Valuative invariants with higher moments

Valuative invariants play significant roles in finding canonical metrics on polarized varieties. A highly notable one is the $\alpha$-invariant that goes back to Tian [38], with the help of which there is now a huge table of Kähler–Einstein Fano varieties that have been discovered by various authors. Another remarkable invariant is the $\delta$-invariant that was recently introduced by Fujita-Odaka [24], which turns out to be a very powerful new tool in the study of K-stability of Fano varieties and now there is a large literature on it; see especially the work of Blum–Jonsson [3] and the references therein. The purpose of this article is to further polish the pertinent field by introducing a family of valuative invariants that interpolates betweent $\alpha$ and $\delta$. As we shall see, these invariants enjoy many properties that were previously only established for $\alpha$ and $\delta$. Our work suggests that the study of valuative invariants can be carried out in a broader context and hopefully this will serve as a new perspective in future research.

To begin with, we first recall the definition of $\delta$. Let $X$ be an $n$-dimensional complex normal projective variety with at worst klt singularities and $L$ is an ample line bundle. Up to a multiple of $L$, we assume throughout that $H^{0}(X,mL)\neq 0$ for any $m\in{\mathbb{Z}}_{>0}$. Put

Consider a basis $s_{1},\cdots,s_{d_{m}}$ of the vector space $H^{0}(X,mL)$, which induces an effective $\mathbb{Q}$-divisor

Any $\mathbb{Q}$-divisor $D$ obtained in this way is called an $m$-basis type divisor of $L$. Let

Then let

This limsup is in fact a limit by [3]. So roughly speaking, $\delta(X,L)$ measures the singularities of basis type divisors of $L$.

The following result demonstrates the importance of the $\delta$-invariant.

Thus by [2, 31], $\delta$-invariant serves as a criterion for the existence of KE metrics on $\mathbb{Q}$-Fano varieties.

Let $\pi:Y\rightarrow X$ be a proper birational morphism from a normal variety $Y$ to $X$ and let $F\subset Y$ be a prime divisor $F$ in $Y$. Such an $F$ will be called a divisor over $X$. Let

denote the expected vanishing order of $L$ along $F$ at level $m$. Here

denotes the pseudo-effective threshold of $L$ along $F$ at level $m$. Then a basic but important linear algebra lemma due to Fujita–Odaka [24] says that

and this supremum is attained by any $m$-basis divisor $D$ arising from a basis $\{s_{i}\}$ that is compatible with the filtration

meaning that each $H^{0}(Y,m\pi^{*}L-jF)$ is spanned by a subset of the $\{s_{i}\}_{i=1}^{d_{m}}$. Then it is easy to deduce that

As $m\rightarrow\infty$, one has

which is called the expected vanishing order of $L$ along $F$. Then Blum–Jonsson [3] further show that the limit of $\delta_{m}(X,L)$ also exists, and is equal to

Another closely related valuative invariant is Tian’s $\alpha$-invariant [38], which can be defined as (cf. [10, Appendix] and [3, Theorem C])

where $\tau(L,F):=\lim\tau_{m}(L,F)/m$, the pseudo-effective threshold of $L$ along $F$.

An important property of $S(L,F)$ is illustrated by the following result of K. Fujita, who shows that $S(L,F)$ can be viewed as the coordinate of the barycenter of certain Newton–Okounkov body along the “$F$-axis”, and hence the well-known Brunn–Minkovski inequality in convex geometry implies the following estimate.

One should think of $\tau$ and $S$ as the non-Archimedean analogues of the $I$ and $I-J$ functionals of Aubin, and it is shown by Boucksom–Jonsson that the above result holds for general valuations as well (cf. [8, Theorem 5.13]). An immediate consequence of Proposition 1.2 is the following:

$S(L,F)$ can be treated as the first moment of the vanishing order of $L$ along $F$. In general for $p\geq 1$, one can also consider the $p$-th moment of the vanishing order of $L$ along $F$. More precisely, given a basis $\{s_{i}\}$ of $H^{0}(X,mL)$ that is compatible with the filtration induced by $F$, put

In the K-stability literature, related $L^{p}$ notions (for test configurations) have been introduced and studied by Donaldson [18], Dervan [16], Hismoto [27] et. al. Our formulation of $S^{(p)}$ is inspired by the recent work of Han-Li [25], where a sequence of Monge–Ampère energies with higher moments was considered. We will give a more formal definition of this in Section 2. As $m\rightarrow\infty$, one has (see Lemma 2.2)

So in particular $S(L,F)=S^{(1)}(L,F).$ One main point of this article is to show that most properties established for $S$ in the literature hold for $S^{(p)}$ as well; see Section 2 for more details.

