Prescribed scalar curvatures for homogeneous toric bundles

Abstract

In this paper, we study the generalized Abreu equation on a Delzant ploytope ∆ ⊂ R 2 and prove the existence of the constant scalar metrics of homogeneous toric bundles under the assumption of an appropriate stability.

Extremal metrics on toric manifolds and homogeneous toric bundles

“…By Theorem 5.5, we need to prove the necessity and sufficiency of uniformly relative K-polystability. By Theorem 2.3 in [5] and Lemma 2.8, uniformly relatively K-polystability is a necessary condition. So we only need to prove the sufficiency of uniformly relatively K-polystability.…”

“…In this section, we generalize our results to homogeneous toric bundles. First, we briefly review homogeneous toric bundles and the generalized Abreu equation, For the details, see [18] and [5]. Then we sketch the proof of Theorem 1.4.…”

“…where the left-invariant vector fields { ∂ ∂x i , H i , V α , W α } form a local basis of G × K M. The vector fields H i , V α , W α and the index R + M are determined by G and K. For more details, see [5], [28]. Let {dx i , ν i , dV α , dW α } 1≤i≤n,α∈R + M be the dual 1-form of the basis.…”

“…K-stabilities for homogeneous toric bundles. Following Donaldson [14] (also see [31], [5]), for any smooth function A on ∆, we can define a functional on C:…”

“…He presented the underlying equation, which we call the generalized Abreu equation (see (5.2)). In [5], Chen-Han-Li-Lian-Sheng solved the problem for extremal metrics for homogeneous toric bundles with two-dimensional fibers under the uniform K-stability condition. In this paper, we prove the following theorem.…”