Quantizing Geodesics in Kähler and Sasaki Geometry

Abstract: The space of Kähler 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 Kähler potentials. Furthermore, we introduce a quantization procedure for geodesics between potentials on normal Kähler varieties and show how this construction extends to the Sasaki setting.