Conventions

For the purpose of this paper, a variety is an irreducible, reduced, separated scheme of finite type over a field $k$ . Furthermore, we will always assume that $k$ is of characteristic zero, algebraically closed, and uncountable. We use the convention that $\mathbf{N}=\{1,2,3,\ldots \}$ .

2.1 Real valuations

Let $X$ be a variety and $K(X)$ denote its function field. A real valuation of $K(X)$ is a group homomorphism

$$\begin{eqnarray}v:K(X)^{\times }\rightarrow \mathbf{R}\end{eqnarray}$$

such that $v$ is trivial on $k$ (the base field) and $v(f+g)\geqslant \min \{v(f),v(g)\}$ . We use the convention that $v(0)=+\infty$ .

A valuation $v$ gives rise to a valuation ring ${\mathcal{O}}_{v}\subset K(X)$ , where ${\mathcal{O}}_{v}:=\{f\in K(X)\mid v(f)\geqslant 0\}$ . Note that if $v$ is a valuation of $K(X)$ and $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ , scaling the outputs of $v$ by $\unicode[STIX]{x1D706}$ gives a new valuation $\unicode[STIX]{x1D706}v$ .

We say that $v$ has a center on $X$ if there exists a map $\unicode[STIX]{x1D70B}:\operatorname{Spec}({\mathcal{O}}_{v})\rightarrow X$ as below.

By [Reference HartshorneHar77, Theorem II.4.3], if such a map $\unicode[STIX]{x1D70B}$ exists, it is necessarily unique. Let $\unicode[STIX]{x1D701}$ denote the unique closed point of $\operatorname{Spec}({\mathcal{O}}_{v})$ . If such a $\unicode[STIX]{x1D70B}$ exists, we define the center of $v$ on $X$ , denoted $c_{X}(v)$ , to be $\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D701})$ . We let $\operatorname{Val}_{X}$ (respectively, $\operatorname{Val}_{X,x}$ ) denote the set of nontrivial real valuations of $K(X)$ with center on $X$ (respectively, center equal to  $x$ ).

Given a valuation $v\in \operatorname{Val}_{X}$ and a nonzero ideal $\mathfrak{a}\subseteq {\mathcal{O}}_{X}$ , we may evaluate $\mathfrak{a}$ along $v$ by setting

$$\begin{eqnarray}v(\mathfrak{a}):=\min \{v(f)\mid f\in \mathfrak{a}\cdot {\mathcal{O}}_{X,c_{X}(v)}\}.\end{eqnarray}$$

When $X$ is affine,

$$\begin{eqnarray}v(\mathfrak{a})=\min \{v(f)\mid f\in \mathfrak{a}(X)\}.\end{eqnarray}$$

It follows from the above definition that if $\mathfrak{a}\subseteq \mathfrak{b}\subset {\mathcal{O}}_{X}$ are nonzero ideals, then $v(\mathfrak{a})\geqslant v(\mathfrak{b})$ . In addition, $v(\mathfrak{a})>0$ if and only if $c_{x}(v)\in \operatorname{Cosupp}(\mathfrak{a})$ .Footnote ²

We endow $\operatorname{Val}_{X}$ with the weakest topology such that, for every ideal $\mathfrak{a}$ on $X$ , the map $\operatorname{Val}_{X}\rightarrow \mathbf{R}\cup \{+\infty \}$ defined by $v\mapsto v(\mathfrak{a})$ is continuous. For information on the space of valuations, see [Reference Jonsson and MustaţǎJM12] and [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15].

2.2 Divisorial valuations

Let $f:Y\rightarrow X$ be a proper birational morphism with $Y$ normal and $E\subset Y$ a prime divisor. The discrete valuation ring ${\mathcal{O}}_{Y,E}$ gives rise to a valuation $\operatorname{ord}_{E}\in \operatorname{Val}_{X}$ that sends $a\in K(X)^{\times }$ to the order of vanishing of $a$ along  $E$ . Note that $c_{X}(\operatorname{ord}_{E})$ is the generic point of $f(E)$ .

We say $v\in \operatorname{Val}_{X}$ is a divisorial valuation if there exists $E$ as above and $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ such that $v=\unicode[STIX]{x1D706}\operatorname{ord}_{E}$ . We write $\operatorname{DVal}_{X}\subset \operatorname{Val}_{X}$ for the set of divisorial valuations on  $X$ .

2.3 Quasimonomial valuations

A quasimonomial valuation is a valuation that becomes monomial on some birational model over $X$ . Specifically, let $f:Y\rightarrow X$ be a proper birational morphism and $p\in Y$ a closed point such that $Y$ is regular at $p$ . Given a regular system of parameters $y_{1},\ldots ,y_{n}\in {\mathcal{O}}_{Y,p}$ at $p$ and $\unicode[STIX]{x1D736}=(\unicode[STIX]{x1D6FC}_{1},\ldots ,\unicode[STIX]{x1D6FC}_{n})\in \mathbf{R}_{{\geqslant}0}^{n}\setminus \{\mathbf{0}\}$ , we define a valuation $v_{\unicode[STIX]{x1D736}}$ as follows. For $r\in {\mathcal{O}}_{Y,p}$ we can write $r$ in $\widehat{{\mathcal{O}}_{Y,p}}$ as $r=\sum _{\unicode[STIX]{x1D6FD}\in \mathbf{Z}_{{\geqslant}0}^{n}}c_{\unicode[STIX]{x1D6FD}}y^{\unicode[STIX]{x1D6FD}}$ , with $c_{\unicode[STIX]{x1D6FD}}\in \widehat{{\mathcal{O}}_{Y,p}}$ either zero or unit. We set

$$\begin{eqnarray}v_{\unicode[STIX]{x1D6FC}}(r)=\min \{\langle \unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}\rangle \mid c_{\unicode[STIX]{x1D6FD}}\neq 0\}.\end{eqnarray}$$

A quasimonomial valuation is a valuation that can be written in the above form. Note that in the above notation, if there exists $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ such that $\unicode[STIX]{x1D706}\cdot \unicode[STIX]{x1D736}\in \mathbf{Z}_{{\geqslant}0}^{r}$ , then $v_{\unicode[STIX]{x1D6FC}}$ is a divisorial valuation. Indeed, take a weighted blowup of $Y$ at $p$ to find the correct exceptional divisor.

2.4 The relative canonical divisor

Let $f:Y\rightarrow X$ be a proper birational morphism of normal varieties. If $X$ is $\mathbf{Q}$ -Gorenstein, that is $K_{X}$ is $\mathbf{Q}$ -Cartier, we define the relative canonical divisor of $f$ to be

$$\begin{eqnarray}K_{Y/X}:=K_{Y}-f^{\ast }(K_{X}),\end{eqnarray}$$

where $K_{Y}$ and $K_{X}$ are chosen so that $f_{\ast }K_{Y}=K_{X}$ . While $K_{Y}$ and $K_{X}$ are only defined up to linear equivalence, $K_{Y/X}$ is a well-defined divisor.

We say that a variety $X$ is klt if $X$ is normal, $\mathbf{Q}$ -Gorenstein, and for any proper birational morphism of normal varieties $Y\rightarrow X$ the coefficients of $K_{Y/X}$ are ${>}-1$ . For further details on klt varieties and the relative canonical divisor, see [Reference Kollár and MoriKM98, § 2.3].

2.5 The log discrepancy of a valuation

Let $X$ be a normal $\mathbf{Q}$ -Gorenstein variety. If $Y\rightarrow X$ is a proper birational morphism with $Y$ normal, and $E\subset Y$ a prime divisor, then the log discrepancy of $\operatorname{ord}_{E}$ is defined by

$$\begin{eqnarray}A_{X}(\operatorname{ord}_{E}):=1+(\text{coefficient of}~E~\text{in}~K_{Y/X}).\end{eqnarray}$$

As explained in [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15] (building upon [Reference Boucksom, Favre and JonssonBFJ08, Reference Jonsson and MustaţǎJM12]), the log discrepancy can be extended to a lower semicontinuous function $A_{X}:\operatorname{Val}_{X}\rightarrow \mathbf{R}\cup \{+\infty \}$ that is homogeneous of order 1, (i.e. $A_{X}(\unicode[STIX]{x1D706}v)=\unicode[STIX]{x1D706}A_{X}(v)$ for $v\in \operatorname{Val}_{X}$ and $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ ). Additionally, $X$ is klt if and only if $A_{X}(v)>0$ for all $v\in \operatorname{Val}_{X}$ .

Note that [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15] defines the log discrepancy function $A_{X}:\operatorname{Val}_{X}\rightarrow \mathbf{R}\cup \{+\infty \}$ when $X$ is normal and $K_{X}$ is not necessarily assumed to be $\mathbf{Q}$ -Cartier. We will not work in this level of generality.

2.6 Graded sequences of ideals

A graded sequence of ideals on a variety $X$ is a sequence of ideals $\mathfrak{a}_{\bullet }=\{\mathfrak{a}_{m}\}_{m\in \mathbf{N}}$ such that $\mathfrak{a}_{m}\cdot \mathfrak{a}_{n}\subseteq \mathfrak{a}_{m+n}$ for all $m,n\in \mathbf{N}$ . By convention, we put $\mathfrak{a}_{0}={\mathcal{O}}_{X}$ . To simplify exposition, we always assume that $\mathfrak{a}_{m}$ is not equal to the zero ideal for all $m\in \mathbf{N}$ .

We provide two examples of graded sequences of ideals.

1. (a) Let $\mathfrak{b}$ be a nonzero ideal on $X$ . We may define a graded sequence $\mathfrak{a}_{\bullet }$ by setting $\mathfrak{a}_{m}:=\mathfrak{b}^{m}$ for all $m\in \mathbf{N}$ . This example is trivial.

