arXiv subjects
Publications and source records attributed to Benoît Claudon.
We investigate the structure of transversely Kähler foliations with quasi-negative tranverse Ricci curvature. In particular, we prove a de Rham type theorem decomposition on the leaf space where we characterize each factor.
We show that a compact Kähler manifold bimeromorphic to a weakly Kähler hyperbolic manifold is weakly Kähler hyperbolic, providing an answer to a problem raised by J. Kollár in his 1995 book "Shafarevic maps and automorphic forms"
In this short note we prove two projectivity criteria for fibrations between mildly singular compact Kähler spaces. They are the relative versions of the celebrated criteria of Kodaira and Moishezon. As an application we obtain that the MRC fibration always has a model that is a projective morphism.
Let $(X, Δ)$ be a compact Kähler klt pair, where $K_X + Δ$ is ample or numerically trivial, and $Δ$ has standard coefficients. We show that if equality holds in the orbifold Miyaoka-Yau inequality for $(X, Δ)$, then its orbifold universal cover is either the unit ball (ample case) or the affine space (numerically trivial case).
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.
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.
This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.
We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an orbicurve.
Let $X$ be a compact K{ä}hler manifold of dimension three. We prove that there exists a projective manifold $Y$ such that $π\_1(X)\simeq π\_1(Y)$. We also prove the bimeromorphic existence of algebraic approximations for compact K{ä}hler manifolds of algebraic dimension $\dim(X)-1$. Together with the work of Graf and the third author, this settles in particular the bimeromorphic Kodaira problem for compact K{ä}hler threefolds.
The proof of the celebrated Viehweg's hyperbolicity conjecture is a consequence of two remarkable results: Viehweg and Zuo's existence results for global pluri-differential forms induced by variation in a family of canonically po-larised manifolds and Campana and Pǎun's vast generalisation of Miyaoka's generic semipositivity result for non-uniruled varieties to the context of pairs. The aim of this chapter is an exposition of Campana-Pǎun's generic semipositivity theorem .
That short note, meant as an addendum to [CCE14], enhances the results contained in loc. cit. In particular it is proven here that a linear K{ä}hler group is already the fundamental group of a smooth complex projective variety. This is achieved by studying the relative deformation of the total space of a smooth family of tori in an equivariant context.
Proven by A. Parshin and S. Arakelov in the early 70's, Shafaverich hyperbolicity conjecture states that a family of curves of genus $g\ge2$ parametrized by a non hyperbolic curve (\emph{i.e.} isomorphic to $\mathbb{P}^1$, $\mathbb{C}$, $\mathbb{C}^*$ or an elliptic curve) has to be isotrivial : the moduli of smooth fibres are constant. In higher dimensions, Viehweg's works on moduli of canonically polarized manifolds led him to generalize this statement in the following way: if a family of canonically polarized manifolds (parametrized by a quasi-projective base) has maximal variation, the base is then of log-general type. It can be thought as an algebraic hyperbolicity property which is expected to hold for the moduli space.Adapting results due to Y. Miyoaka on generic semi-positivity of cotangent bundle to the framework of pairs, F. Campana and M. Păun recently obtained a positive answer to Viehweg's conjecture. We will also take the opportunity of this talk to present the classification of orbifolds as developed in Campana's works. This setting is moreover the right one to state the optimal version of Viehweg's conjecture proven by B. Taji
We show that the general fibres of the Albanese morphism of a projective special manifold are special as well (a question raised by the first-named author). The main ingredient of the proof is a version (established by Birkar and Chen) with boundary of the famous conjecture of Iitaka on the subadditivity of the Kodaira dimension in an algebraic fibre space. Some consequences of our main result are discussed.
In this note we establish the following result (announced in a previous work): if a linear group is the image of a representation of a Kähler group, then it has a finite index subgroup which is the image of a representation of the fundamental group of a smooth projective variety. In particular, a Kähler group which is linear is virtually projective.
We extend to compact Kähler manifolds some classical results on linear representation of fundamental groups of complex projective manifolds. Our approach based on an interversion lemma for fibrations with tori versus general type manifolds as fibers gives a refinement of the classical work of Zuo. We extend to the kahler case some general results on holomorphic convexity of coverings such as the linear shafarevich conjecture. In the first version, the proof of the statement that a linear Kahler group is virtually complex-projective was wrong. We removed it from this new version. The proof will be given in a forthcoming work.
Let X be a compact Kähler manifold such that the universal cover admits a compactification. We conjecture that the fundamental group is almost abelian and reduce it to a classical conjecture of Iitaka.
Using orbifold metrics of the appropriately signed Ricci curvature on orbifolds with negative or numerically trivial canonical bundle and the two-dimensional Log Minimal Model Program, we prove that the fundamental group of special compact Kähler threefolds is almost abelian. This property was conjectured in all dimensions in [Cam04b], and also for orbifolds in [Cam07], where the notion of specialness was introduced. We briefly recall below the definition, basic properties, and the role of special manifolds in classification theory.
We prove that the universal cover of a normal, projective variety X is quasi-projective if and only if a finite, étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.