arXiv subjects

Róbert Szőke

Publications and source records attributed to Róbert Szőke.

8 recordsLinked to original sources

arXiv2026

Polarization complete left invariant connections

Let $(M,\nabla)$ be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood $N$ of the zero section in $TM$ is a complex structure on $N$ such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization $P$ on $N$ is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to $P$. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply $\mathcal P$-complete) if the ac-structure (resp. the ac-polarization) exists on $TM$. Although on a small enough $N$ an ac-structure always exists, entire connections are rear and if a maximal domain of definition $N_{max}$ of an ac-structure exists (different from $TM$), $N_{max}$ is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole $TM$. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are $\mathcal P$-complete.

math.DG↗

arXiv2026

Bergman spaces on algebraic curves

A theorem of Wiegerinck asserts that the Bergman space of an open subset of the complex numbers is either infinite-dimensional or trivial. Recently, this has been generalized to holomorphic vector bundles over the projective line by the third author and later to vector bundles over any compact Riemann surface by Gallagher, Gupta and Vivas. In the present paper we extend the above results to the case of certain singular metrics associated to divisors on a Riemann surface. As corollaries we obtain versions of Wiegerinck's theorem for both projective and affine algebraic curves.

math.CV↗

arXiv2022

On a theorem of Wiegerinck

A theorem of Wiegerinck says that the Bergman space over any domain in $\mathbb C$ is either trivial or infinite dimensional. We generalize this theorem in the following form. Let E be a hermitian, holomorphic vector bundle over $\mathbb P^1$, the later equipped with a volume form and $D$ an arbitrary domain in $\mathbb P^1$. Then the space of holomorphic L2 sections of $E$ over $D$ is either equal to $H^0(M,E)$ or it has infinite dimension.

math.CV↗

arXiv2018

Smooth structures on the field of prequantum Hilbert spaces

When there is a family of complex structures on the phase space, parametrized by a set $S$, the prequantum Hilbert spaces produced by geometric quantization, using the half-form correction, also depends on these parameters. This way we obtain a field of Hilbert spaces $p:H^{pr Q}\rightarrow S$. We show that this field can have natural inequivalent smooth Hilbert bundle structures.

math-ph↗

arXiv2017

Quantization of compact Riemannian symmetric spaces

The phase space of a compact, irreducible, simply connected, Riemannian symmetric space admits a natural family of Kähler polarizations parametrized by the upper half plane $S$. Using this family, geometric quantization, including the half-form correction, produces the field $H^{corr}\rightarrow S$ of quantum Hilbert spaces. We show that projective flatness of $H^{corr}$ implies, that the symmetric space must be isometric to a compact Lie group equipped with a biinvariant metric. In the latter case the flatness of $H^{corr}$ was previously established.

math-ph↗

arXiv2013

Direct images, fields of Hilbert spaces, and geometric quantization

Geometric quantization often produces not one Hilbert space to represent the quantum states of a classical system but a whole family $H_s$ of Hilbert spaces, and the question arises if the spaces $H_s$ are canonically isomorphic. [ADW] and [Hi] suggest to view $H_s$ as fibers of a Hilbert bundle $H$, introduce a connection on $H$, and use parallel transport to identify different fibers. Here we explore to what extent this can be done. First we introduce the notion of smooth and analytic fields of Hilbert spaces, and prove that if an analytic field over a simply connected base is flat, then it corresponds to a Hermitian Hilbert bundle with a flat connection and path independent parallel transport. Second we address a general direct image problem in complex geometry: pushing forward a Hermitian holomorphic vector bundle $E-->Y$ along a non-proper map $Y-->S$. We give criteria for the direct image to be a smooth field of Hilbert spaces. Third we consider quantizing an analytic Riemannian manifold $M$ by endowing $TM$ with the family of adapted Kähler structures from arxiv:1004.4069 [LSz]. This leads to a direct image problem. When $M$ is homogeneous, we prove the direct image is an analytic field of Hilbert spaces. For certain such $M$---but not all---the direct image is even flat; which means that in those cases quantization is unique.

math-ph↗

arXiv2012

Curvature of fields of quantum Hilbert spaces

We show that using the family of adapted Kähler polarizations of the phase space of a compact, simply connected, Riemannian symmetric space of rank-1, the obtained field $H^{corr}$ of quantum Hilbert spaces produced by geometric quantization including the half-form correction is flat if $M$ is the 3-dimensional sphere and not even projectively flat otherwise.

math-ph↗

arXiv2010

A new look at adapted complex structures

Given a closed real analytic Riemannian manifold, we construct and study a one parameter family of adapted complex structures on the manifold of its geodesics.

math.CV↗