Coupled Kähler–Einstein Metrics

Abstract

We propose new types of canonical metrics on Kähler manifolds, called coupled Kähler-Einstein metrics, generalizing Kähler-Einstein metrics. We prove existence and uniqueness results in the cases when the canonical bundle is ample and when the manifold is Kähler-Einstein Fano. In the Fano case we also prove that existence of coupled Kähler-Einstein metrics imply a certain algebraic stability condition, generalizing Kpolystability.

Ricci Iteration for Coupled Kähler–Einstein Metrics

“…Main results. It is known that there always exists a unique CKE metric in the λ = −1 case [HN17]. Correspondingly, we can prove the existence of balanced metrics as well (cf.…”

“…)-geodesic generated by the same holomorphic vector field V (see the arguments in [HN17,Lemma 4.4]). Let f t denotes the flow generated by V .…”

“…For this reason, several generalizations of Kähler-Einstein metrics have been studied. In this paper, we will focus on the "coupled Kähler-Einstein metrics" introduced by Hultgren-Witt Nyström [HN17]: for any N ∈ N, an N -tuple of Kähler classes (Ω [i] ) is called a decomposition of 2πλc 1 (X) if…”

Coupled Sasaki-Ricci solitons

“…Further, since if a (real) Hamiltonian vector field is holomorphic it is necessarily Killing and the group of all isometries is compact, we obtain the following extension of a theorem of Matsushima [22]. This result was stated in [17] for coupled Kähler-Einstein metrics, but our proof would be more elementary.…”

“…Coupled Kähler-Ricci solitons are defined in [16] to be Kähler metrics with Kähler forms ω α representing γ α such that, for each α, f α is a Hamiltonian Killing potential with respect to ω α so that Jgrad α f α is a Hamiltonian Killing vector field where grad α denotes the gradient with respect to ω α . Coupled Kähler-Einstein metrics defined in [17] are the case when f α 's are all constant so that Now we consider Sasakian analogues of the above. Let S be a compact Sasaki manifold with positive basic first Chern class c B 1 (S), which means c B 1 (S) is represented by a positive basic (1, 1)form.…”