Extending Fujita’s barycenter inequality to the $p$-th moment, we have

Letting $p\rightarrow\infty$, one obtains the following

Relating to the $d_{p}$-geometry of maximal geodesic rays, $S^{(p)}(L,F)$ has the following pluri-potential interpretation (in Section 6 we will prove a more general result that holds for linearly bounded filtrations). The proof relies on the main result in [27] and some non-Archimedean ingredients from [7, 8, 9].

The set of moments $\{S^{(p)}\}$ can also be used to construct various kinds of valuative thresholds for $(X,L)$. It turns out that $\alpha$ and $\delta$ are only two special ones. To be more precise, one can put

As we shall see in Proposition 4.4, here one can also take inf over all valuations, which yields the same invariant.

It is interesting to note that $\{\delta^{(p)}(L)\}_{p\geq 1}$ is a decreasing family of valuative invariants with

Moreover, observe that the valuative formulation of $\delta^{(p)}(L)$ also makes sense when $L$ is merely a big $\mathbb{R}$-line bundle. We show that the continuity of $\delta$ established in [42] holds for $\delta^{(p)}$ as well.

Furthermore, we have the following result, generalizing the work of Blum-Liu [4].

And also, in the Fano setting, it turns out that $\delta^{(p)}$ can give sufficient conditions for K-stability.

For $p=1$ this is simply the $\delta$-criterion of Fujita-Odaka [24] (which says that $\delta(-K_{X})>1$ implies uniform K-stability), while for $p=\infty$ this recovers the $\alpha$-criterion of Tian [38] and Odaka–Sano [32] (which says that $\alpha(-K_{X})>\frac{n}{n+1}$ implies uniform K-stability). Thus Theorem 1.8 provides a bridge between $\alpha$ and $\delta$-invariants and gives rise to a family of valuative criterions for the existence of Kähler–Einstein metrics.

To show Theorem 1.8, the key new ingredient is the following monotonicity:

This is deduced from Proposition 5.2, which relies on the Brunn–Minkowski inequality and yields a new result in convex geometry regarding the $p$-th barycenter of a convex body that is probably of independent interest.

Interestingly, the above monotonicity is actually “sharp”, from which we can characterize the borderline case in Theorem 1.8.

When $p=\infty$ this is exactly the main theorem of Fujita [21] (which says that $\alpha(-K_{X})=\frac{n}{n+1}$ implies KE when $X$ is smooth). The proof of Theorem 1.9 uses the strategy of [21], but one major difference is that the equality in our case is more difficult to grasp, which requires some subtle measure-theoretic argument.

Organization. The rest of this article is organized as follows. In Section 2 we introduce $S^{(p)}$ in a more formal way, using filtrations. In Section 3 we prove Theorem 1.6 and in Section 4 we prove Theorem 1.7. Then Theorem 1.3, Theorem 1.8 and Theorem 1.9 are proved in Section 5. Finally in Section 6 we discuss the relation between $S^{(p)}$ and $d_{p}$-geometry and then prove a generalized version of Theorem 1.5.

Acknowledgments. The author would like to thank Tamás Darvas and Mingchen Xia for helpful discussions on Section 6. Special thanks go to Yuchen Liu for valuable comments and for proving Proposition 4.4. He also thanks Yanir Rubinstein for suggesting Theorem 1.9. The author is supported by the China post-doctoral grant BX20190014.

The next definition is a natural generalization of the expected vanishing order introduced in [24, 3].

This definition can be reformulated as follows (which in turn justifies the existence of the above limit).

For simplicity we put

with $R_{m}:=H^{0}(X,mL).$ We say $\mathcal{F}$ is a filtration of $R$ if for any $\lambda\in\mathbb{R}$ and $m\in{\mathbb{N}}$ there is a subspace $\mathcal{F}^{\lambda}R_{m}\subset R_{m}$ satisfying

1. (1)

   $\mathcal{F}^{\lambda^{\prime}}R_{m}\supseteq\mathcal{F}^{\lambda}R_{m}$ for any $\lambda^{\prime}\leq\lambda$;

2. (2)

   $\mathcal{F}^{\lambda}R_{m}=\bigcap_{\lambda^{\prime}<\lambda}\mathcal{F}^{\lambda^{\prime}}R_{m}$;

3. (3)

   $\mathcal{F}^{\lambda_{1}}R_{m_{1}}\cdot\mathcal{F}^{\lambda_{2}}R_{m_{2}}\subseteq\mathcal{F}^{\lambda_{1}+\lambda_{2}}R_{m_{1}+m_{2}}$ for any $\lambda_{1},\lambda_{2}$ and $m_{1},m_{2}\in{\mathbb{N}}$;

4. (4)

   $\mathcal{F}^{\lambda}R_{m}=R_{m}$ for $\lambda\leq 0$ and $\mathcal{F}^{\lambda}R_{m}=\{0\}$ for $\lambda\gg 0$.

