On accumulation points of pseudo-effective thresholds

TL;DR

This paper characterizes accumulation points of pseudo-effective thresholds in algebraic geometry and applies this to advance Fujita's log spectrum conjecture for higher dimensions.

Contribution

It provides a new characterization of accumulation points of pseudo-effective thresholds as invariants of lower-dimensional pairs, linking to Fujita's conjecture.

Findings

Characterization of $k$-th accumulation points as invariants of trivial pairs.

Application of the characterization to Fujita's log spectrum conjecture.

Progress towards understanding the structure of pseudo-effective thresholds.

Abstract

We characterize a $k$-th accumulation point of pseudo-effective thresholds of $n$-dimensional varieties as certain invariant associates to a numerically trivial pair of an $(n-k)$-dimensional variety. This characterization is applied towards Fujita's log spectrum conjecture for large $k$.

Equations

$$\begin{array}{rl}{\mathrm{PET}}_n(I)=\{& \mathrm{pet}(X,\Delta ;M)\mid (X,\Delta )\text{ is lc},\text{ coefficients of }\Delta \text{ are in }I,\\ & M\text{ is a nef and big Cartier divisor},\dim X=n\}.\end{array}$$

$$X\stackrel{f}{\to }{X}^{\prime }\to Z,$$

$$K_X+B+M=f^{\ast }(K_{X^{\prime }}+B^{\prime }+M^{\prime }),$$

$$K_S+B_S+M_S:=(K_X+B+M){|}_S$$

$$K_{S^{\prime }}+B_{S^{\prime }}+M_{S^{\prime }}=(K_{X^{\prime }}+B^{\prime }+M^{\prime }){|}_{S^{\prime }}$$

$$\frac{m-1}{m}+\sum b_i\frac{d_i}{m}+\sum {\mu }_j\frac{e_j}{m},$$

$${\Delta }^{\prime }{\sim }_{\mathbb{R}}{B}^{\prime }+s{A}^{\prime }+M^{\prime }/Z$$

$$\mathrm{pet}(D;M):=\inf \{t\in {\mathbb{R}}_{\ge 0}\mid D+tM\text{ is effective}\}.$$

$$\mathrm{pet}(X,\Delta ;M):=\inf \{t\in {\mathbb{R}}_{\ge 0}\mid K_X+\Delta +tM\text{ is effective}\}.$$

$$\underset{ϵ\to 0}{\lim }\mathrm{pet}(D+ϵB;M)=\mathrm{pet}(D;M).$$

$$D+ϵB\equiv D+\frac{ϵ}{a}(M-E)\equiv D+\frac{ϵ}{a}M-\frac{ϵ}{a}E,$$

$$\mathrm{pet}(D+ϵB;M)\ge \mathrm{pet}(D+\frac{ϵ}{a}M;M)\ge \mathrm{pet}(D;M)-\frac{ϵ}{a}.$$

$$\mathrm{pet}(D+ϵB;M)\le \mathrm{pet}(D-ϵbM;M)\le \mathrm{pet}(D;M)+ϵb.$$

$$\begin{array}{rl}& D(I)=\{\frac{m-1+f}{m}\le 1\mid m\in \mathbb{N},f\in I_{+}\},and\\ & D_c(I)=\{\frac{m-1+f+kc}{m}\le 1\mid m,k\in \mathbb{N},f\in I_{+}\}.\end{array}$$

$$\begin{array}{rl}& D(D(I))=D(I)\cup \{1\},D(D_c(I))\subseteq D_c(I)\cup \{1\},\\ & D_c(D(I))\subseteq D_c(I)\cup \{1\},D_c(D_c(I))\subseteq D_c(I)\cup \{1\}.\end{array}$$

$$D(D_c(I))=D_c(I),D_c(D(I))=D_c(I).$$

$$\begin{array}{rl}{\mathrm{PET}}_n(I)=\{& \mathrm{pet}(X,\Delta ;M)\mid (X,\Delta )\text{ is lc},\Delta \in I,\\ & M\text{ is a nef and big Cartier divisor },\dim X=n\}.\end{array}$$

$$\begin{array}{rl}{\mathfrak{N}}_n(I,c)=& \{(X^{\prime },B^{\prime }+c{M}^{\prime })\mid \dim X=n,\\ & \text{ and }(X^{\prime },B^{\prime }+c{M}^{\prime })\text{ satisfies condition (†)}\}.\end{array}$$

$$\begin{array}{rl}{\mathfrak{K}}_n(I,c)=& \{(X^{\prime },B^{\prime }+c{M}^{\prime })\mid (X^{\prime },B^{\prime }+c{M}^{\prime })\in {\mathfrak{N}}_n(I,c)\\ & \text{ is generalized klt, and }\rho (X^{\prime })=1\}\\ N_n(I)=& \{c\in \mathbb{R}\mid {\mathfrak{N}}_m(I,c)\ne \varnothing ,m\le n\}\\ K_n(I)=& \{c\in \mathbb{R}\mid {\mathfrak{K}}_m(I,c)\ne \varnothing ,m\le n\}.\end{array}$$

