Vincent Guedj

Intrinsic capacities on compact Kähler manifolds

We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally “quasi-pluripolar.”

Viscosity Solutions to Degenerate Complex Monge-Ampère Equations

We develop an alternative approach to Degenerate complex Monge-Ampere equations on compact Kahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the Kahler 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.

Dynamics of polynomial automorphisms of Ck

We study the dynamics of polynomial automorphisms ofCk. To an algebraically stable automorphism we associate positive closed currents which are invariant underf, consideringf as a rational map onPk. These currents give information on the dynamics and allow us to construct a canonical invariant measure which is shown to be mixing.

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

We continue our study of the dynamics of meromorphic mappings with small topological degree ?2(f)<?1(f) on a compact Kahler surface X. Under general hypotheses we are able to construct a canonical invariant measure which is mixing, does not charge pluripolar sets and has 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 C2. They are new even in the birational case (?2(f)=1). 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.

Extension of plurisubharmonic functions with growth control

Abstract 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.

Quasiplurisubharmonic Green functions

Abstract Given a compact Kahler manifold X, a quasiplurisubharmonic function is called a Green function with pole at p ∈ X if its Monge–Ampere 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.

Weak solutions to degenerate complex Monge–Ampère 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–Ampère 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

An introduction to the Kähler-Ricci flow

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