Frédéric Campana
Research Subjects
Frédéric Campana has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Thomas Peternell
Mathematisches Institut, Universität Bayreuth, Bayreuth, Germany
5 shared publications - Ekaterina Amerik
Laboratoire des Mathématiques Université Paris-Sud Bâtiment 425 91405 Orsay France
4 shared publications - Jean-Pierre Demailly
Institut Fourier, Université Grenoble Alpes, Gières, France
4 shared publications - Benoît Claudon
Univ Lorraine, IECL, BP 70239, F-54506 Vandoeuvre Les Nancy, France
3 shared publications - Lionel Darondeau
Sorbonne Université, CNRS, Paris, France
3 shared publications - Andreas Höring
Université Côte d’Azur
2 shared publications
Latest Publications
- Open Access
- 21 December 2023
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Épijournal de Géométrie Algébrique
- Vol. Special um
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 \'etale 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 hyperk\"ahler.Comment: 17 pages. v2: an improvement following a recent work of B. Taji (results valid for fibers with good minimal models rather than semiample canonical bundle). V3: minor corrections, a remark added- 21 December 2023
- Published by Centre pour la Communication Scientifique Directe (CCSD) in Épijournal de Géométrie Algébrique
- Vol. Special um
Open AccessOn 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 \'etale 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 hyperk\"ahler.Comment: 17 pages. v2: an improvement following a recent work of B. Taji (results valid for fibers with good minimal models rather than semiample canonical bundle). V3: minor corrections, a remark added - Open Access
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.
Open AccessErratum 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.
- Open Access
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.Open AccessThe 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. - Open Access
Numerical and kodaira dimensions of cotangent bundles
We conjecture the equality of the numerical and Kodaira dimensions $\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $\nu_1^*(X)=\kappa_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite \'etale cover of $X$. The proof rests on the Beauville-Bogomolov decomposition, and a direct computation for smooth models of quotients $A/G$ of complex tori by finite groups. We conjecture that these equalities hold true, much more generally, when $X$ is `special'. The invariant $\kappa_1^*$ was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when $\kappa_1^*(X)=-dim(X)$ was introduced and studied in [29].Open AccessNumerical and kodaira dimensions of cotangent bundles
We conjecture the equality of the numerical and Kodaira dimensions $\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler manifolds $X$, generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $\nu_1^*(X)=\kappa_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite \'etale cover of $X$. The proof rests on the Beauville-Bogomolov decomposition, and a direct computation for smooth models of quotients $A/G$ of complex tori by finite groups. We conjecture that these equalities hold true, much more generally, when $X$ is `special'. The invariant $\kappa_1^*$ was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when $\kappa_1^*(X)=-dim(X)$ was introduced and studied in [29]. - 1 January 2023
- Published by National Research University, Higher School of Economics (HSE) in Moscow Mathematical Journal
- Vol. 23 (3) , 319-330
Kodaira Additivity, Birational Isotriviality, and Specialness
- 1 January 2023
- Published by National Research University, Higher School of Economics (HSE) in Moscow Mathematical Journal
- Vol. 23 (3) , 319-330
Kodaira Additivity, Birational Isotriviality, and Specialness
- Open Access
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 \'etale 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 hyperk\"ahler.Open AccessOn 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 \'etale 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 hyperk\"ahler. - 24 January 2022
- Published by Springer Nature in European Journal of Mathematics
- Vol. 8 (3) , 815-833
Rational points over complex function fields: remarks on isotriviality and dominatedness
- 24 January 2022
- Published by Springer Nature in European Journal of Mathematics
- Vol. 8 (3) , 815-833
Rational points over complex function fields: remarks on isotriviality and dominatedness
- Open Access
- 1 January 2022
- Published by American Institute of Mathematical Sciences (AIMS) in Electronic Research Archive
- Vol. 30 (4) , 1187-1208
Algebraicity of foliations on complex projective manifolds, applications
This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [1], which is inspired by the former results [2,3,4]. In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or 'orbifold' context.
- 1 January 2022
- Published by American Institute of Mathematical Sciences (AIMS) in Electronic Research Archive
- Vol. 30 (4) , 1187-1208
Open AccessAlgebraicity of foliations on complex projective manifolds, applications
This is an expository text, originally intended for the ANR 'Hodgefun' workshop, twice reported, organised at Florence, villa Finaly, by B. Klingler. We show that holomorphic foliations on complex projective manifolds have algebraic leaves under a certain positivity property: the 'non pseudoeffectivity' of their duals. This permits to construct certain rational fibrations with fibres either rationally connected, or with trivial canonical bundle, of central importance in birational geometry. A considerable extension of the range of applicability is due to the fact that this positivity is preserved by the tensor powers of the tangent bundle. The results presented here are extracted from [1], which is inspired by the former results [2,3,4]. In order to make things as simple as possible, we present here only the projective versions of these results, although most of them can be easily extended to the logarithmic or 'orbifold' context.
- Open Access
On the existence of logarithmic and orbifold jet differentials
Minor typos corrected, source file converted to LaTeX
Open AccessOn the existence of logarithmic and orbifold jet differentials
Minor typos corrected, source file converted to LaTeX