Searcharxiv⌕ Search

arXiv subjects

Roland Knevel

Publications and source records attributed to Roland Knevel.

7 recordsLinked to original sources
arXiv2012

Versal Families of Compact Super Riemann Surfaces

We call every complex connected (1,1)-dimensional supermanifold a super Riemann surface and construct versal super families of compact ones, where the base spaces are allowed to be certain ringed spaces including all complex supermanifolds. Furthermore we choose maximal supersymmetric sub super families which turn out to be versal among all supersymmetric super families. In the cases where special divisors occur we prove the non-existence of versal super families and instead construct locally complete ones. For an accurate study of supersymmetric super families we introduce the duality functor, a covariant involution of the category of super families of compact super Riemann surfaces, and show that the supersymmetric super families are essentially the self-dual ones. As an application of the classification results it is shown that on a supersymmetric super family of compact super Riemann surfaces locally in the base the supersymmetry is uniquely determined up to pullback by automorphisms with identity as body.

math.CV↗
arXiv2012

Stability of the space of Automorphic Forms under Local Deformations of the Lattice

First we explain the concept of local deformation over a 'parameter' algebra P, in particular the notion of a P-lattice in a Lie group. Purpose of this article is to define the spaces of automorphic resp. cusp forms on the upper half plane H for a P- (!) lattice of SL(2, R) and to investigate their structure. It turns out that in almost all cases these spaces are free modules over the complexified P of rank equal to the dimension of the spaces of automorphic resp. cusp forms for the body, which is the associated ordinary lattice in SL(2, R) . In other words almost every automorphic resp. cusp form admits an 'adaption' to local deformations of the lattice. This is shown by giving the quotient of H by the P-lattice together with the cusps the structure of a P- Riemann surface and writing the spaces of automorphic resp. cusp forms as global sections of holomorphic P- (!) line bundles on this quotient.

math.CV↗
arXiv2012

Super Automorphic Forms on the Super Upper Half Plane

The super upper half plane, this is the ordinary upper half plane with additional odd (anticommuting) directions, admits a transitive super action of a certain super Lie group G . First we define the spaces of super automorphic and cusp forms on the super upper half plane for an ordinary lattice in G and give an asymptotic formula for their dimensions for high weight. For involving also the odd directions of G we introduce local super deformation of lattices in G and show that for high weight the spaces of super automorphic and cusp forms are stable under such local super deformations.

math.CV↗
arXiv2012

Integrating P- super vectorfields and the super geodesic flow

Aim of this article is to introduce the notion of integral and geodesic flows on P-supermanifolds as certain partial actions of R . First I introduce the concept of parametrization over a `small' super algebra P, which leads to the notion of P-objects and is superized local deformation theory. It is shown how parametrization makes the theory much easier. A version of Palais' theorem for P-supermanifolds is obtained stating that every infinitesimal P-action of a simply connected P-super Lie group G on a P-supermanifold can be integrated to a whole action of G . Furthermore the faithful linearization of affine P-supermorphisms is proven. Finally I show that Newton's, Lagrange's and Hamilton's approach to mechanics can be formulated also for P- Riemannian supermanifolds and are infact equivalent.

math.DG↗
arXiv2009

A Satake type theorem for Super Automorphic forms

Aim of this article is a Satake type theorem for super automorphic forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. This theorem - roughly speaking - says that for large weight k and all p from 1 to infinity (both including) a super automorphic form on B is a super cusp form if and only if its p-norm with respect to a certain measure on the quotient of B is finite. And so in particular all these Lp-spaces coincide! We will give a proof of this theorem using an unbounded realization of B and Fourier decomposition at the cusps of the quotient mapped to infinity via a partial Cayley transform.

math.CV↗
arXiv2009

A Spanning Set for the space of Super Cusp forms

Aim of this article is the construction of a spanning set for the space of super cusp forms on a complex bounded symmetric super domain B of rank 1 with respect to a lattice. The main ingredients are a generalization of the Anosov closing lemma for partially hyperbolic diffeomorphisms and an unbounded realization of B, in particular Fourier decomposition at the cusps mapped to infinity via a partial Cayley transformation. The elements of the spanning set are in finite-to-one correspondence with closed geodesics, the number of elements corresponding to a geodesic growing linearly with its length.

math.CV↗