[Paper Review] Proof of the Yano-Obata Conjecture for holomorph-projective transformations

Vladimir S. Matveev, Stefan Rosemann|arXiv (Cornell University)|Mar 29, 2011

Advanced Topics in Algebra47 references3 citations

TL;DR

This paper proves the Yano-Obata conjecture for holomorph-projective transformations on closed, connected Kähler manifolds: the identity component of the group of holomorph-projective transformations consists only of isometries unless the manifold is locally isometric to complex projective space with Fubini-Study metric. The proof relies on analyzing $h$-projective structures, curvature conditions, and the degree of mobility of Kähler metrics, showing that non-isometric holomorph-projective transformations exist only in the case of constant holomorphic curvature.

ABSTRACT

We prove the classical Yano-Obata conjecture by showing that the connected component of the group of holomorph-projective transformations of a closed, connected Riemannian Kähler manifold consists of isometries unless the metric has constant positive holomorphic curvature.

Motivation & Objective

• To resolve the classical Yano-Obata conjecture regarding holomorph-projective transformations on Kähler manifolds.
• To determine when the identity component of the holomorph-projective transformation group coincides with the isometry group.
• To characterize the structure of $h$-projectively equivalent Kähler metrics and their associated connections.
• To establish that non-affine holomorph-projective vector fields exist only in the case of constant holomorphic curvature.

Proposed method

• Introduces the concept of $h$-planar curves as curves whose acceleration lies in the complex span of velocity and its $J$-image.
• Defines $h$-projective equivalence between Kähler metrics via preservation of $h$-planar curves.
• Uses the degree of mobility of Kähler metrics to classify $h$-projectively equivalent metrics, especially in the case of mobility two.
• Applies a tensorial equation (92) for symmetric $J$-invariant $(2,0)$-tensors to characterize $h$-projective structures and their connections.
• Analyzes the action of $h$-projective vector fields on the solution space of the tensor equation (92), showing that their Lie derivative preserves the solution space.
• Demonstrates that the Levi-Civita connection of a metric in an $h$-projective structure is uniquely determined by a solution of equation (92), enabling reconstruction of the metric and connection.

Experimental results

Research questions

• RQ1 Under what conditions do holomorph-projective transformations on a Kähler manifold coincide with isometries?
• RQ2 When does the existence of non-affine holomorph-projective vector fields imply that the metric has constant holomorphic curvature?
• RQ3 What is the structure of the space of $h$-projectively equivalent Kähler metrics, particularly when the degree of mobility is two?
• RQ4 How can $h$-projective structures be characterized via solutions to a specific tensor equation (92)?
• RQ5 What is the relationship between the Lie derivative of $h$-projective vector fields and the solution space of the tensor equation (92)?

Key findings

• The identity component of the holomorph-projective transformation group of a closed, connected Kähler manifold consists only of isometries unless the metric has constant positive holomorphic curvature.
• The only Kähler manifolds admitting non-isometric holomorph-projective transformations are those locally isometric to complex projective space with the Fubini-Study metric.
• For metrics of degree of mobility two, the space of solutions to equation (92) is two-dimensional and spanned by the dual metrics scaled by their volume forms.
• The Lie derivative of an $h$-projective vector field on the solution space of (92) satisfies a quadratic matrix equation, implying algebraic constraints on the associated endomorphism.
• The Levi-Civita connection of a metric in an $h$-projective structure is uniquely reconstructed from a non-degenerate solution of equation (92), establishing a one-to-one correspondence.
• The proof shows that $h$-projective structures on Kähler manifolds are determined by solutions of a specific tensor equation, and that non-trivial such structures exist only in the case of constant holomorphic curvature.