2. (b) We fix $v\in \operatorname{Val}_{X}$ and define $\mathfrak{a}_{\bullet }(v)=\{\mathfrak{a}_{m}(v)\}_{m\in \mathbf{N}}$ as follows. If $U\subseteq X$ is an open affine set such that $c_{X}(v)\in U$ , then

   $$\begin{eqnarray}\mathfrak{a}_{m}(v)(U):=\{f\in {\mathcal{O}}_{X}(U)\mid v(f)\geqslant m\}.\end{eqnarray}$$
   If $c_{x}(v)\notin U$ , then $\mathfrak{a}_{m}(v)(U):={\mathcal{O}}_{X}(U)$ . If $c_{X}(v)$ is a closed point $x$ , we have that each ideal $\mathfrak{a}_{m}(v)$ is $\mathfrak{m}_{x}$ -primary,Footnote ³ where $\mathfrak{m}_{x}\subseteq {\mathcal{O}}_{X}$ denotes the ideal of functions vanishing at  $x$ .

Given $v\in \operatorname{Val}_{X}$ and a graded sequence $\mathfrak{a}_{\bullet }$ , we may evaluate $\mathfrak{a}_{\bullet }$ along $v$ by setting

$$\begin{eqnarray}v(\mathfrak{a}_{\bullet }):=\inf _{m\in \mathbf{N}}\frac{v(\mathfrak{a}_{m})}{m}=\lim _{m\rightarrow \infty }\frac{v(\mathfrak{a}_{m})}{m}.\end{eqnarray}$$

See [Reference Jonsson and MustaţǎJM12, Lemma 2.3] for a proof of the previous equality.

2.7 Multiplicities

Let $X$ be a variety of dimension $n$ and $x\in X$ a closed point. Let $\mathfrak{m}_{x}\subseteq {\mathcal{O}}_{X}$ denote the ideal of functions vanishing at  $x$ . We recall that for an $\mathfrak{m}_{x}$ -primary ideal $\mathfrak{a}$ , the Hilbert–Samuel multiplicity of $\mathfrak{a}$ is

$$\begin{eqnarray}\operatorname{e}(\mathfrak{a}):=\lim _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X,x}/\mathfrak{a}^{m})}{m^{n}/n!}.\end{eqnarray}$$

If $\mathfrak{a}\subseteq \mathfrak{b}\subseteq {\mathcal{O}}_{X}$ are $\mathfrak{m}_{x}$ -primary ideals on $X$ , then $\operatorname{e}(\mathfrak{a})\geqslant \operatorname{e}(\mathfrak{b})$ . In addition, $\operatorname{e}(\mathfrak{a})=\operatorname{e}(\overline{\mathfrak{a}})$ where $\overline{\mathfrak{a}}$ denotes the integral closure of  $\mathfrak{a}$ .

We recall the valuative definition of the integral closure of an ideal $\mathfrak{a}$ on a normal variety $X$ [Reference LazarsfeldLaz04, Example 9.6.8]. Let $U\subset X$ affine open subset. We have

$$\begin{eqnarray}\overline{\mathfrak{a}}(U):=\{f\in {\mathcal{O}}_{X}(U)\mid w(f)\geqslant w(\mathfrak{a})~\text{for all}~w\in \operatorname{Val}_{U}~\text{divisorial}\}.\end{eqnarray}$$

2.8 Volumes

Let $\mathfrak{a}_{\bullet }$ be a graded sequence of ideals with the property that each $\mathfrak{a}_{m}$ is $\mathfrak{m}_{x}$ -primary. The volume of $\mathfrak{a}_{\bullet }$ is defined as

$$\begin{eqnarray}\operatorname{vol}(\mathfrak{a}_{\bullet }):=\limsup _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X,x}/\mathfrak{a}_{m})}{m^{n}/n!}.\end{eqnarray}$$

A similar invariant is the multiplicity of $\mathfrak{a}_{\bullet }$ , which is defined as

$$\begin{eqnarray}\operatorname{e}(\mathfrak{a}_{\bullet })=\lim _{m\rightarrow \infty }\frac{\operatorname{e}(\mathfrak{a}_{m})}{m^{n}}.\end{eqnarray}$$

In various degrees of generality, it has been proven that

$$\begin{eqnarray}\operatorname{e}(\mathfrak{a}_{\bullet })=\operatorname{vol}(\mathfrak{a}_{\bullet })\end{eqnarray}$$

[Reference Ein, Lazarsfeld and SmithELS03, Corollary C], [Reference MustaţǎMus02, Theorem 1.7], [Reference Lazarsfeld and MustaţǎLM09, Theorem 3.8], [Reference CutkoskyCut13, Theorem 6.5]. In our setting, the above equality will always hold. In addition, by [Reference CutkoskyCut13, Theorem 1.1], we also have that

$$\begin{eqnarray}\operatorname{vol}(\mathfrak{a}_{\bullet }):=\lim _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X,x}/\mathfrak{a}_{m})}{m^{n}/n!}.\end{eqnarray}$$

For a valuation $v\in \operatorname{Val}_{X,x}$ , the volume of $v$ is given by

$$\begin{eqnarray}\operatorname{vol}(v):=\operatorname{e}(\mathfrak{a}_{\bullet }(v)).\end{eqnarray}$$

Note that if $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ , then $\operatorname{vol}(\unicode[STIX]{x1D706}v)=\operatorname{vol}(v)/\unicode[STIX]{x1D706}^{n}$ .

2.9 Log canonical thresholds

The log canonical threshold is an invariant of singularities that has received considerable interest in the field of birational geometry [Reference KollárKol97, § 8]. For a nonzero ideal $\mathfrak{a}$ on a klt variety  $X$ , the log canonical threshold of $\mathfrak{a}$ is given by

(2.1) $$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}):=\inf _{v\in \operatorname{DVal}_{X}}\frac{A_{X}(v)}{v(\mathfrak{a})}.\end{eqnarray}$$

By [Reference Jonsson and MustaţǎJM12, Lemma 6.7] in the case when $X$ is smooth and a similar argument in the klt case,

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a})=\inf _{v\in \operatorname{Val}_{X}}\frac{A_{X}(v)}{v(\mathfrak{a})}.\end{eqnarray}$$

In the previous expression, we are using the convention that if $v(\mathfrak{a})=0$ or $A(v)<+\infty$ , then $A(v)/v(\mathfrak{a})=+\infty$ . Thus, $\operatorname{lct}({\mathcal{O}}_{X})=+\infty$ . We say that a valuation $v^{\ast }$ computes $\operatorname{lct}(\mathfrak{a})$ if $A_{X}(v)<+\infty$ and $\operatorname{lct}(\mathfrak{a})=A(v^{\ast })/v^{\ast }(\mathfrak{a})$ .

Note the following elementary properties of this invariant. If $m\in \mathbf{Z}_{{>}0}$ , then

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}^{m})=\operatorname{lct}(\mathfrak{a})/m.\end{eqnarray}$$

If $\mathfrak{a}\subseteq \mathfrak{b}$ , then

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a})\leqslant \operatorname{lct}(\mathfrak{b}).\end{eqnarray}$$

The log canonical threshold may be understood in terms of a log resolution of $\mathfrak{a}$ . Recall that $\unicode[STIX]{x1D707}:Y\rightarrow X$ is a log resolution of $\mathfrak{a}$ if the following hold:

1. (a) $\unicode[STIX]{x1D707}$ is a projective birational morphism;

2. (b) $Y$ is smooth and $\operatorname{Exc}(\unicode[STIX]{x1D707})$ is pure codimension 1;

3. (c) $\mathfrak{a}\cdot {\mathcal{O}}_{Y}={\mathcal{O}}_{Y}(-D)$ for an effective divisor $D$ on $Y$ ; and

4. (d) $D_{\text{red}}+\operatorname{Exc}(\unicode[STIX]{x1D707})$ has simple normal crossing.

In this case, we have

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a})=\min _{i=1,\ldots ,r}\frac{A_{X}(\operatorname{ord}_{E_{i}})}{\operatorname{ord}_{E_{i}}(\mathfrak{a})},\end{eqnarray}$$

where $D=\sum _{i=1}^{r}a_{i}E_{i}$ . (Note that $\operatorname{ord}_{E_{i}}(\mathfrak{a})=a_{i}$ .) Thus, there always exists a divisorial valuation that computes $\operatorname{lct}(\mathfrak{a})$ .

For a graded sequence of ideals $\mathfrak{a}_{\bullet }$ on $X$ , the log canonical threshold of $\mathfrak{a}_{\bullet }$ is given by

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}_{\bullet }):=\lim _{m\rightarrow \infty }m\cdot \operatorname{lct}(\mathfrak{a}_{m})=\sup _{m}m\cdot \operatorname{lct}(\mathfrak{a}_{m}).\end{eqnarray}$$

By [Reference Jonsson and MustaţǎJM12, Corollary 6.9] when $X$ is smooth or as a consequence of [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15, Theorem 1.2] when $X$ is klt, we have

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}_{\bullet })=\inf _{v\in \operatorname{Val}_{X}}\frac{A_{X}(v)}{v(\mathfrak{a}_{\bullet })}.\end{eqnarray}$$

As before, we are using the convention that if either $v(\mathfrak{a}_{\bullet })=0$ or $A_{X}(v)=+\infty$ , then $A(v)/v(\mathfrak{a}_{\bullet })=+\infty$ . We say $v^{\ast }\in \operatorname{Val}_{X}$ computes $\operatorname{lct}(\mathfrak{a}_{\bullet })$ if $A(v)<+\infty$ and $\operatorname{lct}(\mathfrak{a}_{\bullet })=A_{X}(v^{\ast })/v^{\ast }(\mathfrak{a}_{\bullet })$ . Such valuations $v^{\ast }$ always exist (see Appendix B). When $X$ is smooth, this is precisely [Reference Jonsson and MustaţǎJM12, Theorem A].

3.1 Multiplier ideals

Let $X$ be a normal $\mathbf{Q}$ -Gorenstein variety. Fix a nonzero ideal $\mathfrak{a}\subseteq {\mathcal{O}}_{X}$ and $f:Y\rightarrow X$ a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\cdot {\mathcal{O}}_{Y}={\mathcal{O}}_{Y}(-D)$ . For a rational number $c>0$ , the multiplier ideal ${\mathcal{J}}(X,c\cdot \mathfrak{a})$ is defined by

$$\begin{eqnarray}{\mathcal{J}}(X,c\cdot \mathfrak{a}):=f_{\ast }{\mathcal{O}}_{Y}(\lceil K_{Y/X}-cD\rceil )\subseteq {\mathcal{O}}_{X}.\end{eqnarray}$$

Note that if $c$ is an integer, then ${\mathcal{J}}(X,c\cdot \mathfrak{a})={\mathcal{J}}(X,\mathfrak{a}^{c})$ . It is a basic fact that ${\mathcal{J}}(X,c\cdot \mathfrak{a})$ is independent of the choice of  $f$ .

