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Chinh H. Lu

Publications and source records attributed to Chinh H. Lu.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2020

Finite entropy vs finite energy

Probability measures with either finite Monge-Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Ampère measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.

math.CV↗
arXiv2020

Comparison of Monge-Ampère capacities

Let $(X,ω)$ be a compact Kähler manifold. We prove that all Monge-Ampère capacities are comparable. Using this we give an alternative direct proof of the integration by parts formula for non-pluripolar products recently proved by M. Xia.

math.CV↗
arXiv2020

Log-concavity of volume and complex Monge-Ampère equations with prescribed singularity

Let $(X,ω)$ be a compact Kähler manifold. We prove the existence and uniqueness of solutions to complex Monge-Ampère equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is dropped, and we work with general model type singularities. We state and prove our theorems in the context of big cohomology classes, however our results are new in the Kähler case as well. As an application we confirm a conjecture by Boucksom-Eyssidieux-Guedj-Zeriahi concerning log-concavity of the volume of closed positive $(1,1)$-currents. Finally, we show that log-concavity of the volume in complex geometry corresponds to the Brunn-Minkowski inequality in convex geometry, pointing out a dictionary between our relative pluripotential theory and $P$-relative convex geometry. Applications related to stability and existence of csck metrics are treated elsewhere.

math.DG↗
arXiv2019

Complex Hessian equations with prescribed singularity on compact Kähler manifolds

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$ and fix $1\leq m\leq n$. 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.

math.CV↗
arXiv2019

The metric geometry of singularity types

Let $X$ be a compact Kähler manifold. Given a big cohomology class $\{θ\}$, there is a natural equivalence relation on the space of $θ$-psh functions giving rise to $\mathcal S(X,θ)$, the space of singularity types of potentials. We introduce a natural pseudometric $d_{\mathcal {S}}$ on $\mathcal S(X,θ)$ that is non-degenerate on the space of model singularity types and whose atoms are exactly the relative full mass classes. In the presence of positive mass we show that this metric space is complete. As applications, we show that solutions to a family of complex Monge-Ampère equations with varying singularity type converge as governed by the $d_\mathcal S$-topology, and we obtain a semicontinuity result for multiplier ideal sheaves associated to singularity types, extending the scope of previous results from the local context.

math.DG↗
arXiv2018

$L^p$ metric geometry of big and nef cohomology classes

Let $(X,ω)$ be a compact Kähler manifold of dimension $n$, and $θ$ be a closed smooth real $(1,1)$-form representing a big and nef cohomology class. We introduce a metric $d_p, p\geq 1$, on the finite energy space $\mathcal{E}^p(X,θ)$, making it a complete geodesic metric space.

math.DG↗
arXiv2018

Pluripotential Theory and Convex Bodies: Large Deviation Principle

We continue the study in a previous work in the setting of weighted pluripotential theory arising from polynomials associated to a convex body $P$ in $({\bf R}^+)^d$. Our goal is to establish a large deviation principle in this setting specifying the rate function in terms of $P-$pluripotential-theoretic notions. As an important preliminary step, we first give an existence proof for the solution of a Monge-Ampère equation in an appropriate finite energy class. This is achieved using a variational approach.

math.CV↗
arXiv2018

Regularity of weak minimizers of the K-energy and applications to properness and K-stability

Let $(X,ω)$ be a compact Kähler manifold and $\mathcal H$ the space of Kähler metrics cohomologous to $ω$. If a cscK metric exists in $\mathcal H$, we show that all finite energy minimizers of the extended K-energy are smooth cscK metrics, partially confirming a conjecture of Y.A. Rubinstein and the second author. As an immediate application, we obtain that existence of a cscK metric in $\mathcal H$ implies J-properness of the K-energy, thus confirming one direction of a conjecture of Tian. Exploiting this properness result we prove that an ample line bundle $(X,L)$ admitting a cscK metric in $c_1(L)$ is $K$-polystable.

math.DG↗
arXiv2018

Quantization in geometric pluripotential theory

The space of Kähler metrics can, on the one hand, be approximated by subspaces of algebraic metrics, while, on the other hand, can be enlarged to finite-energy spaces arising in pluripotential theory. The latter spaces are realized as metric completions of Finsler structures on the space of Kähler metrics. The former spaces are the finite-dimensional spaces of Fubini--Study metrics of Kähler quantization. The goal of this article is to draw a connection between the two. We show that the Finsler structures on the space of Kähler potentials can be quantized. More precisely, given a Kähler manifold polarized by an ample line bundle we endow the space of Hermitian metrics on powers of that line bundle with Finsler structures and show that the resulting path length metric spaces recover the corresponding metric completions of the Finsler structures on the space of Kähler potentials. This has a number of applications, among them a new approach to the rooftop envelopes and Pythagorean formulas of Kähler geometry, a new Lidskii type inequality on the space of Kähler metrics, and approximation of finite energy potentials, as well as geodesic segments by the corresponding smooth algebraic objects.

math.DG↗
arXiv2018

Monotonicity of non-pluripolar products and complex Monge-Ampère equations with prescribed singularity

We establish the monotonicity property for the mass of non-pluripolar products on compact Kahler manifolds, and we initiate the study of complex Monge-Ampere type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kahler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for non-pluripolar products with small unbounded locus.

math.DG↗
arXiv2017

Weak subsolutions to complex Monge-Ampère equations

We compare various notions of weak subsolutions to degenerate complex Monge-Ampère equations, showing that they all coincide. This allows us to give an alternative proof of mixed Monge-Ampère inequalities due to Kolodziej and Dinew.

math.CV↗
arXiv2017

Plurisubharmonic envelopes and supersolutions

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact Kähler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge-Ampère equation. We use these ideas to solve complex Monge-Ampère equations by taking lower envelopes of super-solutions.

math.CV↗
arXiv2017

On the singularity type of full mass currents in big cohomology classes

Let $X$ be a compact Kähler manifold and $\{θ\}$ be a big cohomology class. We prove several results about the singularity type of full mass currents, answering a number of open questions in the field. First, we show that the Lelong numbers and multiplier ideal sheaves of $θ$-plurisubharmonic functions with full mass are the same as those of the current with minimal singularities. Second, given another big and nef class $\{η\}$, we show the inclusion $\mathcal{E}(X,η) \cap {PSH}(X,θ) \subset \mathcal{E}(X,θ).$ Third, we characterize big classes whose full mass currents are "additive". Our techniques make use of a characterization of full mass currents in terms of the envelope of their singularity type. As an essential ingredient we also develop the theory of weak geodesics in big cohomology classes. Numerous applications of our results to complex geometry are also given.

math.DG↗
arXiv2016

From the Kähler-Ricci flow to moving free boundaries and shocks

We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilton-Jacobi equations and hyperbolic conservation laws (notably, in the adhesion model in cosmology). In particular we show how to recover the Hele-Shaw flow (Laplacian growth) of growing 2D domains from the Ricci flow. As we briefly indicate the scaling limit in question arises as the zero-temperature limit of a certain many particle system on X.

math.DG↗