Gauge-Fixing Constant Scalar Curvature Equations on Ruled Manifolds and the Futaki Invariants

Abstract

In this article we introduce and prove the solvability of the gauge-fixing constant scalar curvature equations on ruled Kaehler manifolds. We prove that when some lifting conditions for holomorphic vector fields on the base manifold are satisfied the solutions for the gauge-fixing constant scalar curvature equations are actually solutions for the constant scalar curvature equations provided the corresponding Futaki invariants vanish.