Alternatively, the multiplier ideal can be understood valuatively. If $X$ is an affine variety, then [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15, Theorem 1.2] implies

$$\begin{eqnarray}{\mathcal{J}}(X,c\cdot \mathfrak{a})(X)=\{f\in {\mathcal{O}}_{X}(X)\mid v(f)>cv(\mathfrak{a})-A_{X}(v)~\text{for all}~v\in \operatorname{Val}_{X}\}.\end{eqnarray}$$

When $X$ is not necessarily affine, the above criterion allows us to understand the multiplier ideal locally.

It is important to note the relationship between the log canonical threshold and the multiplier ideal. If $X$ is klt, then

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a})=\sup \{c\mid {\mathcal{J}}(X,c\cdot \mathfrak{a})={\mathcal{O}}_{X}\}.\end{eqnarray}$$

The following lemma provides basic properties of multiplier ideals. The proof is left to the reader. See [Reference LazarsfeldLaz04, Proposition 9.2.32] for the case when $X$ is smooth.

Multiplier ideals satisfy the following ‘subadditivity property’. The property was first observed and proved by Demailly et al. in the smooth case [Reference Demailly, Ein and LazarsfeldDEL00]. The statement was extended to the singular case in [Reference TakagiTak06, Theorem 2.3] and [Reference EisensteinEis11, Theorem 7.3.4].

Theorem 3.2 (Subadditivity).

If $\mathfrak{a},\mathfrak{b}$ are nonzero ideals on $X$ and $c>0$ a rational number, then

$$\begin{eqnarray}\operatorname{Jac}_{X}\cdot {\mathcal{J}}(X,c\cdot (\mathfrak{a}\cdot \mathfrak{b}))\subseteq {\mathcal{J}}(X,c\cdot \mathfrak{a})\cdot {\mathcal{J}}(X,c\cdot \mathfrak{b}),\end{eqnarray}$$

where $\operatorname{Jac}_{X}$ denotes the Jacobian ideal of $X$ .

We recall that the Jacobian ideal of a variety $X$ is $\operatorname{Jac}_{X}:=\operatorname{Fitt}_{n}(X)$ , where $n$ is the dimension of $X$ and $\operatorname{Fitt}_{n}$ denotes the $n$ th fitting ideal as in [Reference EisenbudEis95, 20.2]. The singular locus of $X$ is equal to $\operatorname{Cosupp}(\operatorname{Jac}_{X})$ .

3.2 Asymptotic multiplier ideals

Let $\mathfrak{a}_{\bullet }$ be a graded sequence of ideals on a normal $\mathbf{Q}$ -Gorenstein variety $X$ and $c>0$ a rational number. We recall the definition of the asymptotic multiplier ideal ${\mathcal{J}}(c\cdot \mathfrak{a}_{\bullet })$ . By Lemma 3.1, we have that

$$\begin{eqnarray}{\mathcal{J}}\biggl(X,\frac{1}{p}\cdot \mathfrak{a}_{mp}\biggr)\subseteq {\mathcal{J}}\biggl(X,\frac{1}{pq}\cdot \mathfrak{a}_{pqm}\biggr)\end{eqnarray}$$

for all positive integers $p,q$ . From the above inclusion and Noetherianity, we conclude

$$\begin{eqnarray}\biggl\{{\mathcal{J}}\biggl(X,\frac{1}{p}\cdot \mathfrak{a}_{pm}\biggr)\biggr\}_{p\in \mathbf{N}}\end{eqnarray}$$

has a unique maximal element. The $m$ th asymptotic multiplier ideal ${\mathcal{J}}(X,m\cdot \mathfrak{a}_{\bullet })$ is defined to be this element. Like the standard multiplier ideal, the asymptotic multiplier ideal can also be understood valuatively.

Proposition 3.3 [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15, Theorem 1.2].

If $X$ is affine, $\mathfrak{a}_{\bullet }$ is a graded sequence of ideals on $X$ , and $c>0$ a rational number, then

$$\begin{eqnarray}{\mathcal{J}}(X,c\cdot \mathfrak{a}_{\bullet })=\{f\in {\mathcal{O}}_{X}(X)\mid v(f)>cv(\mathfrak{a}_{\bullet })-A_{X}(v)~\text{for all}~v\in \operatorname{Val}_{X}\}.\end{eqnarray}$$

The asymptotic multiplier ideals satisfy the following property. This property will allow us to approximate valuation ideals.

Proposition 3.4. If $\mathfrak{a}_{\bullet }$ is a graded sequence of ideals on a klt variety $X$ and $m,\ell \in \mathbf{N}$ , then

$$\begin{eqnarray}(\operatorname{Jac}_{X})^{\ell -1}\mathfrak{a}_{m}^{\ell }\subseteq (\operatorname{Jac}_{X})^{\ell -1}\mathfrak{a}_{m\ell }\subseteq {\mathcal{J}}(m\cdot \mathfrak{a}_{\bullet })^{\ell }.\end{eqnarray}$$

3.3 The case of valuation ideals

For a valuation $v\in \operatorname{Val}_{X}$ , we examine the asymptotic multiplier ideals of $\mathfrak{a}_{\bullet }(v)$ . We first prove the following elementary lemma.

Proof. Note that $v(\mathfrak{a}_{m}(v))\geqslant m$ , since $\mathfrak{a}_{m}(v)$ is the ideal of functions vanishing to at least order $m$ along  $v$ . Next, set $\unicode[STIX]{x1D6FC}:=v(\mathfrak{a}_{1}(v))$ . We have $\mathfrak{a}_{1}(v)^{\lceil m/\unicode[STIX]{x1D6FC}\rceil }\subseteq \mathfrak{a}_{m}(v)$ , since $v(\mathfrak{a}_{1}(v)^{\lceil m/\unicode[STIX]{x1D6FC}\rceil })=\unicode[STIX]{x1D6FC}\lceil m/\unicode[STIX]{x1D6FC}\rceil \geqslant m$ . Thus,

$$\begin{eqnarray}v(\mathfrak{a}_{m}(v))\leqslant v(\mathfrak{a}_{1}(v)^{\lceil m/\unicode[STIX]{x1D6FC}\rceil })=\unicode[STIX]{x1D6FC}\lceil m/\unicode[STIX]{x1D6FC}\rceil .\end{eqnarray}$$

The previous two bounds combine to show

$$\begin{eqnarray}1\leqslant \frac{v(\mathfrak{a}_{m}(v))}{m}\leqslant \frac{\unicode[STIX]{x1D6FC}\cdot \lceil m/\unicode[STIX]{x1D6FC}\rceil }{m},\end{eqnarray}$$

and the result follows. ◻

The following results allows us to approximate valuation ideals. In the case when $X$ is smooth and $v$ is an Abhyankhar valuation, the theorem below is a slight strengthening of [Reference Ein, Lazarsfeld and SmithELS03, Theorem A].

Theorem 3.6. If $X$ is a klt variety and $v\in \operatorname{Val}_{X}$ satisfies $A_{X}(v)<+\infty$ , then

$$\begin{eqnarray}(\operatorname{Jac}_{X})^{\ell -1}\cdot \mathfrak{a}_{m}^{\ell }\subseteq (\operatorname{Jac}_{X})^{\ell -1}\cdot \mathfrak{a}_{m\ell }\subseteq \mathfrak{a}_{m-e}^{\ell }\end{eqnarray}$$

for every $m\geqslant e$ and $\ell \in \mathbf{Z}_{{>}0}$ , where $\mathfrak{a}_{\bullet }:=\mathfrak{a}_{\bullet }(v)$ and $e:=\lceil A_{X}(v)\rceil$ .

Proof. By Proposition 3.4, we have that

$$\begin{eqnarray}(\operatorname{Jac}_{X})^{\ell -1}\cdot \mathfrak{a}_{m}^{\ell }\subseteq (\operatorname{Jac}_{X})^{\ell -1}\cdot \mathfrak{a}_{m\ell }\subseteq {\mathcal{J}}(X,m\cdot \mathfrak{a}_{\bullet })^{\ell }.\end{eqnarray}$$

Applying Proposition 3.3 and Lemma 3.5 to $\mathfrak{a}_{\bullet }$ gives that

$$\begin{eqnarray}{\mathcal{J}}(X,m\cdot \mathfrak{a}_{\bullet })\subseteq \mathfrak{a}_{m-e},\end{eqnarray}$$

and the result follows. ◻

3.4 Uniform approximation of volumes

Given a valuation $v\in \operatorname{Val}_{X}$ centered at a closed point on a $n$ -dimensional variety $X$ , we recall

$$\begin{eqnarray}\operatorname{vol}(v)=\lim _{m\rightarrow \infty }\frac{\operatorname{e}(\mathfrak{a}_{m}(v))}{m^{n}},\end{eqnarray}$$

where $n$ is the dimension of $X$ . The following proposition provides a uniform rate of convergence for the terms in the above limit.

Proposition 3.7. Let $X$ be a klt $n$ -dimensional variety and $x\in X$ a closed point. For $\unicode[STIX]{x1D716}>0$ and constants $B,E,r\in \mathbf{Z}_{{>}0}$ , there exists $N=N(\unicode[STIX]{x1D716},B,E,r)$ such that for every valuation $v\in \operatorname{Val}_{X,x}$ with $\operatorname{vol}(v)\leqslant B$ , $A_{X}(v)\leqslant E$ , and $v(\mathfrak{m}_{x})\geqslant 1/r$ , we have

$$\begin{eqnarray}\operatorname{vol}(v)\leqslant \frac{\operatorname{e}(\mathfrak{a}_{m}(v))}{m^{n}}<\operatorname{vol}(v)+\unicode[STIX]{x1D716}\quad \text{for all}~m\geqslant N.\end{eqnarray}$$

Proof of Proposition 3.7.

