Filtrations and test-configurations

Abstract

We introduce a strengthening of K-stability, based on filtrations of the homogeneous coordinate ring. This allows for considering certain limits of families of test-configurations, which arise naturally in several settings. We prove that if a manifold with no automorphisms admits a cscK metric, then it satisfies this stronger stability notion. We also discuss the relation with the birational transformations in the definition of $b$-stability.

Appendix: Asymptotic vanishing orders of graded linear series—S. Boucksom

CNRS-Université Pierre et Marie Curie, e-mail: boucksom@math.jussieu.fr.

1.1 Iitaka dimension and multiplicity

Let $X$ be a projective variety over an algebraically closed field $k$ (of any characteristic), set $n:=\dim X$, and let $L$ be a line bundle on $X$. Denote by

$$\begin{aligned} R=R(X,L):=\bigoplus _{m\in {\mathbb {N}}} H^0(X,mL) \end{aligned}$$

the algebra of sections of $L$. Given a graded subalgebra $S$ of $R$ (aka graded linear series of $L$), set

$$\begin{aligned} {\mathbb {N}}(S):=\{m\in {\mathbb {N}}\mid S_m\ne 0\}, \end{aligned}$$

which is a sub-semigroup of of ${\mathbb {N}}$, hence coincides outside a finite set with the multiples of the gcd $m(S)\in {\mathbb {N}}$ of ${\mathbb {N}}(S)$, sometimes known as the exponent of $S$. Define also the Iitaka dimension of $S$ as $\kappa (S):=\mathrm {tr. deg}(S/k)-1$ if $S\ne k$, and $\kappa (S):=-\infty$ otherwise, so that $\kappa (S)\in \{-\infty ,0,1,...,n\}$.

In this generality, the following result is due to Kaveh and Khovanskii [16] (see also [3]).

Theorem 16

Let $S\ne k$ be a graded subalgebra of $R(X,L)$, and write $\kappa =\kappa (S)$.

• (i) The multiplicity

  $$\begin{aligned} e(S)=\lim _{m\in {\mathbb {N}}(S),\,m\rightarrow \infty }\frac{\kappa !}{m^\kappa }\dim S_m \end{aligned}$$

  exists in $]0,+\infty [$.

• (ii) For each $m\in {\mathbb {N}}(S)$, let $\Phi _m:X\dashrightarrow {\mathbb {P}}(S_m^*)$ be the rational map defined by linear series $S_m$, and denote by $Y_m$ its image. Then we have $\dim Y_m=\kappa$ for all $m\in {\mathbb {N}}(S)$ large enough, and

  $$\begin{aligned} e(S)=\lim _{m\in {\mathbb {N}}(S),\,m\rightarrow \infty }\frac{\deg Y_m}{m^\kappa }. \end{aligned}$$

Note that $L$ is big iff $\kappa (X,L):=\kappa (R)$ is equal to $n:=\dim X$, and we then have $e(R)=\mathrm{vol }(L)$, the volume of $L$.

Definition 17

We say that $S$ contains an ample series if

• (i) $S_m\ne 0$ for all $m\gg 1$, i.e. $S$ has exponent $m(S)=1$.

• (ii) There exists a decomposition $L=A+E$ into ${\mathbb {Q}}$-divisors with $A$ ample and $E$ effective such that $H^0(X,mA)\subset S_m\subset H^0(X,mL)$ for all sufficiently divisible $m\in {\mathbb {N}}$.

This condition immediately implies that the rational map $\Phi _m:X\dashrightarrow {\mathbb {P}}(S_m^*)$ defined by $S_m$ in birational onto its image $Y_m$ for all $m\gg 1$.

Assuming this, let ${\mathfrak {b}}_m\subset {\mathcal {O}}_X$ be the base-ideal of $S_m$, i.e. the image of the evaluation map $S_m\otimes {\mathcal {O}}_X(-mL)\rightarrow {\mathcal {O}}_X$. Let $\mu _m:X_m\rightarrow X$ be any birational morphism with $X_m$ normal and projective and such that ${\mathfrak {b}}_m\cdot {\mathcal {O}}_{X_m}$ is locally principal, hence of the form ${\mathcal {O}}_{X_m}(-F_m)$ for an effective Cartier divisor $F_m$ on $X_m$. We then set

$$\begin{aligned} P_m:=\mu _m^*L-\tfrac{1}{m} F_m, \end{aligned}$$

(49)

which is a nef ${\mathbb {Q}}$-Cartier divisor on $X_m$. If $m$ divides $l$, then we may choose $X_{l}$ to dominate $X_m$, and we have $P_{l}\ge P_m$ after pulling back to $X_l$ (in the sense that the difference is an effective ${\mathbb {Q}}$-divisor). Note also that the intersection number $(P_m^n)$ does not depend on the choice of $X_m$ by the projection formula, and that $(P_{l}^n)\ge (P_m^n)$ when $m$ divides $l$, since $P_m$ and $P_{l}$ are nef with $P_{l}\ge P_m$.

