Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lines in the space of Kähler metrics

Tamás Darvas, Nicholas McCleerey

TL;DR

This work develops a comprehensive framework for weak geodesic lines in the space of Kähler metrics by extending the Ross–Witt Nyström correspondence to lines via Legendre transforms. It establishes a bijection between $L^p$ geodesic lines and zero-mass test lines, linking geometric behavior to dual potential-theoretic data and enabling explicit constructions. The authors provide new examples of geodesic lines on projective manifolds—often not induced by holomorphic vector fields—and show that some lines can be smooth, challenging prior folklore. They also analyze global geometric consequences, including Euclid’s postulates in this infinite-dimensional setting and the emergence of flat embeddings, while sharpening results on Riemann surfaces. Overall, the paper deepens the understanding of the metric geometry of the space of Kähler metrics and opens avenues for further study of parallelism and building-like structures in this context.

Abstract

We establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. We construct a wide range of weak geodesic lines on arbitrary projective Kähler manifolds that are not generated by holomorphic vector fields, in the process disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.

Lines in the space of Kähler metrics

TL;DR

This work develops a comprehensive framework for weak geodesic lines in the space of Kähler metrics by extending the Ross–Witt Nyström correspondence to lines via Legendre transforms. It establishes a bijection between geodesic lines and zero-mass test lines, linking geometric behavior to dual potential-theoretic data and enabling explicit constructions. The authors provide new examples of geodesic lines on projective manifolds—often not induced by holomorphic vector fields—and show that some lines can be smooth, challenging prior folklore. They also analyze global geometric consequences, including Euclid’s postulates in this infinite-dimensional setting and the emergence of flat embeddings, while sharpening results on Riemann surfaces. Overall, the paper deepens the understanding of the metric geometry of the space of Kähler metrics and opens avenues for further study of parallelism and building-like structures in this context.

Abstract

We establish a Ross-Witt Nyström correspondence for weak geodesic lines in the (completed) space of Kähler metrics. We construct a wide range of weak geodesic lines on arbitrary projective Kähler manifolds that are not generated by holomorphic vector fields, in the process disproving a folklore conjecture popularized by Berndtsson. Remarkably, some of these weak geodesic lines turn out to be smooth. In the case of Riemann surfaces, our results can be significantly sharpened. Finally, we investigate the validity of Euclid's fifth postulate for the space of Kähler metrics.

Paper Structure

This paper contains 18 sections, 18 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Ross--Witt Nyström correspondence for lines.
  3. Lines on Riemann surfaces.
  4. Examples on projective manifolds.
  5. Euclid's fifth postulate.
  6. Flat embeddings and spherical buildings.
  7. Acknowledgments.
  8. Preliminaries
  9. Subgeodesics and test curves.
  10. Bounded and finite energy rays.
  11. Maximization of test curves.
  12. Finite energy Ross--Witt Nyström correspondence.
  13. Lines and test lines
  14. Sublinear subgeodesic lines and test lines.
  15. Geodesic turns and maximal test lines.
  16. ...and 3 more sections

Key Result

Theorem 1.1

The map $\{u_t\}_t \to \{\hat{u}_\tau\}_\tau$ between $L^p$ (bounded) geodesic lines and zero mass $L^p$ (bounded) test lines is bijective, with inverse $\{v_\tau\}_\tau \to \{\check v_t\}_t.$

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Example 1.3
  • Example 1.4
  • Remark 1.5
  • Theorem 1.6
  • Example 1.7
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • ...and 30 more