Star products with separation of variables admitting a smooth extension

TL;DR

This paper investigates conditions under which star products with separation of variables, defined on a dense subset of a complex manifold with a pseudo-Kaehler form, can be smoothly extended to the entire manifold.

Contribution

The paper provides a simple criterion for the smooth extension of star products with separation of variables and demonstrates its application to specific examples.

Findings

Identified examples where star products extend smoothly beyond their initial domain.

Proposed a criterion for the existence of smooth extensions of star products.

Showed that certain pseudo-Kaehler structures admit such extensions.

Abstract

Given a complex manifold $M$ with an open dense subset $\Omega$ endowed with a pseudo-Kaehler form $\omega$ which cannot be smoothly extended to a larger open subset, we consider various examples where the corresponding Kaehler-Poisson structure and a star product with separation of variables on $(\Omega, \omega)$ admit smooth extensions to $M$. We suggest a simple criterion of the existence of a smooth extension of a star product and apply it to these examples.