For any valuation $v\in \operatorname{Val}_{X,x}$ , the first inequality is well known. Indeed, the inclusion $\mathfrak{a}_{m}(v)^{p}\subseteq \mathfrak{a}_{mp}(v)$ for $m,p\in \mathbf{N}$ implies that

$$\begin{eqnarray}\frac{\operatorname{e}(\mathfrak{a}_{mp}(v))}{(mp)^{n}}\leqslant \frac{\operatorname{e}(\mathfrak{a}_{m}(v))}{(m)^{n}}.\end{eqnarray}$$

Fixing $m$ and sending $p\rightarrow \infty$ gives

$$\begin{eqnarray}\operatorname{vol}(v)\leqslant \frac{\operatorname{e}(\mathfrak{a}_{m}(v))}{(m)^{n}}.\end{eqnarray}$$

Next, fix $v\in \operatorname{Val}_{X,x}$ satisfying the hypotheses in the statement of Proposition 3.7. We have

$$\begin{eqnarray}(\operatorname{Jac}_{X})^{\ell -1}\mathfrak{a}_{m\ell }(v)\subseteq (\mathfrak{a}_{m-e}(v))^{\ell }\subseteq (\mathfrak{a}_{m-E}(v))^{\ell },\end{eqnarray}$$

where $e=\lceil A_{X}(v)\rceil$ . The first inclusion is the statement in Theorem 3.6, and the second follows from the assumption that $e\leqslant E$ . After replacing $m$ by $m+E$ , we obtain

(3.1) $$\begin{eqnarray}(\operatorname{Jac}_{X})^{\ell }\cdot \mathfrak{a}_{(m+E)\ell }(v)\subseteq (\operatorname{Jac}_{X})^{\ell -1}\cdot \mathfrak{a}_{(m+E)\ell }(v)\subseteq \mathfrak{a}_{m}(v)^{\ell }.\end{eqnarray}$$

On the other hand, the assumption that $v(\mathfrak{m}_{x})\geqslant 1/r$ implies that

(3.2) $$\begin{eqnarray}\mathfrak{m}_{x}^{mr}\subseteq \mathfrak{a}_{m}(v).\end{eqnarray}$$

It follows from (3.1) and (3.2) and the valuative criterion for integral closure (§ 2.7) that

(3.3) $$\begin{eqnarray}(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{\ell }\mathfrak{a}_{(m+E)\ell }(v)\subseteq \overline{\mathfrak{a}_{m}(v)^{\ell }}.\end{eqnarray}$$

Indeed, let $w\in \operatorname{Val}_{X}$ be a divisorial valuation and $f$ and $g$ local sections of $\operatorname{Jac}_{X}^{i}$ and $\mathfrak{m}_{x}^{mrj}$ , respectively, with $i+j=\ell$ . We have $\ell \cdot w(f)+i\cdot w(\mathfrak{a}_{(m+E)\ell }(v))\geqslant i\cdot w(\mathfrak{a}_{m}(v)^{\ell })$ and $w(g)\geqslant j\cdot w(\mathfrak{a}_{m}(v))$ by the two inclusions. Thus,

$$\begin{eqnarray}\displaystyle w(fg)=w(f)+w(g) & {\geqslant} & \displaystyle \frac{i}{\ell }(w(\mathfrak{a}_{m}(v)^{\ell })-w(\mathfrak{a}_{(m+E)\ell }(v)))+\frac{j}{\ell }w(\mathfrak{a}_{m}(v)^{\ell })\nonumber\\ \displaystyle & = & \displaystyle w(\mathfrak{a}_{m}(v)^{\ell })-w(\mathfrak{a}_{(m+E)\ell }(v)).\nonumber\end{eqnarray}$$

From Inclusion (3.3) and Teissier’s Minkowski inequality [Reference TeissierTei77], we see

$$\begin{eqnarray}\operatorname{e}(\mathfrak{a}_{m}(v)^{\ell })^{1/n}\leqslant \operatorname{e}((\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{\ell })^{1/n}+\operatorname{e}(\mathfrak{a}_{(m+E)\ell }(v))^{1/n}.\end{eqnarray}$$

Next, note that if $\mathfrak{a}$ is an $\mathfrak{m}_{x}$ -primary ideal, then $\operatorname{e}(\mathfrak{a}^{m})=m^{n}\operatorname{e}(\mathfrak{a})$ . Applying this property and dividing by $m\cdot \ell$ , gives that

$$\begin{eqnarray}\frac{\operatorname{e}(\mathfrak{a}_{m}(v))^{1/n}}{m}\leqslant \frac{\operatorname{e}(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{1/n}}{m}+\frac{m+E}{m}\cdot \frac{\operatorname{e}(\mathfrak{a}_{(m+E)\ell }(v))^{1/n}}{(m+E)\ell }.\end{eqnarray}$$

After letting $\ell \rightarrow \infty$ , we obtain

$$\begin{eqnarray}\frac{\operatorname{e}(\mathfrak{a}_{m}(v))^{1/n}}{m}\leqslant \frac{\operatorname{e}(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{1/n}}{m}+\frac{m+E}{m}\operatorname{vol}(v)^{1/n}.\end{eqnarray}$$

Since $\operatorname{vol}(v)^{1/n}\leqslant B^{1/n}$ , the assertion will follow if we show that

$$\begin{eqnarray}\lim _{m\rightarrow \infty }\frac{\operatorname{e}(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{1/n}}{m}=0.\end{eqnarray}$$

Choose $h\in \operatorname{Jac}_{X}\cdot {\mathcal{O}}_{X,x}$ that is nonzero and set $R:={\mathcal{O}}_{X,x}/(h)$ and $\tilde{\mathfrak{m}}_{x}:=\mathfrak{m}_{x}\cdot R$ . We have

$$\begin{eqnarray}\lim _{m\rightarrow \infty }\frac{\operatorname{e}(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr})^{1/n}}{m}=\lim _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X,x}/(\operatorname{Jac}_{X}+\mathfrak{m}_{x}^{mr}))^{1/n}}{m}\leqslant \lim _{m\rightarrow \infty }\frac{\operatorname{length}(R/\tilde{\mathfrak{m}}_{x}^{mr})^{1/n}}{m}.\end{eqnarray}$$

The last limit is 0, since

$$\begin{eqnarray}\lim _{m\rightarrow \infty }\frac{\operatorname{length}(R/{\tilde{\mathfrak{m}}_{x}}^{mr})}{m^{n-1}/(n-1)!}=\operatorname{e}({\tilde{\mathfrak{m}}_{x}}^{r})<\infty .\Box\end{eqnarray}$$

4.1 Normalized volume minimizers

Proposition 4.6. If there exists a graded sequence of $\mathfrak{m}_{x}$ -primary ideals $\tilde{\mathfrak{a}}_{\bullet }$ such that

$$\begin{eqnarray}\operatorname{lct}(\tilde{\mathfrak{a}}_{\bullet })^{n}\operatorname{e}(\tilde{\mathfrak{a}}_{\bullet })=\inf _{\mathfrak{a}_{\bullet }\,\mathfrak{m}_{x}\text{-}\text{primary}}\operatorname{lct}(\mathfrak{a}_{\bullet })^{n}\operatorname{e}(\mathfrak{a}_{\bullet }),\end{eqnarray}$$

then there exists $v^{\ast }\in \operatorname{Val}_{X,x}$ that is a minimizer of $\widehat{\operatorname{vol}}_{X,x}$ . Furthermore, if there exists an $\mathfrak{m}_{x}$ -primary ideal $\tilde{\mathfrak{a}}$ such that

$$\begin{eqnarray}\operatorname{lct}(\tilde{\mathfrak{a}})^{n}\operatorname{e}(\tilde{\mathfrak{a}})=\inf _{\mathfrak{a}\,\mathfrak{m}_{x}\text{-}\text{primary}}\operatorname{lct}(\mathfrak{a})^{n}\operatorname{e}(\mathfrak{a}),\end{eqnarray}$$

then we may choose $v^{\ast }$ to be divisorial.

Proof. Assume there exists such a graded sequence $\tilde{\mathfrak{a}}_{\bullet }$ . By Theorem B.1, we may choose a valuation $v^{\ast }$ that computes $\operatorname{lct}(\tilde{\mathfrak{a}}_{\bullet })$ . By Lemma 4.4,

$$\begin{eqnarray}\widehat{\operatorname{vol}}(v^{\ast })\leqslant \operatorname{lct}(\tilde{\mathfrak{a}}_{\bullet })^{n}\operatorname{e}(\tilde{\mathfrak{a}}_{\bullet }).\end{eqnarray}$$

By Proposition 4.3, the result follows.

When there exits such an ideal $\tilde{\mathfrak{a}}$ , the same argument shows that if $v^{\ast }$ computes $\operatorname{lct}(\tilde{\mathfrak{a}})$ , then $v^{\ast }$ is our desired valuation. Furthermore, we may choose $v^{\ast }$ divisorial.◻

Lemma 4.7. If $v^{\ast }$ is a minimizer of $\widehat{\operatorname{vol}}_{X,x}$ , then

$$\begin{eqnarray}A_{X}(v^{\ast })\leqslant \frac{A_{X}(w)}{w(\mathfrak{a}_{\bullet }(v^{\ast }))}\end{eqnarray}$$

for all $w\in \operatorname{Val}_{X,x}$ . Furthermore, equality holds if and only if $w=\unicode[STIX]{x1D706}v^{\ast }$ for some $\unicode[STIX]{x1D706}\in \mathbf{R}_{{>}0}$ .

Proof. We fix $w\in \operatorname{Val}_{X,x}$ and rescale $w$ so that $w(\mathfrak{a}_{\bullet }(v^{\ast }))=1$ . Thus, we are reduced to showing that $A_{X}(v^{\ast })\leqslant A_{X}(w)$ and equality holds if and only if $w=v^{\ast }$ .

