Deformation for coupled Kähler-Einstein metrics

Abstract

The notion of coupled Kähler-Einstein metrics was introduced recently by Hultgren-WittNyström. In this paper we discuss deformation of a coupled Kähler-Einstein metrics on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler-Einstein metric to be deformed to another coupled Kähler-Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher-Einstein metric on a Fano manifold when the complex structure varies.

Anticanonically balanced metrics and the Hilbert-Mumford criterion for the $δ_m$-invariant of Fujita-Odaka

“…They also assume that Y = X 1 = • • • = X k is a normal Q-Gorenstein variety and consider the K-stability as opposed to the Ding stability, but this seems to be a minor difference. While the author believes that the stability condition in Definition 3.17, or its appropriately modified version, is a useful one in studying the coupled Kähler-Einstein metrics, it is important to note that none of the results in [14,29,30,37,38,49,60] seems to be affected when we adopt Definition 3.17 as the relevant notion of stability. The works [14,29,30,37,49], from the point of view of stability, essentially consider the case when (X 1 , L 1 ), .…”