Publications

Differential geometry and homogeneous spaces. Springer Universitext 2024.
Abstract: This textbook offers a rigorous introduction to the foundations of Riemannian Geometry, with a detailed treatment of homogeneous and symmetric spaces, as well as the foundations of the General Theory of Relativity.
Starting with the basics of manifolds, it presents key objects of differential geometry, such as Lie groups, vector bundles, and de Rham cohomology, with full mathematical details. Next, the fundamental concepts of Riemannian geometry are introduced, paving the way for the study of homogeneous and symmetric spaces. As an early application, a version of the Poincaré–Hopf and Chern–Gauss–Bonnet Theorems is derived. The final chapter provides an axiomatic deduction of the fundamental equations of the General Theory of Relativity as another important application. Throughout, the theory is illustrated with color figures to promote intuitive understanding, and over 200 exercises are provided (many with solutions) to help master the material.
The book is designed to cover a two-semester graduate course for students in mathematics or theoretical physics and can also be used for advanced undergraduate courses. It assumes a solid understanding of multivariable calculus and linear algebra.

Analytic torsion forms for fibrations by projective curves. Math. Z. 305;47 (2023).
Abstract: An explicit formula for analytic torsion forms for fibrations by projective curves is given. In particular one obtains a formula for direct images in Arakelov geometry in the corresponding setting. The main tool is a new description of Bismut's equivariant Bott-Chern current in the case of isolated fixed points.

Differentialgeometrie und homogene Räume.
2., vollständig überarbeitete und ergänzte Auflage, Springer Spektrum 2020.

Differentialgeometrie und homogene Räume. Springer Spektrum 2014.
Abstract: Das Ziel dieses Buches ist, im Umfang einer zweisemestrigen Vorlesung die wichtigsten Grundlagen der Riemannschen Geometrie mit allen notwendigen Zwischenresultaten bereitzustellen und die zentrale Beispielklasse der homogenen Räume ausführlich darzustellen. Homogene Räume sind Riemannsche Mannigfaltigkeiten, deren Isometriegruppe transitiv auf ihnen operiert. Alternativ lassen sie sich als Quotienten von Lie-Gruppen durch Untergruppen beschreiben. Homogene Räume spielen in vielen Gebieten der Mathematik eine wichtige Rolle, etwa als Modulräume, deren Punkte Lösungen eines mathematischen Problems parametrisieren. Symmetrische Räume, d.h. Räume, die an jedem Punkt eine Punktspiegelung erlauben, werden als Spezialfall in einem eigenen Kapitel behandelt. Im letzten Kapitel werden als eine wichtige Anwendung der Riemannschen Geometrie einige Grundlagen der allgemeinen Relativitätstheorie axiomatisch deduziert.

A Hirzebruch proportionality principle in Arakelov geometry. in: Number Fields and Function Fields--Two Parallel Worlds , Geer, G. van der; Moonen, B. J.J.; Schoof, R. (Hrsg.), Birkhäuser PM 239 (2005).
2000 MSC: 14G40, 58J52, 20G05, 20G10, 14M17.
Abstract: We describe a tautological subring in the arithmetic Chow ring of bases of abelian schemes. Among the results are an Arakelov version of the Hirzebruch proportionality principle and a formula for a critical power of the first Arakelov Chern class of the Hodge bundle.
Old version (i.e. without the last section): Prépublication de l'institut de mathématiques de Jussieu no. 284 (Avril 2001).

with Gregor Weingart, Quaternionic analytic torsion. (PDF) Adv. Math. 178/2 (2003) 375-395.
Abstract: We define an (equivariant) quaternionic analytic torsion for antiselfdual vector bundles on quaternionic Kähler manifolds, using ideas by Leung and Yi. We compute this torsion for vector bundles on quaternionic homogeneous spaces with respect to any isometry in the component of the identity, in terms of roots and Weyl groups.

with D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry IV: the modular height of C.M. abelian varieties. (PDF) J. reine angew. Math. 556 (2003), 127-148.
Abstract: We give a new proof of a slightly weaker form of a theorem of P. Colmez. This theorem gives a formula for the Faltings height of abelian varieties with complex multiplication by a C.M. field whose Galois group over Q is abelian; it reduces to the formula of Chowla and Selberg in the case of elliptic curves. We show that the formula can be deduced from the arithmetic fixed point formula proved in [KR2]. Our proof is intrinsic in the sense that it does not rely on the computation of the periods of any particular abelian variety.

