Extremal metrics on toric manifolds and homogeneous toric bundles

Abstract

In this paper, we prove the Yau-Tian-Donaldson conjecture of the filtration version for toric manifolds and homogeneous toric bundles.

Special representatives of complexified Kähler classes

“…In our case the situation is slightly complicated from the fact that in (4.1) both sides of the equation depend on ω. Under some simplifying assumptions, however, the proof of [37,Theorem 4.3] gives the required result also in our case. We reproduce the proof for the reader's convenience and in order to emphasise the dependence on sup|A(ω)|.…”

“…The main technical tool to establish these estimates is [7, Theorem 1.2], where it is shown that the norm of any solution to the prescribed scalar curvature equation on a compact Kähler manifold can be estimated in terms of the entropy functional log ω n ω n 0 ω n and a C 0 -bound on the target function. In the toric setting, it is possible to obtain bounds on the entropy from uniform K-stability, see [37,Theorem 4.3]. In our case the situation is slightly complicated from the fact that in (4.1) both sides of the equation depend on ω.…”

A uniform version of the Yau-Tian-Donaldson correspondence for extremal Kähler metrics on polarized toric manifolds

“…Then B is contained in C * and consequently (K) C * implies (K) B . The latter condtion (K) B can be found in [56]. .…”

An effective weighted K-stability condition for polytopes and semisimple principal toric fibratons

“…We end this paragraph by recalling that (1, w)-uniform stability of the lattice polytope [−1, 1] translates to existence of certain canonical Kähler metrics on P 1 thanks to [25]. (1,0)…”