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Quantizing Geodesics in Kähler and Sasaki Geometry

Gilles Courtois, Eleonora Di Nezza, Thomas Franzinetti

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.

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.
Paper Structure (5 sections, 14 theorems, 75 equations, 4 figures)

This paper contains 5 sections, 14 theorems, 75 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper.
  3. The Fubini-Study map $\mathop{\mathrm{FS}}\nolimits_k$
  4. Quantization in the semi-positive case
  5. Quantization in Sasaki geometry

Key Result

Theorem 1

The map $\mathop{\mathrm{FS}}\nolimits_k : (\hat{\mathcal{H}}(L^k) , \hat{d}) \rightarrow (\mathcal{H}_\omega, d_p)$ is Lipschitz. More precisely, Here, $C_k$ is an explicit constant depending only on $k, X$, $\omega$ and $p$.

Figures (4)

  • Figure 1: Step 1
  • Figure : Step 1
  • Figure : Step 2
  • Figure : $\qquad\quad$ Step 3

Theorems & Definitions (25)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 2.4: darvas735geometric
  • Lemma 2.5
  • proof
  • ...and 15 more