- Bismut hypoelliptic Laplacians for manifolds with boundaries, with F. Nier. Preprint.
Boundary conditions for Bismut's hypoelliptic Laplacian which naturally correspond to Dirichlet and Neumann boundary conditions for Hodge Laplacians are considered. Those are related with specific boundary conditions for the differential and its various adjoints. Once the closed realizations of those operators are well understood, the commutation of the differential with the resolvent of the hypoelliptic Laplacian is checked with other properties like the PT-symmetry, which are important for the spectral analysis.
- Coherent sheaves, superconnections, and RRG, with J.-M. Bismut and Z. Wei. Preprint.
Given a compact complex manifold, the purpose of this paper is to construct the Chern character for coherent sheaves with values in Bott-Chern cohomology, and to prove a corresponding Riemann-Roch-Grothendieck formula. Our paper is based on a fundamental construction of Block.
- Complex valued analytic torsion and dynamical zeta function on locally
symmetric spaces. Preprint.
We show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent series at the zero point is related to the regularised determinant of the flat Laplacian of Cappell-Miller. When the flat vector bundle is close to an acyclic and unitary one, we show that the dynamical zeta function is regular at the zero point and that its value is equal to the complex valued analytic torsion of Cappell-Miller. This generalises author's previous results for unitarily flat vector bundles as well as Müller and Spilioti's results on hyperbolic manifolds.
- Intégrales orbitales semi-simples et centre de l'algèbre
enveloppante, with J.-M. Bismut. C. R. Acad. Sci. Paris 357 (2019), 897-906.
Dans une Note antérieure, le premier auteur a donné une formule locale explicite pour les intégrales orbitales semi-simples associées au Casimir. Dans cette Note, nous étendons cette formule à tous les éléments du centre de l'algèbre enveloppante de l'algèbre de Lie considérée.
- Geometric orbital integrals and the center of the enveloping
algebra, with J.-M. Bismut.
The purpose of this paper is to extend the explicit geometric evaluation of semisimple orbital integrals for smooth kernels for the Casimir operator obtained by the first author to the case of kernels for arbitrary elements in the center of the enveloping algebra.
- Analytic torsion, dynamical zeta function, and the Fried conjecture for
admissible twists. Comm. Math. Phys. 387 (2021), 1215–1255.
We show an equality between the analytic torsion and the absolute value at the zero point of the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an acyclic flat vector bundle obtained by the restriction of a representation of the underlying Lie group. This generalises author's previous result for unitarily flat vector bundles, and the results of Bröcker, Müller, and Wotzke on closed hyperbolic manifolds.
- Dynamical zeta functions in the nonorientable case, with Y. Borns-Weil. Nonlinearity 34 (2021), 7322-7334.
We use a simple argument to extend the microlocal proofs of meromorphicity of dynamical zeta functions to the nonorientable case. In the special case of geodesic flow on a connected non-orientable negatively curved closed surface, we compute the order of vanishing of the zeta function at the zero point to be the first Betti number of the surface.
- Analytic torsion and dynamical flow: A survey on the Fried
conjecture. Arithmetic L-Functions and Differential Geometric Methods, 247-299. Progr. Math., 338, (2021).
Given an acyclic and unitarily flat vector bundle on a closed
manifold, Fried conjectured an equality between the analytic
torsion and the value at zero of the Ruelle zeta function
associated to a dynamical flow.
In this survey, we review the Fried conjecture for different
flows, including the suspension flow, the Morse-Smale flow, the geodesic
flow, and the Anosov flow.
- Morse-Smale flow, Milnor metric, and dynamical zeta function, with Jianqing Yu. J. Éc. polytech. Math. 8 (2021), 585-607.
With the help of interactions between the fixed points and the closed orbits of a Morse-Smale flow, we introduce a
Milnor metric on the determinant line of the cohomology of the
underlying closed manifold with coefficients in a flat vector
This allows us to generalise the notion of the absolute value at zero
point of the
Ruelle dynamical zeta function, even in the case where this
value is not well
defined in the classical sense. We give a
formula relating the Milnor metric and the Ray-Singer metric. An
essential ingredient of our proof is Bismut-Zhang's Theorem.
- The Fried conjecture in small dimensions, with N.V. Dang, C. Guillarmou, and G. Rivière.
Invent. Math. 220 (2020), 525–579.
We study the twisted Ruelle zeta function \(\zeta_X(s)\) for smooth Anosov vector fields \(X\) acting on flat vector bundles over smooth compact manifolds.
In dimension \(3\), we prove Fried conjecture, relating Reidemeister torsion and \(\zeta_X(0)\). In higher dimensions, we show more generally that \(\zeta_X(0)\) is locally constant with respect to the vector field \(X\) under a spectral condition.
As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic \(3\)-manifolds.
This gives the first examples of non-analytic Anosov flows and geodesic flows in
variable negative curvature where Fried conjecture holds true.
- Flat vector bundles and analytic torsion on orbifolds, with Jianqing Yu. Comm. Anal. Geom.
This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds.
We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.
- Analytic torsion, dynamical zeta functions, and the Fried
conjecture. Anal. PDE. 11 (2018), 1-74.
We prove the equality of the analytic torsion and the value at zero of a Ruelle dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold. This solves a conjecture of Fried. This article should be read in conjunction with an earlier paper by Moscovici and Stanton.
- Laplacien hypoelliptique, torsion analytique, et théorème de
Cheeger-Müller. J. Funct. Anal. 270 (2016), 2817-2999.
L'objet de cet article est de démontrer une formule reliant les métriques de Ray–Singer hypoelliptique et de Milnor sur le déterminant de la cohomologie d'une variété riemannienne compacte par une déformation à la Witten du laplacien hypoelliptique en théorie de de Rham.
- Analytic torsion, dynamical zeta functions and orbital integrals.
C. R. Acad. Sci. Paris 354 (2016), 433-436.
The purpose of this Note is to prove an identity between the analytic torsion and the value at zero of a dynamical zeta function associated with an acyclic unitarily flat vector bundle on a closed locally symmetric reductive manifold, which solves a conjecture of Fried.
- The hypoelliptic Laplacian, analytic torsion and Cheeger–Müller
Theorem. C. R. Acad. Sci. Paris 352 (2014), 153-156.
The purpose of this Note is to prove a formula relating the hypoelliptic Ray-Singer metric and the Milnor metric on the determinant of the cohomology of a compact Riemannian manifold by a Witten-like deformation of the hypoelliptic Laplacian in de Rham theory.