Berezin-Toeplitz Quantization on K3 Surfaces and Hyperkähler Berezin-Toeplitz Quantization

Given a quantizable Kähler manifold, the Berezin-Toeplitz quantization scheme constructs a quantization in a canonical way. In their seminal paper Martin Bordemann, Eckhard Meinrenken and Martin Schlichenmaier proved that for a compact Kähler manifold such scheme is a well defined quantization which has the correct semiclassical limit.
However, there are some manifolds which admit more than one (non-equivalent) Kähler structure. The question arises then, whether the choice of a different Kähler structure gives rise to a completely different quantizations or the resulting quantizations are related.
An example of such objects are the so called K3 surfaces, which have some extra relations between the different Kähler structures. In this work, we consider the family of K3 surfaces which admit more than one quantizable Kähler structure and we use the relations between the different Kähler structures to study whether the corresponding quantizations are related or not.
In particular, we prove that such K3 surfaces have always Picard number 20, which implies that their moduli space is discrete, and that the resulting quantum Hilbert spaces are always isomorphic, although not always in a canonical way. However, there exists an infinite subfamily of K3 surfaces for which the isomorphism is canonical.
We also define new quantization operators on the product of the different quantum Hilbert spaces and we call this process Hyperkähler quantization.
We prove that these new operators have the semiclassical limit, as well as new properties inherited from the quaternionic numbers.