Erwan Rousseau’s research while affiliated with Laboratoire de Mathématiques de Bretagne Atlantique and other places

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Publications (68)


Entire curves in C-pairs with large irregularity
  • Preprint

October 2024

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1 Read

Stefan Kebekus

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Erwan Rousseau

This paper extends the fundamental theorem of Bloch-Ochiai to the context of C-pairs: If (X, D) is a C-pair with large irregularity, then no entire C-curve in X is ever dense. In its most general form, the paper's main theorem applies to normal K\"ahler pairs with arbitrary singularities. However, it also strengthens known results for compact K\"ahler manifolds without boundary, as it applies to some settings that the classic Bloch-Ochiai theorem does not address. The proof builds on work of Kawamata, Ueno, and Noguchi, recasting parabolic Nevanlinna theory as a "Nevanlinna theory for C-pairs". We hope the approach might be of independent interest.


The Albanese of a C-pair

October 2024

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2 Reads

Written with a view toward applications in hyperbolicity, rational points, and entire curves, this paper addresses the problem of constructing Albanese maps within Campana's theory of C-pairs (or "geometric orbifolds"). It introduces C-semitoric pairs as analogs of the (semi)tori used in the classic Albanese theory and follows Serre by defining the Albanese of a C-pair as the universal map to a C-semitoric pairs. The paper shows that the Albanese exists in relevant cases, gives sharp existence criteria, and conjectures that a "weak Albanese" exists unconditionally.


C-pairs and their morphisms

July 2024

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5 Reads

This paper surveys Campana's theory of C-pairs (or "geometric orbifolds") in the complex-analytic setting, to serve as a reference for future work. Written with a view towards applications in hyperbolicity, rational points, and entire curves, it introduces the fundamental definitions of C-pair-theory systematically. In particular, it establishes an appropriate notion of "morphism", which agrees with notions from the literature in the smooth case, but is better behaved in the singular setting and has functorial properties that relate it to minimal model theory.


On the existence of logarithmic and orbifold jet differentials
  • Article
  • Full-text available

June 2024

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50 Reads

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2 Citations

Annales Henri Lebesgue

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Quasi-positive orbifold cotangent bundles: Pushing further an example by Junjiro Noguchi

April 2024

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13 Reads

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3 Citations

Épijournal de Géométrie Algébrique

In this work, we investigate the positivity of logarithmic and orbifold cotangent bundles along hyperplane arrangements in projective spaces. We show that a very interesting example given by Noguchi (as early as in 1986) can be pushed further to a very great extent. Key ingredients of our approach are the use of Fermat covers and the production of explicit global symmetric differentials. This allows us to obtain some new results in the vein of several classical results of the literature on hyperplane arrangements. These seem very natural using the modern point of view of augmented base loci, and working in Campana's orbifold category.


Simply connectedness and hyperbolicity

August 2023

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7 Reads

We generalize to arbitrary dimension our previous construction of simply connected weakly-special but not special varieties. We show that they satisfy the function field and complex analytic part of Campana's conjecture. Moreover, we give the first examples, in any dimension, of smooth simply connected nonisotrivial projective varieties of general type that satisfy the function field Lang's conjecture.



Divisibility of polynomials and degeneracy of integral points

January 2023

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26 Reads

Mathematische Annalen

We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples of simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results of Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application of the Ru–Vojta’s strategy. We also obtain the analogue results for function fields and Nevanlinna theory with the goal to apply them in a future paper in the context of Campana’s conjectures.



Rational endomorphisms of codimension one holomorphic foliations

August 2022

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26 Reads

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4 Citations

We study dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds and that exhibit non-trivial transverse dynamics.


Citations (29)


... It has been shown by several authors that hyperbolicity properties of X \ D could be investigated through the positivity of the associated logarithmic cotangent bundle Ω X (log D), see e.g. [Nog86], [BD18], [BD19], [DR24], [CDR20], [CDDR21]. ...

Reference:

Hyperbolicity of smooth logarithmic and orbifold pairs in $\mathbb{P}^n
On the existence of logarithmic and orbifold jet differentials

Annales Henri Lebesgue

... In the case of orbifold surfaces (P 2 , (1 − 1 ρ )C) where C is a smooth curve of degree d, such existence results have been obtained in [CDR20] for k = 2, d ⩾ 12 and ρ ⩾ 5 depending on d. In [DR23], the existence of jet differentials is obtained for orbifolds (P n , d i=1 (1 − 1 ρ )H i ) in any dimension for k = 1, ρ ⩾ 3 along an arrangement of hyperplanes of degree d ⩾ 2n( 2n ρ−2 + 1). In [BD19], it is established that the orbifold (P n , (1 − 1 d )D), where D is a general smooth hypersurface of degree d, is hyperbolic i.e. there is no non-constant orbifold entire curve f : ...

