Henri Guenancia

Kahler-Einstein Metrics on Stable Varieties and log Canonical Pairs

Let X be a canonically polarized variety, i.e. a complex projective variety such that its canonical class KX defines an ample

KÄHLER–EINSTEIN METRICS WITH CONE SINGULARITIES ON KLT PAIRS

{mathbb{Q}}

Families of conic Kähler–Einstein metrics

Q -line bundle, and satisfying the conditions G1 and S2. Our main result says that X admits a Kähler–Einstein metric iff X has semi-log canonical singularities i.e. iff X is a stable variety in the sense of Kollár–Shepherd-Barron and Alexeev (whose moduli spaces are known to be compact). By definition a Kähler–Einstein metric in this singular context simply means a Kähler–Einstein on the regular locus of X with volume equal to the algebraic volume of KX, i.e. the top intersection number of KX. We also show that such a metric is uniquely determined and extends to define a canonical positive current in c1(KX). Combined with recent results of Odaka our main result shows that X admits a Kähler–Einstein metric iff X is K-stable, which thus confirms the Yau–Tian–Donaldson conjecture in this general setting of (possibly singular) canonically polarized varieties. More generally, our results are shown to hold in the setting of log minimal varieties and they also generalize some prior results concerning Kähler–Einstein metrics on quasi-projective varieties.

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

In the first part of this paper, we study the properties of some particular plurisubharmonic functions, namely the toric ones. The main result of this part is a precise description of their multiplier ideal sheaves, which generalizes the algebraic case studied by Howald. In the second part, almost entirely independent of the first one, we generalize the notion of the adjoint ideal sheaf used in algebraic geometry to the analytic setting. This enables us to give an analogue of Howald’s theorem for adjoint ideals attached to monomial ideals. Finally, using the local Ohsawa–Takegoshi–Manivel theorem, we prove the existence of the so-called generalized adjunction exact sequence, which enables us to recover a weak version of the global extension theorem of Manivel, for compact Kähler manifolds.

On the boundary behavior of Kähler–Einstein metrics on log canonical pairs

We prove that any Kahler–Einstein metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.

A decomposition theorem for smoothable varieties with trivial canonical class

Abstract In this note, we prove that on a compact Kähler manifold X hskip-0.569055pt{X}hskip-0.569055pt carrying a smooth divisor D such that K X + D {K_{X}+D} is ample, the Kähler–Einstein cusp metric is the limit (in a strong sense) of the Kähler–Einstein conic metrics when the cone angle goes to 0. We further investigate the boundary behavior of those and prove that the rescaled metrics converge to a cylindrical metric on ℂ * × ℂ n - 1 {mathbb{C}^{*}timesmathbb{C}^{n-1}} .

Quasi-projective manifolds with negative holomorphic sectional curvature.

Let