Variation formulas for the complex components of the Bakry-Emery-Ricci   endomorphism
  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. 
Nefton Pali
math.DG