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Henri Guenancia

Publications and source records attributed to Henri Guenancia.

At least 19 recordsLinked to original sources
arXiv2026

Stable Degeneration, Non-degenerate Forms, and Kaledin's Conjecture

We prove that stable degeneration, the canonical degeneration associated to the normalized volume minimizer of a Kawamata log terminal (klt) singularity, preserves non-degenerate reflexive differential forms. In particular, the stable degeneration of a symplectic singularity is again symplectic. Combining this with a deformation-theoretic rigidity result for symplectic degenerations, we confirm Kaledin's conjecture that the formal completion of any symplectic singularity is conical. As applications, we show that the natural base of any normalized nilpotent orbit closure is a K-semistable Fano variety, and that the normalized volume minimizer of a hypertoric singularity is induced by the standard dilation.

math.AG↗
arXiv2026

Families of singular Kähler-Einstein metrics

Refining Yau's and Kolodziej's techniques, we establish very precise uniform a priori estimates for degenerate complex Monge-Ampère equations on compact Kähler manifolds, that allow us to control the blow up of the solutions as the cohomology class and the complex structure both vary. We apply these estimates to the study of various families of possibly singular Kähler varieties endowed with twisted Kähler-Einstein metrics, by analyzing the behavior of canonical densities, establishing uniform integrability properties, and developing the first steps of a pluripotential theory in families. This provides interesting information on the moduli space of stable varieties, extending works by Berman-Guenancia and Song, as well as on the behavior of singular Ricci flat metrics on (log) Calabi-Yau varieties, generalizing works by Rong-Ruan-Zhang, Gross-Tosatti-Zhang, Collins-Tosatti and Tosatti-Weinkove-Yang.

math.CV↗
arXiv2026

A complex analytic approach to orbifold Chern classes on singular varieties and its applications

In this article, we prove the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves on Kähler varieties, generalizing our earlier work \cite{GP25} in dimension three. We also provide a characterization of the equality case, a new purely analytical proof of the numerical characterization of complex torus quotients as well as a novel, complex analytic interpretation of the second orbifold Chern class associated to a $\mathbb Q$-sheaf.

math.AG↗
arXiv2025

Kähler-Ricci flows coming out of metric spaces

Given a compact Kähler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact Kähler manifolds either in a "static" way or at time zero of smooth Kähler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a Kähler-Ricci flow.

math.DG↗
arXiv2025

Beauville-Bogomolov decomposition for klt varieties

These lecture notes present a mostly self-contained proof of the singular version of Beauville-Bogomolov decomposition theorem for compact Kähler varieties with log terminal singularities and zero first Chern class.

math.AG↗
arXiv2025

A note on orbifold regularity of canonical metrics

In this short note, we prove that on a compact Kähler variety $X$ with log terminal singularities and $c_1(X)=0$, any singular Ricci-flat Kähler metric has orbifold singularities in restriction to the orbifold locus of $X$.

math.DG↗
arXiv2025

Log Calabi--Yau manifolds: holomorphic tensors, stability and universal cover

We study various geometric properties of log Calabi-Yau manifolds, i.e. log smooth pairs $(X,D)$ such that $K_X+D=0$. More specifically, we focus on the two cases where $X$ is a Fano manifold and $D$ is either smooth or has two proportional components. Despite the existence of a complete Ricci flat Kähler metric on $X\setminus D$ in both cases, we will show that the geometric properties of the pair $(X,D)$ are vastly different, e.g. validity of Bochner principle, local triviality of the quasi-Albanese map, polystability of $T_X(-\log D)$ and compactifiability of the universal cover of $X\setminus D$. When $D$ has two components we show that the universal cover of $X\setminus D$ is a Calabi-Yau manifold of infinite topological type, and we describe the geometry at infinity from a Riemannian point of view.

math.AG↗
arXiv2025

Degenerating conic Kähler-Einstein metrics to the normal cone

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1α(-K_X)$ with $α>1$. It is well-known that Kähler-Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2πβ$ is bigger than some positive limit value $2πβ_*$. Under the hypothesis that the automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we prove that for $β>β_*$ close enough to $β_*$, such Kähler-Einstein metrics do exist. We identify the limits at various scales when $β\rightarrowβ_*$ and, in particular, we exhibit the appearance of the Tian-Yau metric of $X\setminus D$.

