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Sub-twistor metrics

Dmitrii Korshunov

Abstract

We consider a natural distance function on the period space of a hyperkähler manifold associated to non-holonomic constraints imposed by twistor lines. These metrics were introduced by Verbitsky in the context of the global Torelli theorem for hyperkäler manifolds. We show that they are Finsler and explicitly describe their Finsler norms. To achieve this goal we consider a class of distance functions (called here sub-conic metrics) that slightly generalize sub-Riemann metrics. The main technical result is the statement that every sub-conic metric is sub-Finsler. The content and methods of the paper lie within basic metric geometry. The complex geometrical background and motivation are isolated in the appendix.

Sub-twistor metrics

Abstract

We consider a natural distance function on the period space of a hyperkähler manifold associated to non-holonomic constraints imposed by twistor lines. These metrics were introduced by Verbitsky in the context of the global Torelli theorem for hyperkäler manifolds. We show that they are Finsler and explicitly describe their Finsler norms. To achieve this goal we consider a class of distance functions (called here sub-conic metrics) that slightly generalize sub-Riemann metrics. The main technical result is the statement that every sub-conic metric is sub-Finsler. The content and methods of the paper lie within basic metric geometry. The complex geometrical background and motivation are isolated in the appendix.

Paper Structure

This paper contains 16 sections, 26 theorems, 49 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Sub-conic metrics
  3. Sub-conic and sub-Finsler metrics
  4. Sub-conical metrics are sub-Finsler
  5. Zig-zag approximation
  6. Accessible set
  7. Sub-twistor metric
  8. Homogeneous cones
  9. The period space
  10. Sub-twistor cone
  11. Sub-twistor norm
  12. Global Torelli theorem
  13. Hyperkähler manifolds
  14. Teichmüller and birational Teichmüller spaces
  15. Period map and period domain
  16. ...and 1 more sections

Key Result

Proposition 2.1.1

The norm $\left\lVert\cdot\right\rVert_{g,D}$ is an upper semi-continuous function on $\langle D\rangle$ (viewed as a subset of the total space of $TM$).

Figures (4)

  • Figure 1: Finsler norm of Manhattan metric
  • Figure 2: Manhattan metric
  • Figure 3: Zig-zag approximation
  • Figure 4: Positive sphere metric

Theorems & Definitions (88)

  • Definition 2.1.1
  • Remark 2.1.1
  • Definition 2.1.2
  • Remark 2.1.2
  • Definition 2.1.3
  • Definition 2.1.4
  • Remark 2.1.3
  • Definition 2.1.5
  • Definition 2.1.6
  • Example 2.1.1
  • ...and 78 more