Toric plurisubharmonic functions and analytic adjoint ideal sheaves

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 K\"ahler manifolds.Comment: 24 pages, v2: A minor error fixed in the proof of Theorem 2.13, Two errors partially fixed: coherence of the adjoint ideal needs another assumption (Cor 2.19), Nadel-vanishing with I_+ stated on a compact manifold only (Prop. 2.21 & Cor. 2.23

A decomposition theorem for smoothable varieties with trivial canonical class

In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, \'etale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.Comment: 21 page

Metrics with cone singularities along normal crossing divisors and holomorphic tensor fields

We prove the existence of non-positively curved K\"ahler-Einstein metrics with cone singularities along a given simple normal crossing divisor on a compact K\"ahler manifold, under a technical condition on the cone angles, and we also discuss the case of positively-curved K\"ahler-Einstein metrics with cone singularities. As an application we extend to this setting classical results of Lichnerowicz and Kobayashi on the parallelism and vanishing of appropriate holomorphic tensor fields.Comment: 36 pages, v3: added a section on the log-Fano case. To appear in Annales Scientifiques de l'EN

Degenerating K\"ahler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $\beta\to 0$ of K\"ahler-Einstein metrics $\omega_\beta$ with a cone singularity of angle $2\pi \beta$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $\omega_\beta$ converges to the locally symmetric metric and further give asymptotics of $\omega_\beta$ when $X\setminus D$ is a ball quotient. Our second result deals with the case when $X$ is Fano and $D$ is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of $\omega_\beta$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,\omega_\beta)$ converges to an interval in the Gromov-Hausdorff sense.Comment: 51 pages, v2: exposition improved following the referee's suggestions, to appear in Invent. Mat