math.DG↗
arXiv2024

Geometry of $K$-trivial Moishezon manifolds : decomposition theorem and holomorphic geometric structures

Let $X$ be a compact complex manifold such that its canonical bundle $K_X$ is numerically trivial. Assume additionally that $X$ is Moishezon or $X$ is Fujiki with dimension at most four. Using the MMP and classical results in foliation theory, we prove a Beauville-Bogomolov type decomposition theorem for $X$. We deduce that holomorphic geometric structures of affine type on $X$ are in fact locally homogeneous away from an analytic subset of complex codimension at least two, and that they cannot be rigid unless $X$ is an étale quotient of a compact complex torus. Moreover, we establish a characterization of torus quotients using the vanishing of the first two Chern classes which is valid for any compact complex $n$-folds of algebraic dimension at least $n-1$. Finally, we show that a compact complex manifold with trivial canonical bundle bearing a rigid geometric structure must have infinite fundamental group if either $X$ is Fujiki, $X$ is a threefold, or $X$ is of algebraic dimension at most one.

math.DG↗
arXiv2023

Diameter of Kähler currents

We establish upper bounds on the diameter of compact Kähler manifolds endowed with Kähler metrics whose volume form satisfies an Orlicz integrability condition. Our results extend previous estimates due to Fu-Guo-Song, Y.Li, and Guo-Phong-Song-Sturm. In particular, they do not involve any constraint on the vanishing of the volume form. Moreover, we show that singular Kähler-Einstein currents have finite diameter, provided that their local potentials are Hölder continuous.

math.DG↗
arXiv2023

Strict positivity of Kähler-Einstein currents

Kähler-Einstein currents, also known as singular Kähler-Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces $X$ and their two defining properties are the following: they are genuine Kähler-Einstein metrics on $X_{\rm reg}$ and they admit local bounded potentials near the singularities of $X$. In this note we show that these currents dominate a Kähler form near the singular locus, when either $X$ admits a global smoothing, or when $X$ has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if $X$ is any compact Kähler space of dimension $3$ with log terminal singularities, then any singular Kähler-Einstein metric of non-positive curvature dominates a Kähler form.

math.CV↗
arXiv2022

On subvarieties of singular quotients of bounded domains

Let $X$ be a quotient of a bounded domain in $\mathbb C^n$. Under suitable assumptions, we prove that every subvariety of $X$ not included in the branch locus of the quotient map is of log general type in some orbifold sense. This generalizes a recent result by Boucksom and Diverio, which treated the case of compact, étale quotients. Finally, in the case where $X$ is compact, we give a sufficient condition under which there exists a proper analytic subset of $X$ containing all entire curves and all subvarieties not of general type (meant this time in in the usual sense as opposed to the orbifold sense).

math.AG↗
arXiv2022

Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

Let $X$ be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of $X$: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover $X$ splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of $X$ according to its holonomy representation. In particular, we classify those $X$ which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi--Yau or irreducible holomorphic symplectic. As an application of these results, we show that if $X$ has dimension four, then it satisfies Campana's Abelianity Conjecture.

math.AG↗
arXiv2022

Degenerating Kähler-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 $β\to 0$ of Kähler-Einstein metrics $ω_β$ with a cone singularity of angle $2πβ$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $ω_β$ converges to the locally symmetric metric and further give asymptotics of $ω_β$ 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 $ω_β$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,ω_β)$ converges to an interval in the Gromov-Hausdorff sense.

math.DG↗
arXiv2022

Numerical characterization of complex torus quotients

This article gives a characterization of quotients of complex tori by finite groups acting freely in codimension two in terms of a numerical vanishing condition on the first and second Chern class. This generalizes results previously obtained by Greb--Kebekus--Peternell in the projective setting, and by Kirschner and the second author in dimension three. As a key ingredient to the proof, we obtain a version of the Bogomolov--Gieseker inequality for stable sheaves on singular spaces, including a discussion of the case of equality.

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