and an analogue of the Kiselman minimum principle, which are crucial ingredients in establishing a criterion for geodesic connectivity among $m$-subharmonic functions, expressed in terms of their asymptotic envelopes.

math.CV↗

arXiv2024

Geodesic connectivity and rooftop envelopes in the Cegrell classes

This study examines geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains in $\mathbb{C}^n$. We establish that solutions possessing comparable singularities to the complex Monge-Ampère equation are identical, affirmatively addressing a longstanding open question raised by Cegrell. This achievement furnishes the most general form of the Bedford-Taylor comparison principle within the Cegrell classes. Building on this foundational result, we explore plurisubharmonic geodesics, broadening the criteria for geodesic connectivity among plurisubharmonic functions with connectable boundary values. Our investigation also delves into the notion of rooftop envelopes, revealing that the rooftop equality condition and the idempotency conjecture are valid under substantially weaker conditions than previously established, a finding made possible by our proven uniqueness result. The paper concludes by discussing the core open problems within the Cegrell classes related to the complex Monge-Ampère equation.

math.CV↗

arXiv2023

Relative pluripotential theory on compact Kähler manifolds

Given a compact Kähler manifold, we survey the study of complex Monge-Ampère type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of Guedj--Zeriahi about the finite energy range of the complex Monge-Ampère operator.

math.CV↗

arXiv2023

Quasi-plurisubharmonic envelopes 2: Bounds on Monge-Ampère volumes

In \cite{GL21a} we have developed a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations, when the reference form is closed. This simplifying assumption was used to ensure the constancy of the volumes of Monge-Ampère measures. We study here the way these volumes stay away from zero and infinity when the reference form is no longer closed. We establish a transcendental version of the Grauert-Riemenschneider conjecture, partially answering conjectures of Demailly-Păun \cite{DP04} and Boucksom-Demailly-Păun-Peternell \cite{BDPP13}. Our approach relies on a fine use of quasi-plurisubharmonic envelopes. The results obtained here will be used in \cite{GL21b} for solving degenerate complex Monge-Ampère equations on compact Hermitian varieties.

math.CV↗

arXiv2023

Degenerate complex Hessian equations on compact Hermitian manifolds

In this note we provide uniform a priori estimates for solutions to degenerate complex Hessian equations on compact hermitian manifolds. Our approach relies on the corresponding a priori estimates for Monge-Ampère equations; it provides an extension as well as a short alternative proof to results of Dinew-Kołodziej, Kołodziej-Nguyen and Guo-Phong-Tong.

math.DG↗

arXiv2022

Plurisigned hermitian metrics

Let $(X,ω)$ be a compact hermitian manifold of dimension $n$. We study the asymptotic behavior of Monge-Ampère volumes $\int_X (ω+dd^c φ)^n$, when $ω+dd^c φ$ varies in the set of hermitian forms that are $dd^c$-cohomologous to $ω$. We show that these Monge-Ampère volumes are uniformly bounded if $ω$ is "strongly pluripositive", and that they are uniformly positive if $ω$ is "strongly plurinegative". This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at $6$-dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study $6$-dimensional solvmanifolds with trivial canonical bundle.

math.CV↗

arXiv2022

Geodesic distance and Monge-Ampère measures on contact sets

We prove a geodesic distance formula for quasi-psh functions with finite entropy, extending results by Chen and Darvas. We work with big and nef cohomology classes: a key result we establish is the convexity of the $K$-energy in this general setting. We then study Monge-Ampère measures on contact sets, generalizing a recent result by the first author and Trapani.

math.DG↗

arXiv2021

L^1 metric geometry of big cohomology classes

Suppose $(X,ω)$ is a compact Kähler manifold of dimension $n$, and $θ$ is closed $(1,1)$-form representing a big cohomology class. We introduce a metric $d_1$ on the finite energy space $\mathcal{E}^1(X,θ)$, making it a complete geodesic metric space. This construction is potentially more rigid compared to its analog from the Kähler case, as it only relies on pluripotential theory, with no reference to infinite dimensional $L^1$ Finsler geometry. Lastly, by adapting the results of Ross and Witt Nyström to the big case, we show that one can construct geodesic rays in this space in a flexible manner.

math.DG↗

arXiv2021

Quasi-plurisubharmonic envelopes 3: Solving Monge-Ampère equations on hermitian manifolds

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel \cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry. In \cite{GL21b} we have studied the behavior of Monge-Ampère volumes on hermitian manifolds. We extend here the techniques of \cite{GL21a} to the hermitian setting and use the bounds established in \cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Ampère equations on compact hermitian manifolds.

math.CV↗

arXiv2021

Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds

We develop a new approach to $L^{\infty}$-a priori estimates for degenerate complex Monge-Ampère equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. Our method allows one to obtain new and efficient proofs of several fundamental results in Kähler geometry as we explain in this article. In a sequel we shall explain how this approach also applies to the hermitian setting producing new relative a priori bounds, as well as existence results.

math.CV↗

arXiv2020

Stability and Hölder regularity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds

Let $(X,ω)$ be a compact Hermitian manifold. We establish a stability result for solutions to complex Monge-Ampère equations with right-hand side in $L^p$, $p>1$. Using this we prove that the solutions are Hölder continuous with the same exponent as in the Kähler case \cite{DDGKPZ14}. Our techniques also apply to the setting of big cohomology classes on compact Kähler manifolds.

math.CV↗