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Olivier Biquard

Publications and source records attributed to Olivier Biquard.

At least 19 recordsLinked to original sources
arXiv2025

Degenerating conic Kähler-Einstein metrics to the normal cone

Let $X$ be a Fano manifold of dimension at least $2$ and $D$ be a smooth divisor in a multiple of the anticanonical class, $\frac1α(-K_X)$ with $α>1$. It is well-known that Kähler-Einstein metrics on $X$ with conic singularities along $D$ may exist only if the angle $2πβ$ is bigger than some positive limit value $2πβ_*$. Under the hypothesis that the automorphisms of $D$ are induced by the automorphisms of the pair $(X,D)$, we prove that for $β>β_*$ close enough to $β_*$, such Kähler-Einstein metrics do exist. We identify the limits at various scales when $β\rightarrowβ_*$ and, in particular, we exhibit the appearance of the Tian-Yau metric of $X\setminus D$.

math.DG↗
arXiv2025

Instability of conformally Kähler, Einstein metrics

We prove the instability of conformally Kähler, compact or ALF Einstein 4-manifolds with nonnegative scalar curvature which are not half conformally flat. This applies to all the known examples of gravitational instantons which are not hyperKähler and to the Chen-Lebrun-Weber metric in particular.

math.DG↗
arXiv2023

Gravitational Instantons, Weyl Curvature, and Conformally Kaehler Geometry

In a previous paper, the first two authors classified complete Ricci-flat ALF Riemannian 4-manifolds that are toric and Hermitian, but non-Kaehler. In this article, we consider general Ricci-flat deformations of such spaces, assuming only suitable fall-off conditions. Quite generally, we are able to show that such a deformation must be Hermitian, and must carry a non-trivial Killing vector field with fixed asymptotics. With mild additional hypotheses, we are then able to show that the new Ricci-flat metric must in fact belong to the family of previously classified metrics.

math.DG↗
arXiv2023

About a Family of ALF Instantons with Conical Singularities

We apply the techniques developed in our previous article to describe some interesting families of ALF gravitational instantons with conical singularities. In particular, we completely understand the 5-dimensional family of Chen-Teo metrics and prove that only 4-dimensional subfamilies can be smoothly compactified so that the metric has conical singularities.

math.DG↗
arXiv2022

Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric

Let $X$ be a complex projective manifold and let $D\subset X$ be a smooth divisor. In this article, we are interested in studying limits when $β\to 0$ of Kähler-Einstein metrics $ω_β$ with a cone singularity of angle $2πβ$ along $D$. In our first result, we assume that $X\setminus D$ is a locally symmetric space and we show that $ω_β$ converges to the locally symmetric metric and further give asymptotics of $ω_β$ when $X\setminus D$ is a ball quotient. Our second result deals with the case when $X$ is Fano and $D$ is anticanonical. We prove a folklore conjecture asserting that a rescaled limit of $ω_β$ is the complete, Ricci flat Tian-Yau metric on $X\setminus D$. Furthermore, we prove that $(X,ω_β)$ converges to an interval in the Gromov-Hausdorff sense.

math.DG↗
arXiv2022

On toric Hermitian ALF gravitational instantons

We give a classification of toric, Hermitian, Ricci flat, ALF Riemannian metrics in dimension 4, including metrics with conical singularities. The only smooth examples are on one hand the hyperKaehler ALF metrics, on the other hand, the Kerr, Taub-NUT and Chen-Teo metrics. There are examples with conical singularities with infinitely many distinct topologies. We provide explicit formulas.

math.DG↗
arXiv2021

The renormalized volume of a 4-dimensional Ricci-flat ALE space

We introduce a natural definition of the renormalized volume of a 4-dimensional Ricci-flat ALE space. We then prove that the renormalized volume is always less or equal than zero, with equality if and only if the ALE space is isometric to its asymptotic cone. Currently the only known examples of 4-dimensional Ricci-flat ALE spaces are Kronheimer's gravitational instantons and their quotients, which are also known to be the only possible examples of special holonomy. We calculate the renormalized volume of these spaces in terms of Kronheimer's period map.