By definition, we have

$$\begin{eqnarray}1=w(\mathfrak{a}_{\bullet }(v^{\ast })):=\inf _{m\geqslant 0}\frac{w(\mathfrak{a}_{m}(v^{\ast }))}{m},\end{eqnarray}$$

and, thus, $w(\mathfrak{a}_{m}(v^{\ast }))\geqslant m$ . The latter implies that $\mathfrak{a}_{m}(v^{\ast })\subseteq \mathfrak{a}_{m}(w)$ , so

$$\begin{eqnarray}\operatorname{vol}(w)\leqslant \operatorname{vol}(v^{\ast }).\end{eqnarray}$$

If $A_{X}(w)<A_{X}(v^{\ast })$ , then

$$\begin{eqnarray}A_{X}(w)^{n}\operatorname{vol}(w)<A_{X}(v^{\ast })^{n}\operatorname{vol}(v^{\ast })\end{eqnarray}$$

and this would contradict our assumption on $v^{\ast }$ . Furthermore, if $A(v^{\ast })=A(w)$ , then we must have that $\operatorname{vol}(v^{\ast })=\operatorname{vol}(w)$ . Since $v^{\ast }\leqslant w$ and $\operatorname{vol}(v^{\ast })=\operatorname{vol}(w)$ , then $v^{\ast }=w$ by [Reference Li and XuLX16, Proposition 2.12].◻

The previous proposition was independently observed in [Reference Li and XuLX16, Theorem 1.5]. In fact, prior to our contribution, the original draft of [Reference Li and XuLX16] proved that if (a) holds then (b) holds. Our argument is different from that of [Reference Li and XuLX16].

Proof. By Lemma 4.7, it follows that $\operatorname{lct}(\mathfrak{a}_{\bullet }(v^{\ast }))=A(v^{\ast })$ . The finite generation of the desired ${\mathcal{O}}_{X}$ -algebra is a consequence of [Reference BlumBlu16, Theorem 1.4.1]. Additionally, the second sentence of Lemma 4.7 allows us to apply [Reference BlumBlu16, Proposition 4.4]. Thus, $v^{\ast }$ corresponds to a Kollar component.

To show that $\widehat{\operatorname{vol}}(v^{\ast })$ is rational, we note that the finite generation statement of (a) implies there exists $N>0$ so that $\mathfrak{a}_{mN}(v^{\ast })=(\mathfrak{a}_{N}(v^{\ast }))^{m}$ for all $m\in \mathbf{N}$ [Reference GrothendieckGro61, Lemma II.2.1.6.v]. By Lemma 4.4,

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}_{\bullet }(v^{\ast }))^{n}\operatorname{e}(\mathfrak{a}_{\bullet }(v^{\ast }))\leqslant \widehat{\operatorname{vol}}(v^{\ast }).\end{eqnarray}$$

Lemma 4.1 implies

$$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}_{\bullet }(v^{\ast }))^{n}\operatorname{e}(\mathfrak{a}_{\bullet }(v^{\ast }))=\operatorname{lct}(\mathfrak{a}_{N})^{n}\operatorname{e}(\mathfrak{a}_{N}),\end{eqnarray}$$

and the latter is a rational number. ◻

5.1 Parameterizing ideals

We fix an integer $d>0$ and seek to parameterize ideals $\mathfrak{a}\subset A$ containing $\mathfrak{m}_{A}^{d}$ and contained in $\mathfrak{m}_{A}$ . Since $\mathfrak{m}_{R}^{d}\subseteq \unicode[STIX]{x1D711}^{-1}(\mathfrak{m}_{A}^{d})$ , such an ideal $\mathfrak{a}\subseteq A$ can be generated by $\mathfrak{m}_{A}^{d}$ and images of polynomials from $R$ of $\deg <d$ . Since there are $n_{d}=\binom{r+d-1}{r}-1$ monomials of positive degree less than $d$ in  $R$ , any such ideal $\mathfrak{m}_{A}^{d}\subseteq \mathfrak{a}\subseteq \mathfrak{m}_{A}$ can be generated by $\mathfrak{m}_{A}^{d}$ and the image of $n_{d}$ linear combinations of monomials. After setting $N_{d}=n_{d}^{2}$ , we get a map

$$\begin{eqnarray}\{k\text{-}\text{valued points of}~\mathbb{A}^{N_{d}}\}\longrightarrow \{\text{ideals}~\mathfrak{a}\subseteq A~\text{s.t.}~\mathfrak{m}_{A}^{d}\subseteq \mathfrak{a}\subseteq \mathfrak{m}_{A}\},\end{eqnarray}$$

where $\mathbb{A}^{N_{d}}$ parameterizes coefficients and generators of such ideals. The above map is surjective, but not injective (generators of an ideal are not unique). Additionally, we have a universal ideal $\mathscr{A}\subset {\mathcal{O}}_{X\times \mathbb{A}^{N_{d}}}$ such that $\mathscr{A}$ restricted to the fibers of $p:X\times \mathbb{A}^{N_{d}}\rightarrow \mathbb{A}^{N_{d}}$ give us our $\mathfrak{m}_{A}$ -primary ideals.

The construction in the previous paragraph follows the exposition of [Reference de Fernex and MustaţăFM09, § 3].

5.2 Parameterizing graded sequences of ideals

We proceed to parameterize graded sequences of ideals $\mathfrak{a}_{\bullet }$ of $A$ satisfying

We set

$$\begin{eqnarray}H_{d}:=\mathbb{A}^{N_{1}}\times \cdots \times \mathbb{A}^{N_{d}},\end{eqnarray}$$

where $N_{i}$ is chosen as in the previous section. For $d>c$ , let $\unicode[STIX]{x1D70B}_{d,c}:H_{d}\rightarrow H_{c}$ denote the natural projection maps. Our desired object is the following projective limit

$$\begin{eqnarray}H=\varprojlim H_{d}.\end{eqnarray}$$

For $d>0$ , let $\unicode[STIX]{x1D70B}_{d}:H\rightarrow H_{d}$ denote the natural map.

Note that the above projective limit exists in the category of schemes, since the maps in our directed system are all affine morphisms. Indeed, $H$ is isomorphic to an infinite-dimensional affine space.

Since a $k$ -valued point of $H$ is simply a sequence of $k$ -valued points of $\mathbb{A}^{N_{d}}$ for all $d\in \mathbf{N}$ , we have a surjection

$$\begin{eqnarray}\{k\text{-}\text{valued points of}~H\}\longrightarrow \{\text{sequences of ideals}~\mathfrak{b}_{\bullet }~\text{of}~A~\text{satisfying}~(\dagger )\}.\end{eqnarray}$$

Note that the sequences of ideals on the right-hand side are not necessarily graded.

Given a sequence of ideals $\mathfrak{b}_{\bullet }$ , we can construct a graded sequence $\mathfrak{a}_{\bullet }$ inductively by setting $\mathfrak{a}_{1}:=\mathfrak{b}_{1}$ and

$$\begin{eqnarray}\mathfrak{a}_{q}:=\mathfrak{b}_{q}+\mathop{\sum }_{m+n=q}\mathfrak{a}_{m}\cdot \mathfrak{a}_{n}.\end{eqnarray}$$

If $\mathfrak{b}_{\bullet }$ was graded to begin with, then $\mathfrak{a}_{\bullet }=\mathfrak{b}_{\bullet }$ . By the construction, it is clear that $\mathfrak{a}_{m}\cdot \mathfrak{a}_{n}\subseteq \mathfrak{a}_{m+n}$ . Thus, we have our desired map

$$\begin{eqnarray}\{k\text{-}\text{valued points of}~H\}\longrightarrow \{\text{graded sequences of ideals}~\mathfrak{a}_{\bullet }~\text{of}~A~\text{satisfying}~(\dagger )\}.\end{eqnarray}$$

Additionally, we have a universal graded sequence of ideals $\mathscr{A}_{\bullet }=\{\mathscr{A}_{m}\}_{m\in \mathbf{N}}$ on $X\times H$ . We will often abuse notation and refer to similarly defined ideals $\mathscr{A}_{1},\ldots ,\mathscr{A}_{d}$ on $X\times H_{d}$ .

The following technical lemma will be useful in the next proposition. The proof of the lemma relies on the fact that every descending sequence of nonempty constructible subsets of a variety over an uncountable field has nonempty intersection.

Lemma 5.1. If $\{W_{d}\}_{d\in \mathbf{N}}$ is a collection of nonempty subsets of $H_{d}$ such that

1. (a) $W_{d}\subset H_{d}$ is a constructible, and

2. (b)

   $$\begin{eqnarray}W_{d+1}\subseteq \unicode[STIX]{x1D70B}_{d+1,d}^{-1}(W_{d})\end{eqnarray}$$
   for each $d\in \mathbf{Z}_{{>}0}$ ,

then there exists a $k$ -valued point in

$$\begin{eqnarray}\mathop{\bigcap }_{d\in \mathbf{N}}\unicode[STIX]{x1D70B}_{d}^{-1}(W_{d}).\end{eqnarray}$$

Proof. Note that a $k$ -valued point in the above intersection is equivalent to a sequence of closed points $\{x_{d}\in W_{d}\}_{d\in \mathbf{N}}$ such that $\unicode[STIX]{x1D70B}_{d+1,d}(x_{d+1})=x_{d}$ . We proceed to construct such a sequence.

We first look to find a candidate for $x_{1}$ . Assumption (b) implies

$$\begin{eqnarray}W_{1}\supseteq \unicode[STIX]{x1D70B}_{2,1}(W_{2})\supseteq \unicode[STIX]{x1D70B}_{3,1}(W_{3})\supseteq \cdots\end{eqnarray}$$

is a descending sequence of nonempty sets. Note that $W_{1}$ is constructible by assumption and so are $\unicode[STIX]{x1D70B}_{d,1}(W_{d})$ for all $d$ by Chevalley’s theorem [Reference HartshorneHar77, Exercise II.3.9]. Thus,

$$\begin{eqnarray}W_{1}\cap \unicode[STIX]{x1D70B}_{2,1}(W_{2})\cap \unicode[STIX]{x1D70B}_{3,1}(W_{3})\cap \cdots\end{eqnarray}$$

is nonempty and we may choose a point $x_{1}$ lying in the set.

