Direct images as Hilbert fields and their curvatures
Dat Vu Tran · Purdue e-Pubs (Purdue University System) · 2014
When a complex domain varies in a family, its Bergman space of holomorphic functions will also vary and will form what is called a field of Hilbert spaces, which is a generalization of vector bundle. This field of Hilbert spaces will be smooth in certain situations. In this dissertation, we will provide some conditions which will make this Hilbert field smooth. Then, we will compute the curvature of this Hilbert field and relate it to the pseudoconvex property of the variation. After that, we will provide an example illustrating the connection, or more accurately, the possibility of a lack of a connection between Hilbert field and vector bundle. In the end, we will introduce some partial results regarding the abstract smooth Hilbert fields.