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.
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