Product theorem for K-stability

Abstract

We prove a product formula for delta invariant and as an application, we show that product of K-(semi, poly)stable Fano varieties is also K-(semi, poly)stable.

Introduction

K-(poly)stability of complex Fano varieties was first introduced by Tian [30] and later reformulated in a more algebraic way by Donaldson [12]. By the generalized Yau-Tian-Donaldson (YTD) conjecture, K-polystability of (singular) Fano varieties are expected to give algebraic characterization of the existence of (singular) Kähler-Einstein metric. This has been known in the smooth case [30], [3], [9], [31] and the uniformly K-stable case [26].

From this metric point of view, it is easy to see (or at least expect) that products of K-(semi, poly)stable Fano varieties are also K-(semi, poly)stable. Results of this type actually play an important role towards the proof of the quasi-projectivity of the K-moduli [10]. However, no algebraic proof is known for this intuitively simple fact.

The purpose of this note is to give such a proof. Our main result goes as follows.

Theorem 1.1

Let $X_i$ $(i=1,2)$ be $\mathbb{Q}$-Fano varieties and let $X=X_1\times X_2$. Then X is K-semistable (resp. K-polystable, K-stable, uniformly K-stable) if and only if $X_i$ $(i=1,2)$ are both K-semistable (resp. K-polystable, K-stable, uniformly K-stable).

Indeed, our result works for products of log Fano pairs as well (see Corollary 3.4 and Proposition 4.1).

One of the main tools that goes into the proof is the δ-invariant (or adjoint stability threshold) of a big line bundle (see Section 2.4). This invariant was introduced and studied by [14], [4], and one of their main results is that a $\mathbb{Q}$-Fano variety X is K-semistable (resp. uniformly K-stable) if and only if $\delta (-K_X)\ge 1$ (resp. $\delta (-K_X)>1$). This allows us to reduce most parts of Theorem 1.1 to proving a product formula for δ-invariant (cf. [29, Conjecture 1.10], [10, Conjecture 4.9]):

Theorem 1.2 =Theorem 3.1

Let $(X_i,{\mathit{\Delta }}_i)$ be projective klt pairs and let $L_i$ be big line bundles on $X_i$ $(i=1,2)$. Let $X=X_1\times X_2$, $L=L_1⊠L_2$ and $\mathit{\Delta }={\mathit{\Delta }}_1⊠{\mathit{\Delta }}_2$. Then

• $\delta (X,\mathit{\Delta };L)=\min \{\delta (X_1,{\mathit{\Delta }}_1;L_1),\delta (X_2,{\mathit{\Delta }}_2;L_2)\}$.

• If there exists a divisor E over X which computes $\delta (X,\mathit{\Delta };L)$, then for some $i\in \{1,2\}$, there also exists a divisor $E_i$ over $X_i$ that computes $\delta (X_i,{\mathit{\Delta }}_i;L_i)$.

In particular, this takes care of the product of K-(semi)stable and uniformly K-stable Fano varieties. We note that the analogous product formula for Tian's alpha invariant is well known (see e.g. [16, Section 2], [8, Lemma 2.29] or [21, Proposition 8.11]) and indeed our proof takes inspirations from these works.

For the K-polystable case, we study K-semistable special degenerations of the product of K-semistable Fano varieties and with the help of [25], we show that they always arise from special degenerations of the factors:

Theorem 1.3 =Theorem 4.2

Let $(X_i,{\mathit{\Delta }}_i)$ $(i=1,2)$ be K-semistable log Fano pairs and let $(X,\mathit{\Delta })=(X_1\times X_2,{\mathit{\Delta }}_1⊠{\mathit{\Delta }}_2)$. Let $\varphi :(\mathcal{X},\mathcal{D})\to {\mathbb{A}}^1$ be a special test configuration of $(X,\mathit{\Delta })$ with K-semistable central fiber $({\mathcal{X}}_0,{\mathcal{D}}_0)$, then there exist special test configurations ${\varphi }_i:({\mathcal{X}}_i,{\mathcal{D}}_i)\to {\mathbb{A}}^1$ $(i=1,2)$ of $(X_i,{\mathit{\Delta }}_i)$ with K-semistable central fibers such that $(\mathcal{X},\mathcal{D})\cong ({\mathcal{X}}_1{\times }_{{\mathbb{A}}^1}{\mathcal{X}}_2,{\mathcal{D}}_1⊠{\mathcal{D}}_2)$ (as test configurations, where ${\mathbb{G}}_m$ acts diagonally on ${\mathcal{X}}_1{\times }_{{\mathbb{A}}^1}{\mathcal{X}}_2)$.

Let us briefly explain the ideas of proof as well as the organization of the paper. Section 2 put together some preliminary materials on valuations, filtrations, δ-invariant and K-stability. Since δ-invariant is defined using log canonical threshold of basis type divisors, it is not hard to imagine that Theorem 1.2 follows from inversion of adjunction and it suffices to show that any basis type divisors can be reorganized into one that restricts to a convex combination of basis type divisors on one of the factors. This is done in Section 3 using some auxiliary basis type filtrations constructed in Section 2.6. To address K-polystability, we analyze divisors that compute the δ-invariants. We do so by choosing a maximal torus $\mathbb{T}$ in the automorphism group of the Fano variety and restricting to $\mathbb{T}$-invariant divisor. In this setting, equivariant K-polystability behaves somewhat like K-stability and one can give very explicit description of divisors computing δ-invariants. This is made more precise in Section 4. Once we know that product of K-polystable Fano varieties are still K-polystable, since every K-semistable Fano variety has a unique K-polystable degeneration by [25], the K-semistable degenerations in Theorem 1.3 can be obtained by deforming the K-polystable degenerations (which is a product). But deformations of product of Fano varieties are still product of Fano varieties (see Section 2.7), this gives the proof of Theorem 1.3.

The author would like to thank his advisor János Kollár for constant support, encouragement and numerous inspiring conversations. He also wishes to thank Yuchen Liu and Chenyang Xu for helpful discussions and the anonymous referee for helpful comments. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while the author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2019 semester.