Conformally Einstein-Maxwell Kähler metrics and structure of the automorphism group

Abstract

Let $(M,g)$ be a compact Kähler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K=J{\mathrm{grad}}_{g}f$ for the Kähler form ${\omega }_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell Kähler metric if the conformal metric $\tilde{g}=f^{-2}g$ has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.

arXiv comment20 pages

Cited by in corpus (6)

• Kähler metrics with constant weighted scalar curvature and weighted K-stability
• NUTs, Bolts, and Spindles
• Conformally Kähler, Einstein--Maxwell metrics and boundedness of the modified Mabuchi-functional
• The moduli space of Fano manifolds with Kähler-Ricci solitons
• Volume minimization and obstructions to solving some problems in Kähler geometry
• Einstein--Maxwell equations on 4-dimensional Lie algebras