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Vincent Guedj

Publications and source records attributed to Vincent Guedj.

At least 37 records · Page 2Linked to original sources
arXiv2021

Monge-Ampère equations on compact Hessian manifolds

We consider degenerate Monge-Ampère equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Ampère operators. We then use the Perron method to solve Monge-Ampère equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delanoë, Caffarelli-Viaclovsky and Hultgren-Önnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.

math.DG↗
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

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↗
arXiv2018

The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{è}re equations, modelled on the K{ä}hler-Ricci flow evolving on complex algebraic varieties with Kawamata log-terminal singularities. Under natural assumptions on the Cauchy-Dirichlet boundary data, we show that the envelope of pluripotential subsolutions is semi-concave in time and continuous in space, and provides the unique pluripotential solution with such regularity.

math.DG↗
arXiv2018

Stability of solutions to complex Monge-Ampere flows

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

math.CV↗
arXiv2018

Pluripotential kahler-ricci flows

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log terminal singularities.

math.CV↗
arXiv2018

Toric Pluripotential Theory

We study finite energy classes of quasiplurisubharmonic (qpsh) functions in the setting of toric compact K{ä}hler manifolds. We characterize toric qpsh functions and give necessary and sufficient conditions for them to have finite (weighted) energy, both in terms of the associated convex function in R n , and through the integrability properties of its Legendre transform. We characterize Log-Lipschitz convex functions on the Delzant polytope, showing that they correspond to toric qpsh functions which satisfy a certain exponential integrability condition. In the particular case of dimension one, those Log-Lipschitz convex functions of the polytope correspond to H{ö}lder continuous toric quasisubharmonic functions.

math.CV↗
arXiv2017

Geometry and Topology of the space of Kähler metrics on singular varieties

Let $Y$ be a compact Kähler normal space and $α\in H^{1,1}(Y,\mathbb{R})$ a Kähler class. We study metric properties of the space $\mathcal{H}_α$ of Kähler metrics in $α$ using Mabuchi geodesics. We extend several results by Calabi, Chen, Darvas previously established when the underlying space is smooth. As an application we analytically characterize the existence of Kähler-Einstein metrics on $\mathbb{Q}$-Fano varieties, generalizing a result of Tian, and illustrate these concepts in the case of toric varieties.

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↗
arXiv2016

Kähler-Einstein metrics and the Kähler-Ricci flow on log Fano varieties

We prove the existence and uniqueness of Kähler-Einstein metrics on Q-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler-Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler-Ricci flow provides weak convergence independently of Perelman's celebrated estimates.

math.CV↗
arXiv2016

Choquet-Monge-Ampere Classes

We introduce and study Choquet-Monge-Ampere classes on compact Kahler manifolds. They consist of quasi-plurisubharmonic functions whose sublevel sets have small enough asymptotic Monge-Ampere capacity. We compare them with finite energy classes, which have recently played an important role in Kahler Geometry.

math.CV↗
arXiv2015

Open problems in pluripotential theory

We propose a list of open problems in pluripotential theory partially motivated by their applications to complex differential geometry. The list includes both local questions as well as issues related to the compact complex manifold setting.

math.CV↗
arXiv2014

Weak solutions to degenerate complex Monge-Ampére flows I

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the first of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Ampère flows in domains of $\C^n$.

math.CV↗
arXiv2014

Weak solutions to degenerate complex Monge-Ampére Flows II

Studying the (long-term) behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampére equations. The purpose of this article, the second of a series on this subject, is to develop a viscosity theory for degenerate complex Monge-Ampére flows on compact Kähler manifolds. Our general theory allows in particular to define and study the (normalized) Kähler-Ricci flow on varieties with canonical singularities, generalizing results of Song and Tian.

math.CV↗
arXiv2014

The metric completion of the Riemannian space of Kähler metrics

Let $X$ be a compact Kähler manifold and $\a \in H^{1,1}(X,\R)$ a Kähler class. We study the metric completion of the space $\HH_\a$ of Kähler metrics in $\a$, when endowed with the Mabuchi $L^2$-metric $d$. Using recent ideas of Darvas, we show that the metric completion $(\overline{\HH}_\a,d)$ of $(\HH_\a,d)$ is a CAT(0) space which can be identified with $\E^2(\a)$, a subset of the class $\E^1(\a)$ of positive closed currents with finite energy. We further prove, in the toric setting, that $\overline{\HH}_{\a,tor}=\E_{tor}^2(\a)$.

math.DG↗