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The uniqueness of Poincaré type constant scalar curvature Kähler metric

Yulun Xu, Kai Zheng

Abstract

Let $D$ be a smooth divisor on a closed Kähler manifold $X$. First, we prove that Poincaré type constant scalar curvature Kähler (cscK) metric with a singularity at $D$ is unique up to a holomorphic transformation on $X$ that preserves $D$, if there are no nontrivial holomorphic vector fields on $D$. For the general case, we propose a conjecture relating the uniqueness of Poincaré type cscK metric to its asymptotic behavior near $D$. We give an affirmative answer to this conjecture for those Poincaré type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of $X$ that preserve $D$. We also show that this conjecture can be reduced to a fixed point problem.

The uniqueness of Poincaré type constant scalar curvature Kähler metric

Abstract

Let be a smooth divisor on a closed Kähler manifold . First, we prove that Poincaré type constant scalar curvature Kähler (cscK) metric with a singularity at is unique up to a holomorphic transformation on that preserves , if there are no nontrivial holomorphic vector fields on . For the general case, we propose a conjecture relating the uniqueness of Poincaré type cscK metric to its asymptotic behavior near . We give an affirmative answer to this conjecture for those Poincaré type cscK metrics whose asymptotic behavior is invariant under any holomorphic transformation of that preserve . We also show that this conjecture can be reduced to a fixed point problem.

Paper Structure

This paper contains 28 sections, 42 theorems, 343 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. Background metric of Poincaré type
  5. Quasi coordinates
  6. Function spaces
  7. Poincaré type $C^{1,1}$ geodesic
  8. Energy functionals
  9. Fiber bundle structure of a neighbourhood of D
  10. Holomorphic vector fields
  11. Convexity of the K-energy
  12. Subharmonicity of the K-energy
  13. Continuity of the K-energy
  14. Solvability of the Lichnerowicz operator
  15. Kernel and range
  16. ...and 13 more sections

Key Result

Theorem 1.1

Suppose that $\mathbf{h}^D=0$, where $\mathbf{h}^D$ denotes the set of holomorphic vector fields on $D$. Given two Poincaré type cscK metrics $\omega_1$ and $\omega_2$ in a given cohomology class. Then there exists an element $g\in Aut_0^D(X)$ such that $\omega_1 =g^* \omega_2$.

Theorems & Definitions (82)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.1
  • Remark 1.3
  • Definition 1.4
  • Theorem 1.2
  • Remark 1.6
  • Theorem 1.3
  • Lemma 3.1
  • proof
  • ...and 72 more