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Vincent Guedj

Publications and source records attributed to Vincent Guedj.

At least 55 records · Page 3Linked to original sources
arXiv2013

Continuous approximation of quasi-plurisubharmonic functions

Let $X$ be a compact Kähler manifold and $θ$ a smooth closed $(1,1)$-real form representing a big cohomology class $α\in H^{1,1}(X,\R)$. The purpose of this note is to show, using pluripotential and viscosity techniques, that any $θ$-plurisubharmonic function $\f$ can be approximated from above by a decreasing sequence of continuous $θ$-plurisubharmonic functions with minimal singularities, assuming that there exists a single such function.

math.CV↗
arXiv2011

Hölder continuous solutions to Monge-Ampère equations

Let $(X,ω)$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $\MAH(X,ω)$ of the complex Monge-Ampère operator acting on $ω$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej's result that measures with $L^p$-density belong to $\MAH(X,ω)$ and proving that $\MAH(X,ω)$ has the "$L^p$-property", $p>1$. We also describe accurately the symmetric measures it contains.

math.CV↗
arXiv2011

Kähler-Einstein fillings

We show that on an open bounded smooth strongly pseudoconvex subset of $\CC^{n}$, there exists a Kähler-Einstein metric with positive Einstein constant, such that the metric restricted to the Levi distribution of the boundary is conformal to the Levi form. To achieve this, we solve an associated complex Monge-Ampère equation with Dirichlet boundary condition. We also prove uniqueness under some more assumptions on the open set.

math.CV↗
arXiv2011

A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

In this article, we prove a Lichnerowicz estimate for a compact convex domain of a Kähler manifold whose Ricci curvature satisfies $\Ric \ge k$ for some constant $k>0$. When equality is achieved, the boundary of the domain is totally geodesic and there exists a nontrivial holomorphic vector field. We show that a ball of sufficiently large radius in complex projective space provides an example of a strongly pseudoconvex domain which is not convex, and for which the Lichnerowicz estimate fails.

math.DG↗
arXiv2010

Extension of plurisubharmonic functions with growth control

Suppose that $X$ is an analytic subvariety of a Stein manifold $M$ and that $φ$ is a plurisubharmonic (psh) function on $X$ which is dominated by a continuous psh exhaustion function $u$ of $M$. Given any number $c>1$, we show that $φ$ admits a psh extension to $M$ which is dominated by $cu$ on $M$. We use this result to prove that any $ω$-psh function on a subvariety of the complex projective space is the restriction of a global $ω$-psh function, where $ω$ is the Fubini-Study Kähler form.

math.CV↗
arXiv2010

Viscosity solutions to degenerate Complex Monge-Ampère equations

We develop an alternative approach to Degenerate complex Monge-Ampère equations on compact Kähler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the Kähler case a theorem due to Dinew and Zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous.

math.AG↗
arXiv2009

Dynamics of meromorphic maps with small topological degree II: Energy and invariant measure

We continue our study of the dynamics of meromorphic mappings with small topological degree on a compact Kähler surface $X$. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and admits a natural geometric description. Our hypotheses are always satisfied when $X$ has Kodaira dimension zero, or when the mapping is induced by a polynomial endomorphism of $\mathbf{C}^2$. They are new even in the birational case. We also exhibit families of mappings where our assumptions are generically satisfied and show that if counterexamples exist, the corresponding measure must give mass to a pluripolar set.

math.DS↗
arXiv2009

Dynamics of meromorphic maps with small topological degree III: geometric currents and ergodic theory

We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points are equidistributed towards this measure. This generalize results that were known in the invertible case and is, to our knowledge, one among not very many instances in which a natural invariant measure for a non-invertible dynamical system is well-understood.

math.DS↗
arXiv2009

Quasiplurisubharmonic Green functions

Given a compact Kähler manifold $X$, a quasiplurisubharmonic function is called a Green function with pole at $p\in X$ if its Monge-Ampère measure is supported at $p$. We study in this paper the existence and properties of such functions, in connection to their singularity at $p$. A full characterization is obtained in concrete cases, such as (multi)projective spaces.