Quasi-positive orbifold cotangent bundles: Pushing further an example by Junjiro Noguchi

Épijournal de Géométrie Algébrique

... Other cases in which Conjecture 1.2 is known to be true are the case of closed subvarieties of abelian varieties and more generally, proper varieties of maximal Albanese dimension (where it follows from work of Kawamata, Ueno and Yamanoi, see [JR22, Theorem 3.5 and Corollary 3.10] for details), and the case of symmetric powers of curves and those of certain surfaces (see [BJL24]). Lastly, in [PRT22] it is shown that a smooth projective variety X admitting a line bundle L ⊆ Ω 1 X of numerical dimension 1 is neither Campana-special nor geometrically special. ...

Numerically non-special varieties
  • Citing Article
  • June 2022

Compositio Mathematica

... It is however also natural to study the hyperbolicity of such symmetric powers. For example, if S is a smooth projective hyperbolic variety over C, then one can show that Sym m (S) is also hyperbolic, under suitable assumptions (see [CCR22,GFP]). ...

Hyperbolicity and specialness of symmetric powers

Journal de l’École polytechnique — Mathématiques

... geometrically) pseudo-hyperbolic varieties agrees with the class of varieties of log-general type, Conjecture 1.2 appears to be rather reasonable. For more details on the motivation behind Conjecture 1.2 and its relation to the analogous number theoretic conjecture dealing with potential density of integral points, we refer the reader to the introduction of [JR22]. ...

Albanese Maps and Fundamental Groups of Varieties With Many Rational Points Over Function Fields
  • Citing Article
  • September 2021

International Mathematics Research Notices

... These threefolds do not have a dense set of rational points over any number field, assuming Vojta's higherdimensional abc conjecture [Voj98]; this can be deduced from [AVA18, Corollary 3.3] and the fact that the Bogomolov-Tschinkel threefolds come with an elliptic fibration whose orbifold base is of general type. However, the non-density of rational points on their threefolds (and the examples constructed afterwards in [CP07] and [RTW21]) seemingly cannot be proved, assuming only the abc conjecture. Indeed, these threefolds all have a two-dimensional core (see [Cam04,Definition 3.1]), so that the abc conjecture does not seem to suffice to prove the nondensity of integral points on the relevant orbifold base. ...

Nonspecial varieties and generalised Lang–Vojta conjectures

Forum of Mathematics Sigma

... The Green-Griffiths-Lang-Vojta conjectures predict that a quasi-projective variety X over C is of log-general type if and only if there is a proper closed subset ⊂ X such that X is Brody hyperbolic modulo (i.e., every non-constant holomorphic map C → X (C) factors through (C)); see [3,4,10,18,28,37,41]. For example, this conjecture is known when X is a closed subvariety of an abelian variety by the celebrated theorem of Bloch-Ochiai-Kawamata [9,26,36]. ...

KAWA lecture notes on complex hyperbolic geometry
  • Citing Article
  • June 2018

Annales de la faculté des sciences de Toulouse Mathématiques

... The next step consists precisely of finding sufficient conditions that ensure the existence of global sections P ∈ H 0 (X, E k,m V * D ⊗ O X (−A)). Recall that it has been shown in [CDR20,Proposition 5.1] that the general type assumption is not a sufficient condition for the existence of global jet differentials. ...

Orbifold hyperbolicity

Compositio Mathematica

... Applied to V = C, this refinement can be used to provide a hyperbolicity result in the case where X = X is compact. We obtain the following partial generalization to the non-symmetric case of a previous work of the first author with Rousseau and Taji [CRT19], cf. Theorem 5.15. ...

Hyperbolicity of singular spaces
  • Citing Article
  • October 2017

Journal de l’École polytechnique — Mathématiques

... In recent years the intensive work on product-quotient varieties has produced several interesting examples, e.g. the mentioned example of a surface of general type with canonical map of degree 32; new topological types for surface of general type, in particular a family of surfaces of general type with K 2 = 7, p g = q = 2 (see [CF18] and the references therein); and recently the first examples of rigid but not infinitesimally rigid compact complex manifolds ( [BP18]). For other interesting applications see [Cat17], [FGP18], [GRVR18], [LP16] and [LP18]. ...

On Lang's conjecture for some product-quotient surfaces
  • Citing Article
  • November 2016

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