As a consequence of Theorem 16 above (see also [15, Theorem C]), we get the following version of the Fujita approximation theorem:

Corollary 18

Let $S$ be a graded subalgebra of $R$, and assume that $S$ contains an ample series. Then $e(S)=\lim _{m\rightarrow \infty }(P_m^n)$.

Proof

With the notation of Theorem 16, the rational map $\Phi _m$ lifts to a morphism $f_m:X_m\rightarrow {\mathbb {P}}(S_m^*)$ which is birational onto its image $Y_m$ and such that $f_m^*{\mathcal {O}}(1)=\mu _m^*(mL)-F_m=mP_m$. We thus see that

$$\begin{aligned} (P_m^n)=\frac{\deg Y_m}{m^n}, \end{aligned}$$

and the result follows from (ii) in Theorem 16. $\square$

Remark 19

The special case of Theorem 16 where $S$ contains an ample series, which is what is being used in the previous corollary, was first established in [18].

1.2 Asymptotic vanishing orders and multiplicities

Our goal is to prove the following result.

Theorem 20

Let $X$ be a smooth projective variety over an algebraically closed field $k$, and let $L$ be a line bundle on $X$. Let $S$ be a graded subalgebra of $R=R(X,L)$, and assume that $S$ contains an ample series. Assume also that $e(S)<e(R)=\mathrm{vol }(L)$. Then there exists $\varepsilon >0$ and a (closed) point $x\in X$ with maximal ideal $\mathfrak {m}_x\subset {\mathcal {O}}_{X,x}$ such that $S_m\subset H^0\left( X,mL\otimes \mathfrak {m}_x^{\lfloor m\varepsilon \rfloor }\right)$ for all $m$.

Recall that a divisorial valuation (aka discrete valuation of rank $1$) on $X$ is a valuation $v:k(X)^*\rightarrow {\mathbb {R}}$ of the form $v=c\mathrm{ord }_E$ with $c>0$ and $E$ a prime divisor on a birational model $X'$ of $X$, which can always be assumed to be normal, projective and to dominate $X$. In particular, since $X$ is smooth, every scheme theoretic point $\xi \in X$ defines a divisorial valuation $\mathrm{ord }_\xi$. If we denote by $V=\overline{\{\xi \}}$ the subvariety of $X$ having $\xi$ as its generic point, then we have for all $f\in {\mathcal {O}}_{X,x}$

$$\begin{aligned} \mathrm{ord }_\xi (f)=\min _{x\in V}\mathrm{ord }_x(f). \end{aligned}$$

(50)

If we still denote by ${\mathfrak {b}}_m$ the base-ideal of $S_m$, then each divisorial valuation $v$ on $X$ defines a subadditive sequence

$$\begin{aligned} v({\mathfrak {b}}_m):=\min \{v(f)\mid f\in {\mathfrak {b}}_m{\setminus }\{0\}\}, \end{aligned}$$

and we may thus define the asymptotic vanishing order of $S$ along $v$ (cf. [10]) as

$$\begin{aligned} v(S):=\lim _{m\rightarrow \infty }\frac{v({\mathfrak {b}}_m)}{m}\in [0,+\infty [. \end{aligned}$$

In this language, the conclusion of Theorem 20 amounts to the existence of a closed point $x\in X$ such that $\mathrm{ord }_x(S)>0$. We begin with the following consequence of Izumi’s theorem on divisorial valuations.

Lemma 21

If there exists a divisorial valuation $v$ on $X$ such that $v(S)>0$, then $\mathrm{ord }_x(S)>0$ for some closed point $x\in X$.

Proof

Let $\xi \in X$ be the center of $v$ on $X$ (concretely, there exists a birational morphism $\mu :X'\rightarrow X$ with $X'$ projective and a prime divisor $E\subset X'$ such that $v=c\mathrm{ord }_E$, $c>0$, and $\xi$ is then the generic point of $\mu (E)\subset X$). Since the divisorial valuations $\mathrm{ord }_\xi$ and $v$ share the same center $\xi$ on $X$, the version of Izumi’s theorem proved in [14, Theorem 1.2] implies that there exists $C>0$ such that

$$\begin{aligned} C^{-1}v(f)\le \mathrm{ord }_\xi (f)\le C v(f) \end{aligned}$$

for all $f\in {\mathcal {O}}_{X,\xi }$. Applying this to $f\in {\mathfrak {b}}_m$ yields in the limit as $m\rightarrow \infty$

$$\begin{aligned} \mathrm{ord }_\xi (S)\ge C^{-1}v(S)>0. \end{aligned}$$

But for any closed point $x\in \overline{\{\xi \}}$ we also have $\mathrm{ord }_x\ge \mathrm{ord }_\xi$ on ${\mathcal {O}}_{X,x}$ by (50), and this similarly implies $\mathrm{ord }_x(S)\ge \mathrm{ord }_\xi (S)$, hence $\mathrm{ord }_x(S)>0$. $\square$

As a consequence of Corollary 18, we next prove:

Lemma 22

Let $S,S'$ be two graded subalgebras of $R$ containing an ample series. If $v(S)\ge v(S')$ for all divisorial valuations $v$, then $e(S)\le e(S')$.

