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Frédéric Campana

Publications and source records attributed to Frédéric Campana.

At least 19 recordsLinked to original sources
arXiv2026

Weakly Special Manifolds with no rational curves

Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi--Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an étale quotient of an Abelian variety. The same conclusion holds true if $X$ contains a Zariski dense entire curve, assuming Lang's conjecture. This implies that any non-hyperbolic complex projective manifold contains the image of an Abelian variety, according to another conjecture of Lang. We illustrate this last conjecture by producing examples of canonically polarised submanifolds of abelian varieties containing no subvariety of general type, except for a finite number of disjoint copies of some simple abelian variety, which can be chosen arbitrarily. We also show, more generally, that any projective manifold containing a Zariski dense entire curve appears as the `exceptional set' in Lang's sense of some general type manifold.

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arXiv2026

The Weakly Special Conjecture contradicts orbifold Mordell, and hence the abc conjecture

Starting from an Enriques surface over $\mathbb{Q}(t)$ considered by Lafon, we give the first examples of smooth projective weakly special threefolds which fibre over the projective line in Enriques surfaces (resp. K3 surfaces) with nowhere reduced, but non-divisible, fibres and general type orbifold base. We verify that these families of Enriques surfaces (resp. K3 surfaces) are non-isotrivial and compute their fundamental groups by studying the behaviour of local points along certain étale covers. The existence of the above threefolds implies that the Weakly Special Conjecture formulated in 2000 contradicts the Orbifold Mordell Conjecture, and hence the abc conjecture. Using these examples, we can also easily disprove several complex-analytic analogues of the Weakly Special Conjecture. Finally, the existence of such threefolds shows that Enriques surfaces and K3 surfaces can have non-divisible but nowhere reduced degenerations, thereby answering a question raised in 2005.

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arXiv2023

On algebraically coisotropic submanifolds of holomorphic symplectic manifolds

We investigate algebraically coisotropic submanifolds $X$ in a holomorphic symplectic projective manifold $M$. Motivated by our results in the hypersurface case, we raise the following question: when $X$ is not uniruled, is it true that up to a finite étale cover, the pair $(X,M)$ is a product $(Z\times Y, N\times Y)$ where $N, Y$ are holomorphic symplectic and $Z\subset N$ is Lagrangian? We prove that this is indeed the case when $M$ is an abelian variety, and give some partial answer when the canonical bundle $K_X$ is semi-ample. In particular, when $K_X$ is nef and big, $X$ is Lagrangian in $M$ (in fact this also holds without nefness assumption). We also remark that Lagrangian submanifolds do not exist on a sufficiently general Abelian variety, in contrast to the case when $M$ is irreducible hyperkähler.

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arXiv2023

Erratum and addendum to the paper: Abundance for Kähler threefolds

In this text we signal a serious gap in the proof of the main theorem of our paper and explain which parts of the statement remain valid. In fact, the main theorem remains valid unless possibly the variety does not admit positive-dimensional subvarieties through a very general point and is not bimeromorphic to a quotient of a torus. This latter case would be ruled out by a Chern class inequality which holds in the algebraic case but is still unknown in the Kähler setting.

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arXiv2023

The nilpotent quotients of normal quasi-projective varieties with proper quasi-Albanese map

We show that if $X$ is a normal complex quasi-projective variety, the quasi-Albanese map of which is proper, then the torsionfree nilpotent quotients of $π_1(X)$ are, up to a controlled finite index, the same ones as those of the normalisation of its quasi-Albanese image. When $X$ is quasi-Kähler smooth, we get the same conclusion, but only for the smooth models of the quasi-Albanese image. In this second case, the proof is elementary, as the one given in [Cam95] for $X$ compact. In the normal quasi-projective case, the étale Galois cover of $X$ associated to the nilpotent completion of $π_1(X)$ is thus holomorphically convex. This is proved in the smooth case by $3$ other methods in [GGK22], which motivated the present text. When $X$ is `special' in the sense of [Cam11], we deduce that the torsion free nilpotent quotients of $π_1(X)$ are abelian. Examples show that this property fails (as first observed in [CDY22]) when the quasi-Albanese map is not proper. This leads to replace our previous `Abelianity conjecture' in the compact case by an `Nilpotency conjecture' in the non-compact quasi-Kähler context.

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arXiv2021

On the existence of logarithmic and orbifold jet differentials

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V[D] in the general orbifold case-with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.

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arXiv2021

Subharmonicity of direct images and applications

In this article we establish new positivity properties for direct images of twisted pluricanonical bundle of an algebraic fiber space. As a corollary we obtain an algebraicity criteria for holomorphic foliations which partly confirms a conjecture of Pereira-Touzet.

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arXiv2020

Hyperbolicity and Specialness of Symmetric Powers

Inspired by the computation of the Kodaira dimension of symmetric powers Xm of a complex projective variety X of dimension n $\ge$ 2 by Arapura and Archava, we study their analytic and algebraic hyperbolic properties. First we show that Xm is special if and only if X is special (except when the core of X is a curve). Then we construct dense entire curves in (suf-ficiently hig) symmetric powers of K3 surfaces and product of curves. We also give a criterion based on the positivity of jet differentials bundles that implies pseudo-hyperbolicity of symmetric powers. As an application, we obtain the Kobayashi hyperbolicity of symmetric powers of generic projective hypersur-faces of sufficiently high degree. On the algebraic side, we give a criterion implying that subvarieties of codimension $\le$ n -- 2 of symmetric powers are of general type. This applies in particular to varieties with ample cotangent bundles. Finally, based on a metric approach we study symmetric powers of ball quotients.

