Ricci Iteration for Coupled Kähler–Einstein Metrics

Abstract

We study the quantization of coupled Kähler-Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, so called the "balanced metrics". We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce some algebro-geometric obstruction which interpolates between the Donaldson-Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.

Deformation for coupled Kähler-Einstein metrics

“…The Matsushima type obstruction theorem states that if (ω i ) N i=1 is a coupled Kähler-Einstein metric, then the identity component of the holomorphic automorphism group Aut 0 (X) is the complexification of the identity component of the isometry group Isom 0 (X, ω 1 ) of ω 1 , and in particular, it is then reductive [9,6]. Note that, in this case, Isom 0 (X, ω 1 ) = Isom 0 (X, ω 2 ) = • • • = Isom 0 (X, ω N ) as pointed out in [16]. On the other hand, for the Futaki type obstruction, the Futaki type invariant is defined in the following way.…”

“…In particular, the existence theorem were developed. Pingali [14] and Takahashi [16] introduced a continuity method and a Ricci iteration method respectively to construct coupled Kähler-Einstein metrics. Hutgren [8] developed a detailed study for the existence of such metrics on toric Fano manifolds, and Delcroix-Hultgren [5] extended it to more general settings.…”

On coupled constant scalar curvature Kähler metrics

“…Our aim in this paper is to study a set of metrics satisfying some coupled equations on a Kähler manifold, that generalise constant scalar curvature Kähler (cscK) metrics, and the coupled Kähler-Einstein metrics studied in [16,17,4,13,24,28]. Throughout the paper we fix a polarized tuple (M, (L i )), i.e., an n-dimensional Kähler manifold M with ample line bundles L 0 , L 1 , • • • , L m , and we denote the line bundle ⊗ m i=0 L i by L. We are interested in metrics ω i ∈ 2πc 1 (L i ) for i = 0, • • • , m that satisfy…”