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Frédéric Rochon

Publications and source records attributed to Frédéric Rochon.

At least 19 recordsLinked to original sources
arXiv2026

A gluing construction of $Z$-extremal Kähler metrics and new examples of Bismut Hermitian-Einstein $3$-folds

We present a gluing construction for a class of 6th order geometric non-linear PDE's on a Kähler manifold or orbifold, arising in the study of Bismut Hermitian Einstein complex $3$-folds, the supersymmetric ${\rm AdS}_3 \times Y_7$ and ${\rm AdS}_2 \times Y_9$ solutions of type IIB and $D=11$ supergravity models in physics, and in Dervan's Z-critical Kähler metrics. As applications, we obtain a desingularization theorem for orbifolds with crepant singularities admitting a solution of the PDE, as well as a blowing-up of points theorem for smooth complex $3$-folds admitting such solutions. These results yield new examples of regular Bismut Hermitian Einstein structures on the smooth manifolds $3(S^2\times S^4)\sharp 4(S^3\times S^3)$, $5(S^2\times S^4)\sharp 6(S^3\times S^3)$ and $S^1\times 9(S^2\times S^3)$, as well as infinite topological types of smooth compact $7$-dimensional manifolds $Y_7$, providing new solutions to the supersymmetric ${\rm AdS}_3 \times Y_7$ type IIB and ${\rm AdS}_2 \times T^2 \times Y^7$ type $D=11$ supergravity models.

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arXiv2026

Warped quasi-asymptotically conical Calabi-Yau metrics

We construct many new examples of complete Calabi-Yau metrics of maximal volume growth on certain smoothings of Cartesian products of Calabi-Yau cones with smooth cross-sections. A detailed description of the geometry at infinity of these metrics is given in terms of a compactification by a manifold with corners obtained through the notion of weighted blow-up for manifolds with corners. A key analytical step in the construction of these Calabi-Yau metrics is to derive good mapping properties of the Laplacian on some suitable weighted Hölder spaces. Our methods also produce singular Calabi-Yau metrics with an isolated conical singularity modelled on a Calabi-Yau cone distinct from the tangent cone at infinity, in particular yielding a transition behavior between different Calabi-Yau cones as conjectured by Yang Li. This is used to exhibit many examples where the tangent cone at infinity does not uniquely specify a complete Calabi-Yau metric with exact Kähler form.

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arXiv2026

On the growth of torsion in the cohomology of some arithmetic groups of $\mathbb{Q}$-rank one

Given a number field $F$ with ring of integers $\mathcal{O}_{F}$, one can associate to any torsion free subgroup of $\operatorname{SL}(2,\mathcal{O}_{F})$ of finite index a complete Riemannian manifold of finite volume with fibered cusp ends. For natural choices of flat vector bundles on such a manifold, we show that analytic torsion is identified with the Reidemeister torsion of the Borel-Serre compactification. This is used to obtain exponential growth of torsion in the cohomology for sequences of congruence subgroups.

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arXiv2025

Géométrie à l'infini des variétés hyperkählériennes toriques

We show that simply connected toric hyperKähler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In the asymptotically conical case, we also provide a fine description of the geometry at infinity of their Taub-NUT deformations of order 1 in terms of a compactification by a manifold with corners, which allows us to show that those deformations are of bounded geometry, to estimate their curvature at infinity and their volume growth and to identify uniquely their tangent cone at infinity. In many instances, the dimension of this tangent cone at infinity is strictly smaller than the order of the volume growth. Finally, our methods show that the Taub-NUT deformations of maximal order of the Euclidean metric are quasi-fibered boundary metrics, which allows us to identify uniquely their tangent cone at infinity, to compute their reduced $L^2$ cohomology groups and to prove that Sen's S-duality conjecture holds for centered monopoles of type $(1,1,\ldots,1)$.

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arXiv2025

Asymptotic geometry at infinity of quiver varieties

Using an approach developed by Melrose to study the geometry at infinity of the Nakajima metric on the reduced Hilbert scheme of points on $\mathbb{C}^2$, we show that the Nakajima metric on a quiver variety is quasi-asymptotically conical (QAC) whenever its defining parameters satisfy an appropriate genericity assumption. As such, it is of bounded geometry and of maximal volume growth. Being QAC is one of two main ingredients allowing us to use the work of Kottke and the second author to compute its reduced $L^2$-cohomology and prove the Vafa-Witten conjecture. The other is a vanishing theorem in $L^2$-cohomology for exact wedge $3$-Sasakian metrics generalizing a result of Galicki and Salamon for closed $3$-Sasakian manifolds.

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arXiv2024

Quasi-fibered boundary pseudodifferential operators

We develop a pseudodifferential calculus for differential operators associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics including the quasi-asymptotically conical metrics (QAC metrics) of Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE metrics) of Joyce. Introducing various principal symbols, we introduce the notion of fully elliptic QFB operators and show that those are Fredholm when acting on QFB Sobolev spaces. For QAC metrics, we also develop a pseudodifferential calculus for the conformally related class of Qb metrics. We use these calculi to construct a parametrix for the Hodge-deRham operator of certain QFB metrics, allowing us to show that it is Fredholm on suitable Sobolev spaces and that the space of $L^2$ harmonic forms is finite dimensional. Our parametrix is obtained by inverting certain model operators at infinity, inversions that we achieve in part through a fine understanding of the low energy limit of the resolvent of the Hodge-deRham operator. Our parametrix also implies that $L^2$ harmonic forms decay faster at infinity than an arbitrary $L^2$ form, the extra decay being quantified in terms of a small negative power of the distance function. This decay of $L^2$ harmonic forms is used in a companion paper to study the $L^2$ cohomology of some $QFB$ metrics.