with Ch. Kaiser, A fixed point formula of Lefschetz type in Arakelov geometry III: representations of Chevalley schemes and heights of flag varieties. (PDF) Invent. Math. 147 (2002), 633-669.
Abstract: We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.

with D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula. (PDF) Ann. Inst. Fourier 52 (2002), 81--103.
Abstract: This is the second of a series of papers dealing with an Arakelovian analog of the holomorphic Lefschetz fixed point formula. We use the main result of the first paper to prove a residue formula for arithmetic characteristic classes living on arithmetic varieties acted upon by a diagonalisable torus; here recent results of Bismut-Goette on the equivariant analytic torsion play a key role.

with D. Roessler, A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof (PDF), Invent. Math. 145 (2001), 333-396 ; announced in C. R. Acad. Sci. 326 (1998), 719-722.
Abstract: We consider arithmetic varieties endowed with an action of the group scheme of n-th roots of unity and we define equivariant arithmetic K0-theory for these varieties. We use the equivariant analytic torsion to define direct image maps in this context and we prove a Riemann-Roch theorem for the natural transformation of equivariant arithmetic K0-theory induced by the restriction to the fixed point scheme; this theorem can be viewed as an analog, in the context of Arakelov geometry, of the regular case of the theorem proved by P. Baum, W. Fulton and G. Quart. We show that it implies an equivariant refinement of the arithmetic Riemann-Roch theorem, in a form conjectured by J.-M. Bismut.

Complex analytic torsion forms for torus fibrations and moduli spaces, (PDF) 166--195 in: Regulators in Analysis, Geometry and Number Theory, N. Schappacher, A. Reznikov (ed.), Progress in Math. 171, Birkhäuser 2000.
Abstract: We construct analytic torsion forms for line bundles on holomorphic fibrations by tori, which are not necessarily Kähler fibrations. This is done by double transgressing the top Chern class. The forms are given in terms of Epstein zeta functions. Also, we establish a corresponding double transgression formula and an anomaly formula. The forms are investigated more closely for the universal bundle over the moduli space of polarized abelian varieties and for the bundle of Jacobians over the Teichmüller space.

Equivariant Reidemeister torsion on symmetric spaces, (PDF) Math. Ann. 307 (1997), 57-69.
Abstract: We calculate explicitly the equivariant real-analytic Ray-Singer torsion for all symmetric spaces G/K of compact type with respect to the action of G. We show that it equals zero except for the odd-dimensional Graßmannians and the space SU(3)/SO(3). As a corollary, we classify up to diffeomorphism all isometries of these spaces which are homotopic to the identity; also, we classify their quotients by finite group actions up to homeomorphism.

Holomorphic torsion on Hermitian symmetric spaces, J. reine angew. Math. 460 (1995), 93-116; announced (PDF) in C. R. Acad. Sci. 319 (1994), 247-252.
Abstract: We calculate explicitly the equivariant holomorphic Ray-Singer torsion for all equivariant Hermitian vector bundles over Hermitian symmetric spaces G/K of the compact type with respect to any isometry $g\in G$. In particular, we obtain the value of the usual non-equivariant torsion. The result is shown to provide very strong support for Bismut's conjecture of an equivariant arithmetic Grothendieck-Riemann-Roch theorem.

Equivariant analytic torsion on PnC (PDF,) Math. Ann. 297 (1993), 553-565; announced (PDF) in C. R. Acad. Sci. 316 (1993), 471-476.
Abstract: The subject of the paper is to calculate an equivariant version of the complex Ray-Singer torsion for all bundles on the P1C and for the trivial line bundle on PnC, for isometries which have isolated fixed points. The result can for all n be expressed with a special function, which is very similar to the series defining the Gillet-Soulé R-genus.

Torsion analytique complexe, thesis, Université de Paris-Sud (Orsay), no. 1445, 2e trimestre 1993.
Abstract: In this thesis we are studying some aspects of the complex Ray-Singer torsion and of its generalizations.

with J.-M. Bismut, Higher analytic torsion forms for direct images and anomaly formulas, J. Alg. Geom. 1 (1992), 647-684.
Abstract: In this paper, we construct analytic torsion forms associated to Kähler fibrations, and we establish corresponding anomaly formulas.

Über den stochastischen Beweis des Atiyah-Singer-Indextheorems, Diplomarbeit, Schriftenreihe des SFB 256 no. 118 (1990), Universität Bonn.
Abstract: We give a new version of Bismut's proof of the Atiyah-Singer index theorem. Our method is close to Leandres, but more intrinsic. We do not use local coordinates and the computation of the derivation of the stochastic parallel transport is simplified. The paper contains an elementary introduction to stochastic analysis.

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