Quantizing Geodesics in K\"ahler and Sasaki Geometry

TL;DR

This paper investigates the quantization of geodesics in K"ahler and Sasaki geometry, establishing conditions under which finite-dimensional approximations correspond to infinite-dimensional geodesic paths and extending these methods to more general settings.

Contribution

It provides a precise characterization of when Fubini-Study images of geodesics are quasi-geodesics and introduces a new quantization procedure for geodesics on K"ahler and Sasaki varieties.

Findings

Characterization of Fubini-Study images as quasi-geodesics

Quantization procedure for geodesics on K"ahler varieties

Extension of quantization methods to Sasaki geometry

Abstract

The space of K\"ahler potentials can be quantized through the classical Fubini-Study map, relating infinite-dimensional geometric structures to finite-dimensional symmetric spaces. We prove (exactly) when the Fubini-Study image of a geodesic line in the space of positive definite Hermitian matrices gives rise to a quasi-geodesic in the space of K\"ahler potentials. Furthermore, we introduce a quantization procedure for geodesics between potentials on normal K\"ahler varieties and show how this construction extends to the Sasaki setting.