Christophe Mourougane

  | Enseignement | Recherche |  

BCOV invariants of Calabi--Yau manifolds and degenerations of Hodge structures
(en collaboration avec Dennis Eriksson et Gerard Freixas)

Calabi--Yau manifolds have risen to prominence in algebraic geometry, in part because of mirror symmetry and enumerative geometry. After Bershadsky--Cecotti--Ooguri--Vafa (BCOV), it is expected that genus 1 curve counting on a Calabi--Yau manifold is related to a conjectured invariant, only depending on the complex structure of the mirror, and built from Ray--Singer holomorphic analytic torsions. To this end, extending work of Fang--Lu--Yoshikawa in dimension~3, we introduce and study the BCOV invariant of Calabi--Yau manifolds of arbitrary dimension. To determine it, knowledge of its behaviour at the boundary of moduli spaces is imperative. To address this problem, we prove general results on degenerations of L^2 metrics on Hodge bundles and their determinants, refining the work of Schmid. We express the singularities of these metrics in terms of limiting Hodge structures, and derive consequences for the dominant and subdominant singular terms of the BCOV invariant.

On genus one mirror symmetry in higher dimension and the BCOV conjectures
(en collaboration avec Dennis Eriksson et Gerard Freixas)

The mathematical physicists Bershadsky--Cecotti--Ooguri--Vafa (BCOV) proposed, in a seminal article from '94, an extension of genus zero mirror symmetry to higher genera. We offer a mathematical treatment of the BCOV conjecture at genus one, based on the usage of the arithmetic Riemann--Roch theorem. As an application of our previous results on the BCOV invariant, we establish this conjecture for Calabi--Yau hypersurfaces in projective space. This seems to be the first example of higher dimensional mirror symmetry, of BCOV type, at genus one. The case of quintic threefolds was studied by Fang--Lu--Yoshikawa. Our contribution takes place on the $B$-side, and the relation to the $A$-side is provided by Zinger. Our approach also lends itself to arithmetic considerations of the BCOV invariant, and we study a Chowla--Selberg type theorem expressing it in terms of special $\Gamma$ values for certain Calabi--Yau manifolds with complex multiplication. Finally, we put forward a variant of the BCOV program at genus one, as a conjectured functorial Grothendieck--Riemann--Roch relationship.

Dernières publications
Singularities of metrics on Hodge bundles and their topological invariants
(en collaboration avec Dennis Eriksson et Gerard Freixas)

We consider degenerations of complex projective Calabi--Yau varieties and study the singularities of L^2, Quillen and BCOV metrics on Hodge and determinant bundles.
The dominant and subdominant terms in the expansions of the metrics close to non-smooth fibers are shown
to be related to well-known topological invariants of singularities, such as limit Hodge structures, vanishing cycles and log-canonical thresholds.
We also describe corresponding invariants for more general degenerating families in the case of the Quillen metric.

Stability of the tangent bundle of G/P in positive characteristics
(en collaboration avec Indranil Biswas et Pierre-Emmanuel Chaput)

We prove that the tangent bundle of the quotient of an almost simple simply-connected affine algebraic group
over an algebraically closed field k of characteristic p by a parabolic sub-group is Frobenius stable with respect to the anticanonical polarization if p>3.

Stability of restrictions of cotangent bundles of irreducible hermitian symmetric spaces of compact type
Notes d'exposé
(en collaboration avec Indranil Biswas et Pierre-Emmanuel Chaput)

It is known that the cotangent bundle ΩY of an irreducible Hermitian symmetric space Y of compact type is stable.
Except for a few obvious exceptions, we show that if X ⊂ Y is a complete intersection such that Pic(Y) → Pic(X) is surjective, then the restriction Ω Y |X is stable.
We then address some cases where the Picard group increases by restriction.

Hessian of the metric form on twistor spaces
(en collaboration avec Guillaume Deschamps et Noël Le Du)

We compute the hessian of the natural Hermitian form successively on the Calabi family T(M,g,(I,J,K)) of a hyperkahler manifold (M,g,(I,J,K)),
on the twistor space T(M,g) of a 4-dimensional anti-self-dual Riemannian manifold (M,g) and on the twistor space T(M,g,D) of a quaternionic Kähler manifold (M,g,D).
We show a strong convexity property of the cycle space of twistor lines on the Calabi family T(M,g,(I,J,K)) of a hyperkahler manifold.
We also prove convexity properties of the 1-cycle space of the twistor space T(M,g) of a 4-dimensional anti-self-dual Einstein manifold (M,g) of non-positive scalar curvature
and of the 1-cycle space of the twistor space T(M,g,D) of a quaternionic Kähler manifold (M,g,D) of non-positive scalar curvature.
We check that no non-Kahler strong KT manifold occurs as such a twistor space.

Bulletin de la SMF 145, fascicule 1 (2017), 1-27

Families of hypersurfaces of large degree
Grauert and Manin showed that a non-isotrivial family of compact  complex hyperbolic curves has finitely many sections.
 We consider a general moving enough family of high enough degree hypersurfaces in a complex projective space.
We show the existence of a strict closed subset of its total space that contains the image of all its sections.

Journal of the European Mathematical Society

 Publications sur Mathscinet

Textes généraux
*Le texte de ma thèse de doctorat
*Un texte sur les systèmes pluri-canoniques sur les surfaces
(d'après Zariski, Mumford, Kodaira, Bombieri, Reider)
*Un texte sur l'amplitude et la positivité des fibrés vectoriels
*Un cours de géométrie d'Arakelov (Aspects hermitiens)
*Un texte sur les métriques de Hodge
*Un texte sur les métriques L² et métriques de Quillen
*Un texte sur les dimensions cachées en musique

Noël Le du "Hessien de la forme métrique sur les espaces de twisteurs".
Damian Brotbek "Projective varieties with ample cotangent bundle".
Basile Pillet "Géométrie complexe globale et infinitésimale de l’espace des twisteurs d’une variété hyperkählérienne"
Jie Liu (en co-direction avec Andreas Höring) "Géométrie des variétés de Fano : sous-faisceaux du fibré tangent et diviseur fondamental".

Rencontres annuelles du GDR GAGC

Groupe de travail "Singularités du MMP (Rennes, Juin 2019)
Conférence de géométrie analytique et différentielle (en l'honneur de Jean-Pierre Demailly, juin 2017)
Semestre autour des espaces de modules (Janvier-Juillet 2014)
Ecole d'été de théorie de Hodge complexe et p-adique (Rennes, Mai 2014)
Rencontre Stabilité (Roscoff, décembre 2013)
Groupe de travail GIT (Rennes, septembre-décembre 2013)
Rencontre Vojta (Chevaleret, Octobre 2012)
Rencontre Faltings-Nadel (Jussieu, Janvier 2012)
Notes d'un cours à l'Institut Chern (Tianjin, Juin 2011)
Rencontre Hyperbolicité (Paris, les 4 et 5 novembre 2010)
Journées de géométrie d'Arakelov à Rennes : 18–19 décembre 2008
Rencontre positivité (Rennes 10-12 Septembre 2007)

Le plan de conservation partagé en mathématique
Equipe de géométrie et singularités
Equipe de géométrie arithmétique
CIRGET (Montréal)
Labex Lebesgue
Anr positive
GDR Géométrie Algébrique/ Géométrie Complexe
GDR CNRS 2945 "Singularités et Applications"
Recherches à Paris