arXiv subjects

Franz W. Kamber

Publications and source records attributed to Franz W. Kamber.

9 recordsLinked to original sources

arXiv2022

Orbifold-like and proper $\mathfrak g$-manifolds

In [4] and [5], we generalized the concept of completion of an infinitesimal group action $ζ: {\mathfrak g} \to \mathfrak X (M)$ to an actual group action on a (non-compact) manifold $M$, originally introduced by R. Palais [9], and showed by examples that this completion may have quite pathological properties (much like the leaf space of a foliation). In the present paper, we introduce and investigate a tamer class of $\mathfrak g$-manifolds, called orbifold--like, for which the completion has an orbifold structure. This class of $\mathfrak g$-manifolds is reasonably well-behaved with respect to its local topological and smooth structure to allow for many geometric constructions to make sense. In particular, we investigate proper $\mathfrak g$-actions and generalize many of the usual properties of proper group actions to this more general setting.

math.DG↗

arXiv2010

The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations

In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years. This paper summarizes the work in the papers arXiv:1005.3845 [math.DG] and arXiv:1008.1757 [math.DG].

math.DG↗

arXiv2010

Index theory for basic Dirac operators on Riemannian foliations

In this paper we prove a formula for the analytic index of a basic Dirac-type operator on a Riemannian foliation, solving a problem that has been open for many years. We also consider more general indices given by twisting the basic Dirac operator by a representation of the orthogonal group. The formula is a sum of integrals over blowups of the strata of the foliation and also involves eta invariants of associated elliptic operators. As a special case, a Gauss-Bonnet formula for the basic Euler characteristic is obtained using two independent proofs.

math.DG↗

arXiv2008

A generalization of Abel's Theorem and the Abel--Jacobi map

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative to a triangulation of $X$, we define a (simplicial) $(n-d-1)$--gerbe $Λ_{Z}$, the Abel gerbe determined by $Z$, whose vanishing as a Deligne cohomology class generalizes the notion of `linear equivalence to zero'. In this setting, Abel's theorem remains valid. Moreover we generalize the classical Inversion Theorem for the Abel--Jacobi map, thereby proving that the moduli space of Abel gerbes is isomorphic to the harmonic Deligne cohomology; that is, gerbes with harmonic curvature.

math.DG↗

arXiv2004

Gerbes, simplicial forms and invariants for families of foliated bundles

The notion of a gerbe with connection is conveniently reformulated in terms of the simplicial deRham complex. In particular the usual Chern-Weil and Chern-Simons theory is well adapted to this framework and rather easily gives rise to `characteristic gerbes' associated to families of bundles and connections. In turn this gives invariants for families of foliated bundles. A special case is the Quillen line bundle associated to families of flat SU(2)-bundles

math.DG↗

arXiv2004

Dimensional reduction of the perturbed Hermitian-Einstein equation

Given a Kaehlerian holomorphic fiber bundle whose fiber is a compact homogeneous Kaehler manifold, we describe the perturbed Hermitian-Einstein equations relative to certain holomorphic vector bundles. With respect to special metrics on the holomorphic bundles, there is a dimensional reduction procedure which reduces these equations to a system of equations on the base, known as the twisted coupled vortex equations.

math.DG↗

arXiv2004

A Fourier-Mukai transform for real torus bundles

We construct a Fourier--Mukai transform for smooth complex vector bundles $E$ over a torus bundle $π:M \to B,$ the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles $E$ with a flat partial unitary connection, that is families or deformations of flat vector bundles (or unitary local systems) on the torus $T.$ This leads to a correspondence between such objects on $M$ and relative skyscraper sheaves $\cS$ supported on a spectral covering $Σ\hra \what M,$ where $\hatπ:\what{M} \to B$ is the flat dual fiber bundle. Additional structures on $(E,\nabla)$ (flatness, anti-self-duality) will be reflected by corresponding data on the transform $(\cS, Σ).$ Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture

math.DG↗

arXiv2003

Completing Lie algebra actions to Lie group actions

For a finite dimensional Lie algebra $\g$ of vector fields on a manifold $M$ we show that $M$ can be completed to a $G$-space in a unversal way, which however is neither Hausdorff nor $T_1$ in general. Here $G$ is a connected Lie group with Lie-algebra $\g$. For a transitive $\g$-action the completion is of the form $G/H$ for a Lie subgroup $H$ which need not be closed. In general the completion can be constructed by completing each $\g$-orbit.

math.DG↗

arXiv2000

The flow completion of a manifold with vector field

For a vector field $X$ on a smooth manifold $M$ there exists a smooth but not necessarily Hausdorff manifold $M_\Bbb R$ and a complete vector field $X_\Bbb R$ on it which is the universal completion of $(M,X)$.

math.DG↗