K-stability for varieties with a big anticanonical class

Abstract

We extend the algebraic K-stability theory to projective klt pairs with a big anticanonical class. While in general such a pair could behave pathologically, it is observed in this note that K-semistability condition will force them to have a klt anticanonical model, whose stability property is the same as the original pair.

A relative Yau-Tian-Donaldson conjecture and stability thresholds

“…It would be interesting to see if the proof of the implication (ii) ⇒ (iv) of Theorem D can be adapted to the big setting. Together with [28] and [22] this would give an uniform Ding version of the Yau-Tian-Donaldson conjecture in the case −K X is big, without passing to the klt anticanonical model as in [60].…”

“…Supposing −K X big, Xu in [60] proved that the K-stability condition (given in terms of the δ-invariant) of (X, −K X ) forced to have a klt anticanonical model, whose stability properties are essentially the same as that of (X, −K X ). Moreover, in [60] it is also showed that the uniform Ding stability introduced in [28] implies δ(X) > 1. This concludes the equivalence between the existence of weak Kähler-Einstein, the uniform Ding stability and δ(X) > 1 thanks to [22,28].…”