On compact holomorphically pseudosymmetric K\"ahlerian manifolds

TL;DR

This paper investigates conditions under which holomorphic pseudosymmetry in compact Kähler manifolds implies local symmetry, and constructs examples of non-compact manifolds to illustrate the necessity of compactness assumptions.

Contribution

It establishes that holomorphic pseudosymmetry reduces to local symmetry under certain conditions and provides new examples of non-compact manifolds with this property.

Findings

Holomorphic pseudosymmetry implies local symmetry in compact cases with constant scalar curvature.

Constructs non-compact examples showing compactness is essential for the theorem.

Examples by Jelonek demonstrate sign-changing structure functions on manifolds.

Abstract

For compact K\"ahlerian manifolds, the holomorphic pseudosymmetry reduces to the local symmetry if additionally the scalar curvature is constant and the structure function is non-negative. Similarly, the holomorphic Ricci-pseudosymmetry reduces to the Ricci-symmetry under these additional assumptions. We construct examples of non-compact essentially holomorphically pseudosymmetric K\"ahlerian manifolds. These examples show that the compactness assumption cannot be omitted in the above stated theorem. Recently, the first examples of compact, simply connected essentially holomorphically pseudosymmetric K\"ahlerian manifolds are discovered by W. Jelonek. In his examples, the structure functions change their signs on the manifold.