Next, we look at

$$\begin{eqnarray}W_{2}\cap \unicode[STIX]{x1D70B}_{2,1}^{-1}(x_{1})\supseteq \unicode[STIX]{x1D70B}_{3,2}(W_{3})\cap \unicode[STIX]{x1D70B}_{2,1}^{-1}(x_{1})\supseteq \unicode[STIX]{x1D70B}_{4,2}(W_{4})\cap \unicode[STIX]{x1D70B}_{2,1}^{-1}(x_{1}),\end{eqnarray}$$

and note that for $d\geqslant 2$ each $\unicode[STIX]{x1D70B}_{d,2}(W_{d})\cap \unicode[STIX]{x1D70B}_{2,1}^{-1}(x_{1})$ is nonempty by our choice of $x_{1}$ . By the same argument as before, we see

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{2,1}^{-1}(x_{1})\cap W_{2}\cap \unicode[STIX]{x1D70B}_{3,2}(W_{3})\cap \unicode[STIX]{x1D70B}_{4,2}(W_{4})\cap \cdots\end{eqnarray}$$

is nonempty and contains a closed point $x_{2}$ . Continuing in this manner, we construct the desired sequence.◻

5.3 Finding limit points

The proof of the following proposition relies on the previous construction of a space that parameterizes graded sequences of ideals. The proof is inspired by arguments in [Reference KollárKol08, Reference de Fernex, Ein and MustaţăFEM10, Reference de Fernex, Ein and MustaţăFEM11].

Proposition 5.2. Let $X$ be a klt variety and $x\in X$ a closed point. Assume there exists a collection of graded sequences of $\mathfrak{m}_{x}$ -primary ideals $\{\mathfrak{a}_{\bullet }^{(i)}\}_{i\in \mathbf{N}}$ and $\unicode[STIX]{x1D706}\in \mathbf{R}$ such that the following hold.

1. (a) (Convergence from above) For every $\unicode[STIX]{x1D716}>0$ , there exists positive constants $M,N$ so that

   $$\begin{eqnarray}\operatorname{lct}(\mathfrak{a}_{m}^{(i)})^{n}\operatorname{e}(\mathfrak{a}_{m}^{(i)})\leqslant \unicode[STIX]{x1D706}+\unicode[STIX]{x1D716}\end{eqnarray}$$
   for all $m\geqslant M$ and $i\geqslant N$ .

2. (b) (Boundedness from below) For each $m,i\in \mathbf{N}$ , we have

   $$\begin{eqnarray}\mathfrak{m}_{x}^{m}\subseteq \mathfrak{a}_{m}^{(i)}.\end{eqnarray}$$

3. (c) (Boundedness from above) There exists $\unicode[STIX]{x1D6FF}>0$ such that

   $$\begin{eqnarray}\mathfrak{a}_{m}^{(i)}\subseteq \mathfrak{m}^{\lfloor m\unicode[STIX]{x1D6FF}\rfloor }\end{eqnarray}$$
   for all $m,i\in \mathbf{N}$ .

Then, there exists a graded sequence of $\mathfrak{m}_{x}$ -primary ideals $\tilde{\mathfrak{a}}_{\bullet }$ on $X$ such that

$$\begin{eqnarray}\operatorname{lct}(\tilde{\mathfrak{a}}_{\bullet })^{n}\operatorname{e}(\tilde{\mathfrak{a}}_{\bullet })\leqslant \unicode[STIX]{x1D706}.\end{eqnarray}$$

Proof. It is sufficient to consider the case when $X$ is affine. Thus, we may assume that $X=\operatorname{Spec}A$ and $A=k[x_{1},\ldots ,x_{r}]/\mathfrak{p}$ as in the beginning of this section. In addition, we may assume that $x\in X$ corresponds to the maximal ideal $\mathfrak{m}_{A}$ . We recall that § 5.2 constructs a variety $H$ parameterizing graded sequences of ideals on $X$ satisfying (†). In addition, we have finite-dimensional truncations $H_{d}$ that parameterize the first $d$ elements of such a sequence.

Each graded sequence $\mathfrak{a}_{\bullet }^{(i)}$ satisfies (†) by assumptions (b) and (c). Thus, we may choose a point $p_{i}\in H$ corresponding to $\mathfrak{a}_{\bullet }^{(i)}$ . Note that $\unicode[STIX]{x1D70B}_{d}(p_{i})\in H_{d}$ corresponds to the first $d$ -terms of $\mathfrak{a}_{\bullet }^{(i)}$ .

Claim 1. We may choose infinite subsets $\mathbf{N}\supset I_{1}\supset I_{2}\supset \cdots \,$ and set

$$\begin{eqnarray}Z_{d}:=\overline{\{\unicode[STIX]{x1D70B}_{d}(p_{i})\mid i\in I_{d}\}}\end{eqnarray}$$

such that (∗∗) holds:

To prove Claim 1, we construct such a sequence inductively. First, we set $I_{1}=\mathbf{N}$ . Since $H_{1}\simeq \mathbb{A}^{0}$ is a point, (∗∗) is trivially satisfied for $d=1$ . After having chosen $I_{d}$ , choose $I_{d+1}\subset I_{d}$ so that (∗∗) is satisfied for $Z_{d+1}$ . By the Noetherianity of $H_{d}$ , such a choice is possible.

Claim 2. We have the inclusion $Z_{d+1}\subseteq \unicode[STIX]{x1D70B}_{d+1,d}^{-1}(Z_{d})$ for all $d\geqslant 1$ .

The proof of Claim 2 follows from the definition of $Z_{d}$ . Since $\unicode[STIX]{x1D70B}_{d}(p_{i})\in Z_{d}$ for all $i\in I_{d}$ and $I_{d}\supseteq I_{d+1}$ , it follows that $\unicode[STIX]{x1D70B}_{d+1,d}^{-1}(Z_{d})$ is a closed set containing $\unicode[STIX]{x1D70B}_{d+1}(p_{i})$ for $i\in I_{d+1}$ . The closure of the latter set of points is precisely $Z_{d+1}$ .

Claim 3. If $p\in Z_{d}$ is a closed point, we have $\mathscr{A}_{d}|_{p}\subseteq \mathfrak{m}^{\lfloor d\unicode[STIX]{x1D6FF}\rfloor }$ .

We now prove Claim 3. The set $\{p\in H_{d}\mid {\mathcal{A}}_{d}|_{p}\subseteq \mathfrak{m}^{\lfloor d\unicode[STIX]{x1D6FF}\rfloor }\}$ is a closed in $H_{d}$ . By assumption (c), $\unicode[STIX]{x1D70B}_{d}(p_{i})$ lies in the above closed set for all $i\in I_{d}$ . Thus, $Z_{d}\subseteq \{p\in H_{d}\mid {\mathcal{A}}_{d}|_{p}\subseteq \mathfrak{m}^{\lfloor d\unicode[STIX]{x1D6FF}\rfloor }\}$ .

We now return to the proof of the proposition. We look at the normalized multiplicity of the ideals parameterized by  $Z_{d}$ . By Propositions A.1 and A.2, for each  $d$ , we may choose a nonempty open set $U_{d}\subseteq Z_{d}$ such that

$$\begin{eqnarray}\operatorname{lct}(\mathscr{A}_{d}|_{p})^{n}\operatorname{e}(\mathscr{A}_{d}|_{p})=\unicode[STIX]{x1D706}_{d}\end{eqnarray}$$

is constant for $p\in U_{d}$ . Set

$$\begin{eqnarray}I_{d}^{\circ }=\{i\in I_{d}\mid \unicode[STIX]{x1D70B}_{d}(p_{i})\in U_{d}\}\subseteq I_{d},\end{eqnarray}$$

and note that $I_{d}\setminus I_{d}^{\circ }$ is finite. If this was not the case, then (∗∗) would not hold.

The finiteness of $I_{d}\setminus I_{d}^{\circ }$ has two consequences. First,

$$\begin{eqnarray}\lim _{d\rightarrow \infty }\sup \unicode[STIX]{x1D706}_{d}\leqslant \unicode[STIX]{x1D706},\end{eqnarray}$$

since $\unicode[STIX]{x1D70B}_{d}(p_{i})\in U_{d}$ for all $i\in I_{d}^{\circ }$ and assumption (a) of this proposition. Second, since $I_{d+1}\subset I_{d}$ for $d\in \mathbf{N}$ , we have

$$\begin{eqnarray}I_{d}^{\circ }\cap I_{d-1}^{\circ }\cdots \cap I_{1}^{\circ }\neq \emptyset .\end{eqnarray}$$

Claim 4. There exists a $k$ -valued point $\tilde{p}\in H$ such that $\unicode[STIX]{x1D70B}_{d}(\tilde{p})\in U_{d}$ for all $d\in \mathbf{Z}_{{>}0}$ .

Proving this claim will complete the proof. Indeed, a point $\tilde{p}\in H$ corresponds to a graded sequence of $\mathfrak{m}_{x}$ -primary ideals $\tilde{\mathfrak{a}}_{\bullet }$ . Since $\unicode[STIX]{x1D70B}_{d}(\tilde{p})\in U_{d}$ , we will have $\operatorname{lct}(\tilde{\mathfrak{a}}_{d})^{n}\operatorname{e}(\tilde{\mathfrak{a}}_{d})=\unicode[STIX]{x1D706}_{d}$ . In addition, Claim 2 implies and $\tilde{\mathfrak{a}}_{d}\subset \mathfrak{m}_{x}^{\lfloor d\unicode[STIX]{x1D6FF}\rfloor }$ for all $d\in \mathbf{Z}_{{>}0}$ . Thus,

$$\begin{eqnarray}\operatorname{lct}(\tilde{\mathfrak{a}}_{\bullet })^{n}\operatorname{e}(\tilde{\mathfrak{a}}_{\bullet })=\lim _{d\rightarrow \infty }\operatorname{lct}(\tilde{\mathfrak{a}}_{d})\operatorname{e}(\tilde{\mathfrak{a}}_{d})\leqslant \lim _{d\rightarrow \infty }\sup \unicode[STIX]{x1D706}_{d}\leqslant \unicode[STIX]{x1D706},\end{eqnarray}$$

and the proof will be complete.