math.DG↗
arXiv2021

Arakelov-Milnor inequalities and maximal variations of Hodge structure

In this paper we study the $\mathbb{C}^*$-fixed points in moduli spaces of Higgs bundles over a compact Riemann surface for a complex semisimple Lie group and its real forms. These fixed points are called Hodge bundles and correspond to complex variations of Hodge structure. We introduce a topological invariant for Hodge bundles that generalizes the Toledo invariant appearing for Hermitian Lie groups. A main result of this paper is a bound on this invariant which generalizes both the Milnor-Wood inequality of the Hermitian case and the Arakelov inequalities of classical variations of Hodge structure. When the generalized Toledo invariant is maximal, we establish rigidity results for the associated variations of Hodge structure which generalize known rigidity results for maximal Higgs bundles and their associated maximal representations in the Hermitian case.

math.AG↗
arXiv2019

Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

math.DG↗
arXiv2018

Ricci flat Kähler metrics on rank two complex symmetric spaces

We obtain Ricci flat Kähler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metrics.

math.DG↗
arXiv2017

Higgs bundles, the Toledo invariant and the Cayley correspondence

We define the Toledo invariant of a G-Higgs bundle on a Riemann surface, where G is a real semisimple group of Hermitian type, and we prove a Milnor-Wood type bound for this invariant when the bundle is semistable. We prove rigidity results when the Toledo invariant is maximal, establishing in particular a Cayley correspondence when the symmetric space defined by G is of tube type. This gives a new proof of the Milnor-Wood inequality of Burger-Iozzi-Wienhard for representations of the fundamental group of a Riemann surface into G. Compared to previous results using Higgs bundles, it uses general theory and avoids any case by case study.

math.DG↗
arXiv2015

Métriques hyperkählériennes pliées

N. Hitchin recently introduced the notion of folded hyperKähler metrics, in relation with SL(\infty,R) Higgs bundles. We provide a construction of such metrics, and prove the local existence of the Hitchin component for SL(\infty,R).

math.DG↗
arXiv2013

Desingularisation de metriques d'Einstein. II

We calculate a wall crossing formula for 4-dimensional Poincare-Einstein metrics, through a wall made of orbifold Poincare-Einstein metrics with A1 singularities. This is based on a formalism which enables to deal with higher order terms of the Einstein equation in this setting. Some other consequences are deduced.

math.DG↗
arXiv2013

Smoothing singular extremal Kähler surfaces and minimal Lagrangians

We consider smoothings of a complex surface with singularities of class T and no nontrivial holomorphic vector field. Under an hypothesis of non degeneracy of the smoothing at each singular point, we prove that if the singular surface admits an extremal metric, then the smoothings also admit extremal metrics in nearby Kähler classes. In addition, we construct small Lagrangian stationary spheres which represent Lagrangian vanishing cycles for surfaces close to the singular one.

math.DG↗
arXiv2011

A construction of conformal-harmonic maps

Conformal harmonic maps from a 4-dimensional conformal manifold to a Riemannian manifold are maps satisfying a certain conformally invariant fourth order equation. We prove a general existence result for conformal harmonic maps, analogous to the Eells-Sampson theorem for harmonic maps. The proof uses a geometric flow and relies on results of Gursky-Viaclovsky and Lamm.

math.DG↗
arXiv2011

Désingularisation de métriques d'Einstein. I

We find a new obstruction for a real Einstein 4-orbifold with an A1-singularity to be a limit of smooth Einstein 4-manifolds. The obstruction is a curvature condition at the singular point. For asymptotically hyperbolic metrics, with boundary at infinity a conformal metric, we prove that if the obstruction vanishes, one can desingularize Einstein orbifolds with such singularities. The Dirichlet problem consists in finding Einstein metrics with given conformal infinity on the boundary: we prove that our obstruction defines a wall in the space of conformal metrics on the boundary, and that all the Einstein metrics must have their conformal infinity on one side of the wall.

math.DG↗