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

Abstract

We compute first variation formulas for the complex components of the Bakry-Emery-Ricci endomorphism along Kähler 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 Kähler metric. We show as application that the Soliton-Kähler-Ricci flow generated by the Soliton-Ricci flow represents a complex strictly parabolic system of the complex components of the variation of the Kähler metric.

Keywords:

• Bakry-Emery-Ricci tensor
• variations of Kähler structures
• complex parabolic systems

AMS Subject Classifications:

• 53C55
• 35K60