Róbert Szőke

The Tangent Bundle of an Almost Complex Manifold

Motivated by deformation theory of holomorphic maps between almost complex manifolds we endow, in a natural way, the tangent bundle of an almost complex manifold with an almost complex structure. We describe various properties of this structure.

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 Hs of Hilbert spaces, and the question arises if the spaces Hs are canonically isomorphic. Axelrod etxa0al. (J. Diff. Geo. 33:787–902, 1991) and Hitchin (Commun. Math. Phys. 131:347–380, 1990) suggest viewing Hs 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

Quantization of compact Riemannian symmetric spaces

{E to Y}

A family of Kahler-Einstein manifolds and metric rigidity of Grauert tubes

E→Y along a non–proper map

Adapted complex structures and Riemannian homogeneous spaces

{Y to S}

Uniqueness in geometric quantization

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 Lempert and Szőke (Bull. Lond. Math. Soc. 44:367–374, 2012). 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.