Variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism

TL;DR

This paper derives variation formulas for the complex parts of the Bakry-Emery-Ricci endomorphism in Kähler geometry, revealing standard differential operators and symmetry properties, and applies these to analyze the Soliton-Kähler-Ricci flow.

Contribution

It provides explicit first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism within Kähler structures, advancing understanding of their geometric behavior.

Findings

Principal parts of variations are standard complex differential operators.

Variations exhibit particular symmetry properties.

The Soliton-Kähler-Ricci flow is shown to be a complex strictly parabolic system.

Abstract

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along K\"ahler structures. Our formulas show that the principal parts of the variations are quite standard complex differential operators with particular symmetry properties on the complex decomposition of the variation of the K\"ahler metric. We show as application that the Soliton-K\"ahler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the K\"ahler metric.