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arXiv2024

$L^2$-cohomology of quasi-fibered boundary metrics

We develop new techniques to compute the weighted $L^2$-cohomology of quasi-fibered boundary metrics (QFB-metrics). Combined with the decay of $L^2$-harmonic forms obtained in a companion paper, this allows us to compute the reduced $L^2$-cohomology for various classes of QFB-metrics. Our results applies in particular to the Nakajima metric on the Hilbert scheme of $n$ points on $\mathbb{C}^2$, for which we can show that the Vafa-Witten conjecture holds. Using the compactification of the monopole moduli space announced by Fritzsch, the first author and Singer, we can also give a proof of the Sen conjecture for the monopole moduli space of magnetic charge 3.

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arXiv2023

Products of manifolds with fibered corners

Manifolds with fibered corners arise as resolutions of stratified spaces, in many body compactifications of vector spaces, moduli spaces, and other settings. We define a category of fibered corners manifolds which has products and transverse fiber products, generalizing both the resolutions of products of stratified spaces and many body products, which are special cases. The product in the fibered corners category is a resolution of the cartesian product by blow-up which we call the 'ordered product'. This ordered product is a natural product for wedge (aka incomplete edge) metrics and quasi-fibered boundary metrics, a class which includes QAC and QALE metrics.

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arXiv2021

Low energy limit for the resolvent of some fibered boundary operators

For certain Dirac operators $ð_ϕ$ associated to a fibered boundary metric $g_ϕ$, we provide a pseudodifferential characterization of the limiting behavior of $(ð_ϕ+kγ)^{-1}$ as $k\searrow 0$, where $γ$ is a self-adjoint operator anti-commuting with $ð_ϕ$ and whose square is the identity. This yields in particular a pseudodifferential characterization of the low energy limit of the resolvent of $ð_ϕ^2$, generalizing a result of Guillarmou and Sher about the low energy limit of the resolvent of the Hodge Laplacian of an asymptotically conical metric. As an application, we use our result to give a pseudodifferential characterization of the inverse of some suspended version of the operator $ð_ϕ$. One important ingredient in the proof of our main theorem is that the Dirac operator $ð_ϕ$ is Fredholm when acting on suitable weighted Sobolev spaces. This result has been known to experts for some time and we take this as an occasion to provide a complete explicit proof.

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arXiv2020

A Cheeger-Müller theorem for manifolds with wedge singularities

We study the spectrum and heat kernel of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold degenerating to a manifold with wedge singularities. Provided the Hodge Laplacians in the fibers of the wedge have an appropriate spectral gap, we give uniform constructions of the resolvent and heat kernel on suitable manifolds with corners. When the wedge manifold and the base of the wedge are odd dimensional, this is used to obtain a Cheeger-Müller theorem relating analytic torsion with the Reidemeister torsion of the natural compactification by a manifold with boundary.

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arXiv2019

Geometrically finite Poincaré-Einstein metrics

We construct new examples of Einstein metrics by perturbing the conformal infinity of geometrically finite hyperbolic metrics and by applying the inverse function theorem in suitable weighted Hölder spaces.

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arXiv2019

Quasi-asymptotically conical Calabi-Yau manifolds

We construct new examples of quasi-asymptotically conical (QAC) Calabi-Yau manifolds that are not quasi-asymptotically locally Euclidean (QALE). We do so by first providing a natural compactification of QAC-spaces by manifolds with fibred corners and by giving a definition of QAC-metrics in terms of an associated Lie algebra of smooth vector fields on this compactification. Thanks to this compactification and the Fredholm theory for elliptic operators on QAC-spaces developed by the second author and Mazzeo, we can in many instances obtain Kähler QAC-metrics having Ricci potential decaying sufficiently fast at infinity. This allows us to obtain QAC Calabi-Yau metrics in the Kähler classes of these metrics by solving a corresponding complex Monge-Ampère equation.

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arXiv2019

Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

On an odd-dimensional oriented hyperbolic manifold of finite volume with strongly acyclic coefficient systems, we derive a formula relating analytic torsion with the Reidemeister torsion of the Borel-Serre compactification of the manifold. In a companion paper, this formula is used to derive exponential growth of torsion in cohomology of arithmetic groups.

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arXiv2018

Resolvent, heat kernel and torsion under degeneration to fibered cusps

Manifolds with fibered cusps are a class of complete noncompact Riemannian manifolds including all locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary.

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arXiv2017

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

We establish a Cheeger-Muller theorem for unimodular representations satisfying a Witt condition on a noncompact manifold with cusps. This class of spaces includes all non-compact hyperbolic spaces of finite volume, but we do not assume that the metric has constant curvature nor that the link of the cusp is a torus. We use renormalized traces in the sense of Melrose to define the analytic torsion and we relate it to the intersection R-torsion of Dar of the natural compactification to a stratified space. Our proof relies on our recent work on the behavior of the Hodge Laplacian spectrum on a closed manifold undergoing degeneration to a manifold with fibered cusps.

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