math.CV↗
arXiv2009

Dynamics of meromorphic maps with small topological degree I: from cohomology to currents

We consider the dynamics of a meromorphic map on a compact kahler surface whose topological degree is smaller than its first dynamical degree. The latter quantity is the exponential rate at which its iterates expand the cohomology class of a kahler form. Our goal in this article and its sequels is to carry out a conjectural program for constructing and analyzing a natural measure of maximal entropy for each such map. Here we take the first step, converting information about the linear action of the map on cohomology to invariant currents with special geometric structure. We also give some examples and identify some additional properties of maps on irrational surfaces and of maps whose invariant cohomology classes have vanishing self-intersection.

math.CV↗
arXiv2008

Singular Kahler-Einstein metrics

We study degenerate complex Monge-Ampère equations of the form $(ω+dd^c φ)^n = e^{t φ} μ$ where $ω$ is a big semi-positive form on a compact Kähler manifold $X$ of dimension $n$, $t \in \R^+$, and $μ=fω^n$ is a positive measure with density $f\in L^p(X,ω^n)$, $p>1$. We prove the existence and unicity of bounded $ω$-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition. In case $X$ is projective and $ω=ψ^*ω'$, where $ψ:X\to V$ is a proper birational morphism to a normal projective variety, $[ω']\in NS_{\R} (V)$ is an ample class and $μ$ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation. We use these results to construct singular Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

math.AG↗
arXiv2008

Regularity of Dynamical Green Functions

For meromorphic maps of complex manifolds, ergodic theory and pluripotential theory are closely related. In nice enough situations, dynamically defined Green's functions give rise to invariant currents which intersect to yield measures of maximal entropy. `Nice enough' is often a condition on the regularity of the Green's function. In this paper we look at a variety of regularity properties that have been considered for dynamical Green's functions. We simplify and extend some known results and prove several others which are entirely new. We also give some examples indicating the limits of what one can hope to achieve in complex dynamics by relying solely on the regularity of a dynamical Green's function.

math.CV↗
arXiv2007

Domains of definition of Monge-Ampère operators on compact Kähler manifolds

Let $(X,ω)$ be a compact Kähler manifold. We introduce and study the largest set $DMA(X,ω)$ of $ω$-plurisubharmonic (psh) functions on which the complex Monge-Ampère operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all $\om$-psh functions. We prove that certain twisted Monge-Ampère operators are well defined for all $ω$-psh functions. As a consequence, any $\om$-psh function with slightly attenuated singularities has finite weighted Monge-Ampère energy.

math.CV↗
arXiv2007

The weigthed Monge-Ampère energy of quasiplurisubharmonic functions

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold $(X,ω)$. We show that the complex Monge-Ampère operator $(ω+ dd^c \cdot)^n$ is well-defined on the class ${\mathcal E}(X,ω)$ of $ω$-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class ${\mathcal E}(X,ω)$ is the largest class of $ω$-psh functions on which the Monge-Ampère operator is well-defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the Monge-Ampère operator $(ω+dd^c \cdot)^n$ on ${\mathcal E}(X,ω)$, as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular Kähler-Einstein metrics.

math.CV↗
arXiv2006

Propriétés ergodiques des applications rationnelles

This is a survey article with focus on the following problem. Given $f:X \to X$ a meromorphic endomorphism of some compact Kähler manifold $X$, construct and study - under natural numerical conditions - a canonical invariant probability measure with remarkable ergodic properties (mixing, hyperbolicity, maximal entropy, etc).

math.CV↗
arXiv2006

Hölder continuous solutions to Monge-Ampère equations

We study the regularity of solutions to complex Monge-Ampère equations $(dd^c u)^n=f dV$, on bounded strongly pseudoconvex domains $ Ω\subset \C^n$. We show, under a mild technical assumption, that the unique solution $u$ to such an equation is Hölder continuous if the boundary values $ϕ$ are Hölder continuous and the density $f$ belongs to $L^p(Ω)$ for some $p>1$. This improves previous results by Bedford-Taylor and Kolodziej.

math.CV↗