We are left to prove Claim 4. In order to do so, we will apply Lemma 5.1 to find such a point $\tilde{p}\in H$ . First, we define constructible sets $W_{d}\subseteq H_{d}$ inductively. Set $W_{1}=U_{1}$ . After having chosen $W_{d}$ , set $W_{d+1}=\unicode[STIX]{x1D70B}_{d+1,d}^{-1}(W_{d})\cap U_{d}$ . For each $d\in \mathbf{N}$ we have the following:

• ∙ $W_{d}$ is open in $Z_{d}$ and, thus, constructible in $H_{d}$ ;

• ∙ $W_{d}$ is nonempty, since $W_{d}$ contains $\unicode[STIX]{x1D70B}_{d}(p_{i})$ for all $i\in I_{d}^{\circ }\cap I_{d-1}^{\circ }\cap \cdots \cap I_{1}^{\circ }$ , which is nonempty.

By Lemma 5.1, there exists a $k$ -valued point $\tilde{p}\in H$ such that $\unicode[STIX]{x1D70B}_{d}(\tilde{p})\in W_{d}\subset U_{d}$ for all $d\in \mathbf{Z}_{{>}0}$ . This completes the proof.◻

Remark 5.3. In the previous proof, we construct a graded sequence of ideal $\tilde{\mathfrak{a}}_{\bullet }$ based on a collection of graded sequences $\{\mathfrak{a}_{\bullet }^{(i)}\}_{i\in \mathbf{N}}$ . While the construction of $\tilde{\mathfrak{a}}_{\bullet }$ is inspired by past constructions of generic limits, $\tilde{\mathfrak{a}}_{\bullet }$ is not a generic limit in the sense of [Reference KollárKol08, 28].

We construct the precise analog as follows. We set

$$\begin{eqnarray}Z:=\mathop{\bigcap }_{d}\unicode[STIX]{x1D70B}_{d}^{-1}(Z_{d})\subseteq H,\end{eqnarray}$$

with $Z_{d}$ defined in the previous proof. The generic point of $Z$ gives a map $\operatorname{Spec}(K(Z))\rightarrow H$ , where $K(Z)$ is the function field of $Z$ . Thus, we get a graded sequence of ideals $\widehat{\mathfrak{a}}_{\bullet }$ on $X_{K(Z)}$ , the base change of $X$ by  $K(Z)$ .

In the previous proof, we wanted to construct a graded sequence on  $X$ , not a base change of  $X$ . Thus, $\tilde{\mathfrak{a}}_{\bullet }$ was chosen to be a graded sequence corresponding to a very general point in  $Z$ .

7.1 Log discrepancies

If $(X,\unicode[STIX]{x1D6E5})$ is a log pair, the log discrepancy function $A_{(X,\unicode[STIX]{x1D6E5})}:\operatorname{Val}_{X}\rightarrow \mathbf{R}\cup \{+\infty \}$ is defined as follows. If $Y\rightarrow X$ is a proper birational morphism with $Y$ normal, and $E\subset Y$ a prime divisor, then the log discrepancy of $\operatorname{ord}_{E}$ over $(X,\unicode[STIX]{x1D6E5})$ is

$$\begin{eqnarray}A_{(X,\unicode[STIX]{x1D6E5})}(\operatorname{ord}_{E}):=1+(\text{coefficient of}~E~\text{in}~K_{Y}-f^{\ast }(K_{X}+\unicode[STIX]{x1D6E5})).\end{eqnarray}$$

As in [Reference Boucksom, de Fernex, Favre and UrbinatiBFFU15], $A_{(X,\unicode[STIX]{x1D6E5})}$ can then be extended to $\operatorname{Val}_{X}$ . If $(X,\unicode[STIX]{x1D6E5})$ is a log pair, we say $(X,\unicode[STIX]{x1D6E5})$ is klt if $A_{X}(v)>0$ for all divisorial valuations $v\in \operatorname{Val}_{X}$ .

7.2 Normalized volume minimizers

If $(X,\unicode[STIX]{x1D6E5})$ is a klt log pair and $x\in X$ is a closed point, the normalized volume of a valuation $v\in \operatorname{Val}_{X,x}$ over the pair $(X,\unicode[STIX]{x1D6E5})$ is defined to be

$$\begin{eqnarray}\widehat{\operatorname{vol}}_{(X,\unicode[STIX]{x1D6E5}),x}(v):=A_{(X,\unicode[STIX]{x1D6E5})}(v)^{n}\operatorname{vol}(v).\end{eqnarray}$$

We claim that if $(X,\unicode[STIX]{x1D6E5})$ is a klt pair and $x\in X$ is a closed point, then there exists a minimizer of $\widehat{\operatorname{vol}}_{(X,\unicode[STIX]{x1D6E5}),x}$ .

The main subtlety in extending our arguments to the log setting is in extending Theorem 3.6, which is a consequence of the subadditivity theorem. Takagi proved the following subadditivity theorem for log pairs.

Theorem 7.1 (Takagi [Reference TakagiTak13]).

Let $(X,\unicode[STIX]{x1D6E5})$ be a log pair, $\mathfrak{a},\mathfrak{b}$ ideals on $X$ , and $s,t\in \mathbf{Q}_{{\geqslant}0}$ . For $r\in \mathbf{Z}_{{>}0}$ so that $r(K_{X}+\unicode[STIX]{x1D6E5})$ , we have

$$\begin{eqnarray}\operatorname{Jac}_{X}\cdot {\mathcal{J}}((X,\unicode[STIX]{x1D6E5}),\mathfrak{a}^{s}\mathfrak{b}^{s}{\mathcal{O}}_{X}(-r\unicode[STIX]{x1D6E5})^{1/r})\subseteq {\mathcal{J}}((X,\unicode[STIX]{x1D6E5}),\mathfrak{a}^{s}){\mathcal{J}}((X,\unicode[STIX]{x1D6E5}),\mathfrak{b}^{t}).\end{eqnarray}$$

Takagi’s result implies the following generalization of Theorem 3.6 for log pairs. The remaining arguments in the paper extend to this setting.

Theorem 7.2. Let $(X,\unicode[STIX]{x1D6E5})$ be a log pair and $r\in \mathbf{Z}_{{>}0}$ such that $r(K_{X}+\unicode[STIX]{x1D6E5})$ is Carter. If $v\in \operatorname{Val}_{X}$ satisfies $A_{X}(v)<+\infty$ , then

$$\begin{eqnarray}(\operatorname{Jac}_{X}\cdot {\mathcal{O}}_{X}(-r\unicode[STIX]{x1D6E5}))^{\ell -1}\cdot \mathfrak{a}_{m}^{\ell }\subseteq (\operatorname{Jac}_{X}{\mathcal{O}}_{X}(-r\unicode[STIX]{x1D6E5}))^{\ell -1}\cdot \mathfrak{a}_{m\ell }\subseteq \mathfrak{a}_{m-e}^{\ell }\end{eqnarray}$$

for every $m\geqslant e$ and $\ell \in \mathbf{Z}_{{>}0}$ , where $\mathfrak{a}_{\bullet }:=\mathfrak{a}_{\bullet }(v)$ and $e:=\lceil A_{(X,\unicode[STIX]{x1D6E5})}(v)\rceil$ .

8.1 Deformation to the initial ideal

As explained in [Reference EisenbudEis95], when $X_{\unicode[STIX]{x1D70E}}\simeq \mathbb{A}^{n}$ and $I\subset R_{\unicode[STIX]{x1D70E}}$ , there exists a deformation of $I$ to a monomial ideal. A similar argument works in our setting.

Following the approach of [Reference Kaveh and KhovanskiiKK14, § 6], we put a $\mathbf{Z}_{{\geqslant}0}^{n}$ order on the monomials of $R_{\unicode[STIX]{x1D70E}}$ . Fix $y_{1},\ldots ,y_{n}\in N\,\cap \,\unicode[STIX]{x1D70E}$ that are linearly independent in  $M_{\mathbf{R}}$ . Thus, we get an injective map $\unicode[STIX]{x1D70C}:M\rightarrow \mathbf{Z}^{n}$ by sending

$$\begin{eqnarray}v\longmapsto (\langle y_{1},v\rangle ,\ldots ,\langle y_{n},v\rangle ).\end{eqnarray}$$

Since each $y_{i}\in \unicode[STIX]{x1D70E}$ , we have $\unicode[STIX]{x1D70C}(M\cap \unicode[STIX]{x1D70E}^{\vee })\subseteq \mathbf{Z}_{{\geqslant}0}^{n}$ . After putting the lexigraphic order on $\mathbf{Z}_{{\geqslant}0}^{n}$ , we get an order ${>}$ on the monomials of  $R_{\unicode[STIX]{x1D70E}}$ .

An element $f\in R_{\unicode[STIX]{x1D70E}}$ may be written as a sum of scalar multiples of distinct monomials. The initial term of  $f$ , denoted $\operatorname{in}_{{>}}f$ , is the greatest term of $f$ with respect to the order  ${>}$ . For an ideal $I\subset R_{\unicode[STIX]{x1D70E}}$ , the initial ideal of $I$ is

$$\begin{eqnarray}\operatorname{in}_{{>}}I=(\operatorname{in}_{{>}}f\mid f\in I).\end{eqnarray}$$

Note that if $I$ is $\mathfrak{m}_{x}$ -primary, then so is $\operatorname{in}_{{>}}I$ . Also, if $\{I_{m}\}_{m\in \mathbf{N}}$ is a graded sequence of ideals of $R_{\unicode[STIX]{x1D70E}}$ , then so is $\{\operatorname{in}_{{>}}I_{m}\}_{m\in \mathbf{N}}$ . This follows from the fact that $\operatorname{in}_{{>}}f\cdot \operatorname{in}_{{>}}g=\operatorname{in}_{{>}}fg$ .

