Integrating holomorphic sectional curvatures

TL;DR

This paper uses representation theory to analyze the holomorphic sectional curvature of Kähler and Hermitian metrics, proving it determines the entire curvature tensor and exploring its implications.

Contribution

It provides a new proof that holomorphic sectional curvature determines the full curvature tensor using representation-theoretic methods.

Findings

Calculated the $L^2$-norm of holomorphic sectional curvature for Kähler metrics

Proved holomorphic sectional curvature determines the entire curvature tensor

Extended analysis to holomorphic bisectional curvature

Abstract

We calculate the $L^2$-norm of the holomorphic sectional curvature of a K\"ahler metric by representation-theoretic means. This yields a new proof that the holomorphic sectional curvature determines the whole curvature tensor. We then investigate what the holomorphic sectional curvature of a Hermitian metric determines and calculate the $L^2$-norm of the holomorphic bisectional curvature.