Quantizing Geodesics in K\"ahler and 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.