We say $\mathcal{F}$ is linearly bounded if there exists $C>0$ such that $\mathcal{F}^{Cm}R_{m}=\{0\}$ for any $m\in{\mathbb{N}}$. We say $\mathcal{F}$ is a filtration of $R_{m}$ if only items $(1),(2)$ and $(4)$ are satisfied. We call $\mathcal{F}$ trivial if $\mathcal{F}^{\lambda}R_{m}=\{0\}$ for any $\lambda>0$.

Following [3], the definition of $S_{m}^{(p)}$ also extends to filtrations of $R_{m}$. More precisely, let $\mathcal{F}$ be a filtration of $R_{m}$, the jumping numbers of $\mathcal{F}$ are given by

where

Then we put (see also [25, (81)])

Thus $S_{m}^{(1)}(L,\mathcal{F})=S_{m}(L,\mathcal{F})$ is the rescaled sum of jumping numbers studied in [3].

A filtration is called an ${\mathbb{N}}$-filtration if all its jumping numbers are non-negative integers. For instance the filtration induced by a divisor over $X$ is ${\mathbb{N}}$-filtration. Given any filtration $\mathcal{F}$ of $R_{m}$, its induced ${\mathbb{N}}$-filtration $\mathcal{F}_{\mathbb{N}}$ is given by

Then one has

The next result is a simple generalization of [3, Proposition 2.11]

Now as in [3, Section 2.2] let us fix a local system $(z_{1},...,z_{n})$ around a regular closed point of $X$, which then yields a Newton–Okounkov body $\Delta\subset\mathbb{R}^{n}$ for $L$ and denote the Lebesgue measure on $\Delta$ by $\rho$. Then any filtration $\mathcal{F}$ of $R=R(X,L)=\oplus_{m\geq 0}R_{m}$ induces a family of graded linear series $V_{\bullet}^{t}$ ($t\in\mathbb{R}_{\geq 0}$) and also a concave function $G$ on $\Delta$. More precisely, $V_{\bullet}^{t}=\oplus_{m}V^{t}_{m}$ with

which induces a Newton–Okounkov body $\Delta^{t}\subset\Delta$ and

Then define

and also put

One then can deduce that

This generalizes [3, Lemma 2.6] (see also the proof of Theorem 1.3). A simple consequence is that

The next result naturally generalizes Lemma 2.9, Corollary 2.10 and Proposition 2.11 in [3]. Since its proof is largely verbatim, we omit it.

Let $\mathrm{Val}_{X}$ be the set of real valuations on the function field of $X$ that are trivial on the ground field ${\mathbb{C}}$. Any $v\in\mathrm{Val}_{X}$ induces a filtration $\mathcal{F}_{v}$ on $R$ via

for $m\in{\mathbb{N}}$ and $\lambda\in\mathbb{R}_{\geq 0}$. Then put

Note that $\mathcal{F}_{v}$ is saturated in the sense of [20, Definition 4.4]. To be more precise, let

be the base ideal of $\mathcal{F}^{\lambda}_{v}R_{m}$. Let $\overline{\mathfrak{b}(|\mathcal{F}^{\lambda}_{v}R_{m}|)}$ denote its integral closure (i.e., $\overline{\mathfrak{b}(|\mathcal{F}^{\lambda}_{v}R_{m}|)}$ is the set of elements $f\in\mathcal{O}_{X}$ satisfying a monic equation $f^{d}+a_{1}f^{d-1}+...+a_{d}=0$ with $a_{i}\in\mathfrak{b}(|\mathcal{F}_{v}^{\lambda}R_{m}|)^{i}$).

For $t\in\mathbb{R}_{\geq 0}$ and $l\in{\mathbb{N}}$, set

The previous lemma implies that $V^{t}_{m,\bullet}:=\bigoplus_{l\in{\mathbb{N}}}V^{t}_{m,l}$ is a subalgebra of $V^{t}_{\bullet}$. Put

As illustrated in [3, Section 5], the graded linear series $V^{t}_{m,\bullet}$ can effectively approximate $V^{t}_{\bullet}$. The argument therein extends to our $L^{p}$ setting in a straightforward way. So we record the following result, which generalizes [3, Theorem 5.3], and leave its proof to the interested reader.