Lemma 8.1. If $I\subset R_{\unicode[STIX]{x1D70E}}$ is an $\mathfrak{m}_{x}$ -primary ideal, then

$$\begin{eqnarray}\operatorname{length}(R_{\unicode[STIX]{x1D70E}}/I)=\operatorname{length}(R_{\unicode[STIX]{x1D70E}}/\text{in}_{{>}}I).\end{eqnarray}$$

Similar to the argument in [Reference EisenbudEis95], we construct a deformation of $I$ to $\operatorname{in}_{{>}}I$ . Since $R_{\unicode[STIX]{x1D70E}}$ is Noetherian, we may choose elements $g_{1},\ldots ,g_{s}\in I$ such that

$$\begin{eqnarray}I=(g_{1},\ldots ,g_{s})\quad \text{and}\quad \operatorname{in}_{{>}}I=(\operatorname{in}_{{>}}g_{1},\ldots ,\operatorname{in}_{{>}}g_{s}).\end{eqnarray}$$

Fix an integral weight $\unicode[STIX]{x1D706}:M\cap \unicode[STIX]{x1D70E}^{\vee }\rightarrow \mathbf{Z}_{{\geqslant}0}$ such that

$$\begin{eqnarray}\operatorname{in}_{{>}_{\unicode[STIX]{x1D706}}}(g_{i})=\operatorname{in}_{{>}}(g_{i})\end{eqnarray}$$

for all $i$ . Note that ${>}_{\unicode[STIX]{x1D706}}$ denotes the order on the monomials induced by the weight function  $\unicode[STIX]{x1D706}$ .

Let $R_{\unicode[STIX]{x1D70E}}[t]$ denote the polynomial ring in one variable over $R_{\unicode[STIX]{x1D70E}}$ . For $g=\sum \unicode[STIX]{x1D6FC}_{m}\unicode[STIX]{x1D712}^{m}$ , we write $b:=\max \{\unicode[STIX]{x1D706}(m)\mid \unicode[STIX]{x1D6FC}_{m}\neq 0\}$ and set

$$\begin{eqnarray}\tilde{g}:=t^{b}\sum \unicode[STIX]{x1D6FC}_{m}t^{-\unicode[STIX]{x1D706}(m)}\unicode[STIX]{x1D712}^{m}.\end{eqnarray}$$

Next, let

$$\begin{eqnarray}\tilde{I} =(\tilde{g}_{1}\ldots \tilde{g}_{s})\subset R_{\unicode[STIX]{x1D70E}}[t].\end{eqnarray}$$

For $c\in k$ , we write $I_{c}$ for the image of $\tilde{I}$ under the map $R_{\unicode[STIX]{x1D70E}}[t]\rightarrow R_{\unicode[STIX]{x1D70E}}$ defined by $t\mapsto c$ . It is clear that $I_{1}=I$ and $I_{0}=\operatorname{in}_{{>}}I$ .

8.2 Proof of Theorem 1.4

Theorem 1.4 is a direct consequence of Proposition 8.3. Our proof of Proposition 8.3 is inspired by the main argument in [Reference MustaţǎMus02].

Proposition 8.3. Let $\mathfrak{a}_{\bullet }$ be a graded sequence of $\mathfrak{m}_{x}$ -primary ideals on $X_{\unicode[STIX]{x1D70E}}$ . We have that

$$\begin{eqnarray}\operatorname{lct}(\operatorname{in}_{{>}}(\mathfrak{a}_{\bullet }))^{n}\operatorname{e}(\operatorname{in}(\mathfrak{a}_{\bullet }))\leqslant \operatorname{lct}(\mathfrak{a}_{\bullet })^{n}\operatorname{e}(\mathfrak{a}_{\bullet }).\end{eqnarray}$$

Proof. We first note that

$$\begin{eqnarray}\operatorname{e}(\operatorname{in}_{{>}}(\mathfrak{a}_{\bullet })):=\limsup _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X_{\unicode[STIX]{x1D70E}},x}/\text{in}_{{>}}(\mathfrak{a}_{m}))}{m^{n}/n!}=\limsup _{m\rightarrow \infty }\frac{\operatorname{length}({\mathcal{O}}_{X_{\unicode[STIX]{x1D70E}},x}/\mathfrak{a}_{m})}{m^{n}/n!}=:\operatorname{e}(\mathfrak{a}_{\bullet }),\end{eqnarray}$$

where the second equality follows from Proposition 8.1. By Proposition 8.2,

$$\begin{eqnarray}\operatorname{lct}(\operatorname{in}_{{>}}(\mathfrak{a}_{\bullet }))\leqslant \operatorname{lct}(\mathfrak{a}_{\bullet }).\end{eqnarray}$$

The result follows. ◻

Proof of Theorem 1.4.

Since $\operatorname{Val}_{X,x}^{\text{toric}}\subset \operatorname{Val}_{X,x}$ , we have

$$\begin{eqnarray}\inf _{v\in \operatorname{Val}_{X,x}}\widehat{\operatorname{vol}}(v)\leqslant \inf _{v\in \operatorname{Val}_{x,x}^{\text{toric}}}\widehat{\operatorname{vol}}(v).\end{eqnarray}$$

We proceed to show the reveres inequality. Note that

(8.1) $$\begin{eqnarray}\inf _{v\in \operatorname{Val}_{X,x}}\widehat{\operatorname{vol}}(v)=\inf _{\mathfrak{a}_{\bullet }\,\mathfrak{m}_{x}\text{-}\text{primary}}\operatorname{lct}(\mathfrak{a}_{\bullet })^{n}\operatorname{e}(\mathfrak{a}_{\bullet })=\inf _{\substack{ \mathfrak{a}_{\bullet }\,\mathfrak{m}_{x}\text{-}\text{primary} \\ \text{monomial}}}\operatorname{lct}(\mathfrak{a}_{\bullet })^{n}\operatorname{e}(\mathfrak{a}_{\bullet }),\end{eqnarray}$$

where the first equality follows from Proposition 4.3 and the second from Proposition 8.3. The last infimum in (8.1) is equal to

$$\begin{eqnarray}\inf _{\substack{ \mathfrak{a}\,\mathfrak{m}_{x}\text{-}\text{primary} \\ \text{monomial}}}\operatorname{lct}(\mathfrak{a})^{n}\operatorname{e}(\mathfrak{a})\end{eqnarray}$$

by Lemma 4.1. Thus, it is sufficient to show that

$$\begin{eqnarray}\inf _{v\in \operatorname{Val}_{X,x}^{\text{toric}}}\widehat{\operatorname{vol}}(v)\leqslant \inf _{\substack{ \mathfrak{a}\,\mathfrak{m}_{x}\text{-}\text{primary} \\ \text{monomial}}}\operatorname{lct}(\mathfrak{a})^{n}\operatorname{e}(\mathfrak{a}).\end{eqnarray}$$

Let $\mathfrak{a}$ be an $\mathfrak{m}_{x}$ -primary monomial ideal. Since $\mathfrak{a}$ is a monomial ideal, there exists a toric valuation $v^{\ast }\in \operatorname{Val}_{X,x}^{\text{toric}}$ such that $v^{\ast }$ computes $\operatorname{lct}(\mathfrak{a})$ . (This follows from the fact that there exists a toric log resolution of  $\mathfrak{a}$ .) By Proposition 4.4,

$$\begin{eqnarray}\widehat{\operatorname{vol}}(v^{\ast })\leqslant \operatorname{lct}(\mathfrak{a})^{n}\operatorname{e}(\mathfrak{a}),\end{eqnarray}$$

and the proof is complete. ◻

Figure 1. The cone $\unicode[STIX]{x1D70E}$ is drawn. The toric variety $X_{\unicode[STIX]{x1D70E}}$ is isomorphic to the cone over $\mathbb{P}^{2}$ blown up at a point.

8.3 An example of nondivisorial volume minimizer

Let $V$ denote $\mathbb{P}^{2}$ blown up at a point. Note that $V$ is a Fano variety. The affine cone $C(V,-K_{V})=\operatorname{Spec}(\bigoplus _{m\geqslant 0}H^{0}(V,-K_{V}))$ is isomorphic to the toric variety $X_{\unicode[STIX]{x1D70E}}$ , where $\unicode[STIX]{x1D70E}\subseteq \mathbf{R}^{3}$ is the cone in Figure 1. Let $x$ denote the torus invariant point of $X_{\unicode[STIX]{x1D70E}}$ .

We seek to find a minimizer of the function $\operatorname{Val}_{X_{\unicode[STIX]{x1D70E}},x}^{\text{toric}}\rightarrow \mathbf{R}_{{>}0}$ defined by $v_{u}\mapsto \widehat{\operatorname{vol}}(v_{u})$ . Since the normalized volume is invariant under scaling, it is sufficient to consider elements $u\in \operatorname{Int}(\unicode[STIX]{x1D70E})$ of the form $u=(a,b,1)\in \operatorname{Int}(\unicode[STIX]{x1D70E})$ . We have

$$\begin{eqnarray}A_{X_{\unicode[STIX]{x1D70E}}}(v_{(a,b,1)})=\langle (a,b,1),(0,0,1)\rangle =1.\end{eqnarray}$$

The normalized volume of $v_{(a,b,c)}$ is

$$\begin{eqnarray}\widehat{\operatorname{vol}}(v_{(a,b,1)}):=A_{X}(v_{(a,b,1)})^{3}\operatorname{vol}(v_{(a,b,1)})=3!\operatorname{Vol}(\unicode[STIX]{x1D70E}^{\vee }\setminus H_{(a,b,1)}(1)).\end{eqnarray}$$

After computing the previous volume, we see that the function is minimized at

$$\begin{eqnarray}(a^{\ast },b^{\ast },1)=(4/3-\sqrt{13}/3,4/3-\sqrt{13}/3,1),\end{eqnarray}$$

with $\widehat{\operatorname{vol}}(v_{(a^{\ast },b^{\ast },1)})=\frac{1}{12}(46+13\sqrt{13})$ . By Theorem 1.4, the toric volume minimizer $v^{\ast }=v_{(a^{\ast },b^{\ast },1)}$ is also a minimizer of $\widehat{\operatorname{vol}}_{X_{\unicode[STIX]{x1D70E}},x}$ . Since $\widehat{\operatorname{vol}}(v_{(a^{\ast },b^{\ast },1)})$ is irrational, Proposition 4.9 implies there cannot be a divisorial minimizer of $\widehat{\operatorname{vol}}_{X_{\unicode[STIX]{x1D70E}},x}$ .