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Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface

Rajas Sandeep Sompurkar

Abstract

In this paper we prove that on a special type of minimal ruled surface, which is an example of a `pseudo-Hirzebruch surface', every Kähler class admits a certain kind of `higher extremal Kähler metric', which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a `higher constant scalar curvature Kähler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface.

Existence of Higher Extremal Kähler Metrics on a Minimal Ruled Surface

Abstract

In this paper we prove that on a special type of minimal ruled surface, which is an example of a `pseudo-Hirzebruch surface', every Kähler class admits a certain kind of `higher extremal Kähler metric', which is a Kähler metric whose corresponding top Chern form and volume form satisfy a nice equation motivated by analogy with the equation characterizing an extremal Kähler metric. From an already proven result, it will follow that this specific higher extremal Kähler metric cannot be a `higher constant scalar curvature Kähler (hcscK) metric', which is defined, again by analogy with the definition of a constant scalar curvature Kähler (cscK) metric, to be a Kähler metric whose top Chern form is harmonic. By doing a certain set of computations involving the top Bando-Futaki invariant we will conclude that hcscK metrics do not exist in any Kähler class on this surface.
Paper Structure (16 sections, 29 theorems, 80 equations)

This paper contains 16 sections, 29 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Background of the Study
  3. Overview of the Paper
  4. Constructing Higher Extremal Kähler Metrics on a Minimal Ruled Surface
  5. The Minimal Ruled Surface and its Kähler Cone
  6. Description of the Momentum Construction Method
  7. Analysis of the ODE BVP for the Momentum Profile
  8. Proof of the Main Result
  9. First Part of the Proof
  10. Second Part of the Proof
  11. Bando-Futaki Invariants and hcscK Metrics
  12. Generalization of the Results to All Pseudo-Hirzebruch Surfaces
  13. The Kähler Cone of a Pseudo-Hirzebruch Surface
  14. The Momentum Construction Method Applied to a Pseudo-Hirzebruch Surface
  15. The Case of Positive Degree
  16. ...and 1 more sections

Key Result

Theorem 2.1.1

$\alpha$ is a Kähler class on $X$ if and only if the following conditions are satisfied:

Theorems & Definitions (51)

  • Theorem 2.1.1: Nakai-Moishezon Criterion
  • Corollary 2.1.1: Fujiki, Tønnesen-Friedman
  • Lemma 2.3.1: Pingali
  • Theorem 2.3.1: Pingali
  • Theorem 2.3.2
  • Corollary 2.3.1
  • Remark 2.3.1
  • Corollary 2.3.2
  • Lemma 3.1.1: Pingali
  • Lemma 3.1.2
  • ...and 41 more