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arXiv2020

Orbifold hyperbolicity

We define and study jet bundles in the geometric orbifold category. We show that the usual arguments from the compact and the logarithmic settings do not all extend to this more general framework. This is illustrated by simple examples of orbifold pairs of general type that do not admit any global jet differential, even if some of these examples satisfy the Green-Griffiths-Lang conjecture. This contrasts with an important result of Demailly (2010) proving that compact varieties of general type always admit jet differentials. We illustrate the usefulness of the study of orbifold jets by establishing the hyperbolicity of some orbifold surfaces, that cannot be derived from the current techniques in Nevanlinna's theory. We also conjecture that Demailly's theorem should hold for orbifold pairs with smooth boundary divisors under a certain natural multiplicity condition, and provide some evidence towards it.

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arXiv2019

Projective klt pairs with nef anti-canonical divisor

In this paper, we study a projective klt pair $(X, Δ)$ with the nef anti-log canonical divisor $-(K_X+Δ)$ and its maximally rationally connected fibration $ψ: X \dashrightarrow Y$. We prove that the numerical dimension of the anti-log canonical divisor $-(K_X+Δ)$ on $X$ coincides with that of the anti-log canonical divisor $-(K_{X_y}+Δ_{X_y})$ on a general fiber $X_y$ of $ψ: X \dashrightarrow Y$, which is an analogue of Ejiri-Gongyo's result formulated for the Kodaira dimension. As a corollary, we reveal a relation between positivity of the anti-canonical divisor and the rational connectedness, which gives a sharper estimate than the question posed by Hacon-$\mathrm{M^{c}}$Kernan. Moreover, in the case of $X$ being smooth, we show that a maximally rationally connected fibration $ψ: X \to Y$ can be chosen to be a morphism to a smooth projective variety $Y$ with numerically trivial canonical divisor, and further that it is locally trivial with respect to the pair $(X, Δ)$, which can be seen as a generalization of Cao-Höring's structure theorem to klt pair cases. Finally, we study the structure of the slope rationally connected quotient for a pair $(X, Δ)$ with $-(K_X +Δ)$ nef, and obtain a structure theorem for projective orbifold surfaces.

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arXiv2018

Rationally connected manifolds and semipositivity of the Ricci curvature

This work establishes a structure theorem for compact Kähler manifolds with semipositive anticanonical bundle. Up to finite étale cover, it is proved that such manifolds split holomorphically and isometrically as a product of Ricci flat varieties and of rationally connected manifolds. The proof is based on a characterization of rationally connected manifolds through the non existence of certain twisted contravariant tensor products of the tangent bundle, along with a generalized holonomy principle for pseudoeffective line bundles. A crucial ingredient for this is the characterization of uniruledness by the property that the anticanonical bundle is not pseudoeffective.

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arXiv2017

Specialness and Isotriviality for Regular Algebraic Foliations

We show that an everywhere regular foliation $\mathcal F$ with compact canonically polarized leaves on a quasi-projective manifold $X$ has isotrivial family of leaves when the orbifold base of this family is special. By a recent work of Berndtsson, Paun and Wang, the same proof works in the case where the leaves have trivial canonical bundle. The specialness condition means that the $p$-th exterior power of the logarithmic extension of its conormal bundle does not contain any rank-one subsheaf of maximal Kodaira dimension $p$, for any $p>0$. This condition is satisfied, for example, in the very particular case when the Kodaira dimension of the determinant of the Logarithmic extension of the conormal bundle vanishes. Motivating examples are given by the `algebraically coisotropic' submanifolds of irreducible hyperkähler projective manifolds.

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arXiv2016

Characteristic foliation on non-uniruled smooth divisors on hyperkaehler manifolds

We prove that the characteristic foliation $F$ on a non-singular divisor $D$ in an irreducible projective hyperkaehler manifold $X$ cannot be algebraic, unless the leaves of $F$ are rational curves or $X$ is a surface. More generally, we show that if $X$ is an arbitrary projective manifold carrying a holomorphic symplectic $2$-form, and $D$ and $F$ are as above, then $F$ can be algebraic with non-rational leaves only when, up to a finite étale cover, $X$ is the product of a symplectic projective manifold $Y$ with a symplectic surface and $D$ is the pull-back of a curve on this surface. When $D$ is of general type, the fact that $F$ cannot be algebraic unless $X$ is a surface was proved by Hwang and Viehweg. The main new ingredient for our results is the observation that the canonical bundle of the (apriori, orbifold; but the orbifold structure is actually trivial) base of the family of leaves must be torsion. This implies, in particular, the isotriviality of the family of leaves of $F$. We also make some remarks in the Kähler case and apply this to the Lagrangian conjecture in the last section.

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arXiv2015

Fundamental group and pluridifferentials on compact Kähler manifolds

A compact Kähler manifold is shown to be simply-connected if its `symmetric cotangent algebra' is trivial. Conjecturally, such a manifold should even be rationally connected. The relative version is also shown: a proper surjective connected holomorphic map between connected complex manifolds induces an isomorphism of fundamental groups if its smooth fibres are as above, and if the domain is Kähler.

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arXiv2015

Orbifold generic semi-positivity: an application to families of canonically polarized manifolds

In this article we establish a version of Y. Miyaoka generic semi-positivity theorem in the context of log-canonical orbifold pairs. As an application, we show that the canonical bundle associated to a lc pair is big as soon as there exists a generically injective morphism from an ample line bundle to some symmetric power of the cotangent bundle associated to the orbifold pair.

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arXiv2014

Quelques propriétés de stabilité des variétés spéciales

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.

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