Proof

Let ${\mathfrak {b}}_m,{\mathfrak {b}}'_m\subset {\mathcal {O}}_X$ be the base-ideals of $S_m$ and $S_m'$ respectively, and let $P_m$ and $P'_m$ be the nef ${\mathbb {Q}}$-Cartier divisors they determine on some high enough model $X_m$ over $X$, as in (49).

Given $\varepsilon >0$, Corollary 18 allows to find $m_0\in {\mathbb {N}}$ such that $e(S)\le (P_{m_0}^n)+\varepsilon$, and hence

$$\begin{aligned} e(S)\le (P_{m_1}\cdot P_{m_0}^{n-1})+\varepsilon \end{aligned}$$

(51)

for any multiple $m_1$ of $m_0$, since $P_{m_0}$ is nef and $P_{m_0}\le P_{m_1}$. By the projection formula and the definition of $P_{m_1}$ and $P'_{m_1}$, we have

$$\begin{aligned} (P_{m_1}\cdot P_{m_0}^{n-1})-(P'_{m_1}\cdot P_{m_0}^{n-1})=\sum _{E\subset X_{m_0}}\left( \frac{\mathrm{ord }_E(F'_{m_1})}{m_1}-\frac{\mathrm{ord }_E(F_{m_1})}{m_1}\right) (E\cdot P_{m_0}^{n-1}), \end{aligned}$$

where the sum runs over prime divisors $E$ of $X_{m_0}$ and any $E$ actually contributing to the sum is contained in the support of $F_{m_0}+F'_{m_0}$, hence belongs to a finite set of prime divisors of $X_{m_0}$ independent of $m_1$. Since we have by assumption

$$\begin{aligned} \lim _{m_1\rightarrow \infty }\frac{\mathrm{ord }_E(F_{m_1})}{m_1}=\mathrm{ord }_E(S)\ge \mathrm{ord }_E(S')=\lim _{m_1\rightarrow \infty }\frac{\mathrm{ord }_E(F'_{m_1})}{m_1} \end{aligned}$$

for any such $E$, we may thus choose $m_1$ a large enough multiple of $m_0$ to guarantee that

$$\begin{aligned} (P_{m_1}\cdot P_{m_0}^{n-1})\le (P'_{m_1}\cdot P_{m_0}^{n-1})+\varepsilon , \end{aligned}$$

and hence

$$\begin{aligned} e(S)\le (P'_{m_1}\cdot P_{m_2}\cdot P_{m_0}^{n-2})+2\varepsilon \end{aligned}$$

(52)

for any multiple $m_2$ of $m_1$, by (51) and the fact that $P_{m_0}, P'_{m_1}, P_{m_2}$ are nef with $P_{m_0}\le P_{m_2}$. We similarly have

$$\begin{aligned} \begin{aligned}&(P'_{m_1}\cdot P_{m_2}\cdot P_{m_0}^{n-2})-(P'_{m_1}\cdot P'_{m_2}\cdot P_{m_0}^{n-2})=\\&\quad \quad \quad =\sum _{E\subset X_{m_1}}\left( \frac{\mathrm{ord }_E(F'_{m_2})}{m_2}-\frac{\mathrm{ord }_E(F_{m_2})}{m_2}\right) (P'_{m_1}\cdot E\cdot P_{m_0}^{n-1})\le \varepsilon \end{aligned} \end{aligned}$$

for $m_2$ large enough, hence

$$\begin{aligned} e(S)\le (P'_{m_1}\cdot P'_{m_2}\cdot P_{m_3}\cdot P_{m_0}^{n-3})+3\varepsilon \end{aligned}$$

for any multiple $m_3$ of $m_2$, using (52) and $P_{m_0}\le P_{m_3}$. Continuing in this way, we finally obtain positive integers $m_1,...,m_n$ with $m_i$ dividing $m_{i+1}$ and such that

$$\begin{aligned} e(S)\le (P'_{m_1}\cdot ...\cdot P'_{m_n})+(n+1)\varepsilon , \end{aligned}$$

hence

$$\begin{aligned} e(S)\le (P'^n_{m_n})+(n+1)\varepsilon \end{aligned}$$

since $P_{m_i}\le P_{m_n}$. But $m_n$ can be taken to be as large as desired, thus $(P'^n_{m_n})$ is as close to $e(S')$ as we like by Corollary 18, and we conclude as desired that $e(S)\le e(S')$. $\square$

Proof of Theorem 20

By Lemma 22, the assumption $e(S)<e(R)$ implies that $v(S)>v(R)\ge 0$ for some divisorial valuation $v$. We conclude using Lemma 21. $\square$