Extremal metrics on toric manifolds and homogeneous toric bundles

Abstract

Abstract In this paper, we prove the Yau–Tian–Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles, thus for toric manifolds and homogeneous toric bundles, M admits an extremal metric in c 1 ⁢ ( L ) {c_{1}(L)} if and only if ( M , L ) {(M,L)} is relatively K ^ {\widehat{K}} -stable.