$$N_0(I)=K_0(I)=\{\frac{1-f}{m}\ge 0\mid f\in I_{+},m\in \mathbb{N}\}.$$

$$(K_{X^{\prime }}+B^{\prime }+c{M}^{\prime }){|}_{S^{\prime }}=K_{S^{\prime }}+B_{S^{\prime }}+c{M}_{S^{\prime }}$$

$$f^{\ast }{M}^{\prime }=M+E$$

$$E{|}_S\equiv g^{\ast }(M^{\prime }{|}_{S^{\prime }})\not\equiv 0.$$

$$B^{\prime }=p+B^{\prime \prime }\equiv \frac{1}{2}{p}_1+\frac{1}{2}{p}_2+B^{\prime \prime }.$$

$$K_{X^{\prime \prime }}+B^{\prime \prime }+c{M}^{\prime \prime }=f^{\ast }(K_{X^{\prime }}+B^{\prime }+c{M}^{\prime })\equiv 0,$$

$$K_F+B_Y{|}_F+c{M}_Y{|}_F\equiv 0,$$

$$(K_Y+B_Y+c{M}_Y){|}_S=K_S+B_S+c{M}_S\equiv 0.$$

$$K_{W_i}+{\Delta }_{W_i}+T_i=f_i^{\ast }(K_{X_i}+{\Delta }_i),$$

$$K_{X_i}+{\Delta }_i+{ϵ}_i{A}_i{\sim }_{\mathbb{R}}{K}_{X_i}+{\tilde{\Delta }}_i$$

Full text

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Generalized polarized pairs

4. 2.2 Generalized adjunction

5. 2.3 MMP for generalized polarized pairs

6. 2.4 A collection of relevant results

7. 3 Accumulation points of pseudo-effective thresholds

8. 3.1 A characterization of $k$-th accumulation points

9. 3.2 An application

1. Introduction

It has long been realized that the behavior of certain invariants, such as log canonical threshold and minimal log discrepancy, relates to deep results in birational geometry. A celebrated conjecture of Shokurov [Sho88, K*+*92] predicts that the log canonical thresholds satisfy the ascending chain condition (ACC). This conjecture has been extensively studied before it is fully established in [HMX14]. Another problem is the distribution of log canonical thresholds. It is speculated that the accumulation points of $n$-dimensional log canonical thresholds should lie in the set of $(n-1)$-dimensional log canonical thresholds. This is established in [HMX14] under certain restrictions on the coefficients of boundary divisors.

Pseudo-effective threshold is another invariant of this kind. Roughly speaking, a pseudo-effective threshold is a measurement of how far a divisor is away from being effective with respect to a given divisor. Fujita defines the (log) Kodaira energy [Fuj92, Fuj96] which is nothing but the negative of the corresponding pseudo-effective threshold. From the classification perspective, smooth varieties with large pseudo-effective thresholds have been extensively studied.

Pseudo-effective threshold can be analogized to log canonical threshold in many aspects. Fujita proposed the spectrum conjecture on pseudo-effective threshold ([Fuj96, (3.2)]) which is an analogy for ACC conjecture on log canonical thresholds. One version of this conjecture predicts that the set of pseudo-effective thresholds is an ACC set. This conjecture has been confirmed in [Fuj96] for $3$-folds, in [DC16, DC17] for arbitrary varieties, and in [HL17] for more general setting. Fujita’s log spectrum conjecture (Fujita attributed this to Shokurov in [Fuj96, (3.7)]) can be viewed as an analogy to the aforementioned conjecture on the accumulation points of log canonical thresholds. Notice that the term “log spectrum conjecture” in [DC16, HL17] is used in a different context.

In the terminology of this paper, Fujita’s log spectrum conjecture can be stated as the following (the original conjecture is stated in terms of the Kodaira energy). Let $(X,\Delta)$ be a log smooth variety with $\Delta$ a reduced divisor. Let $M$ be an ample Cartier divisor on $X$ and $\operatorname{pet}(X,\Delta;M)\in\mathbb{R}_{\geq 0}$ be the pseudo-effective threshold of $K_{X}+\Delta$ with respect to $M$ (see Section 3). Let $\operatorname{PET}_{n}$ be the set of all such $\operatorname{pet}(X,\Delta;M)$ with $\dim X=n$. For a subset $S\subseteq\mathbb{R}$, let $\lim^{1}S$ be the set of accumulation points of $S$, and let the set of the $k$-th accumulation points be $\lim^{k}S=\lim^{1}(\lim^{k-1}S)$ for any $k\in\mathbb{N}$.

Conjecture 1.1** (Fujita’s log spectrum conjecture [Fuj96, (3.7)]).**

Under the above notation, $\lim^{k}(\operatorname{PET}_{n})\leq n-k$ for any positive integer $k\leq n$ .

This conjecture is widely open and it seems that no complete answer is known even for surfaces. The main result of this paper is to obtain a characterization of the $k$-th accumulation points under reasonable assumptions on the coefficients of the boundary divisors.

Let $I\subseteq[0,1]$ be a subset, then $I$ is called a DCC set if it satisfies the descending chain condition. Let $I_{+}\coloneqq\{\sum n_{i}a_{i}\leq 1\mid a_{i}\in I,n_{i}\in\mathbb{N}\}\cup\{0\}$, and

[TABLE]

Theorem 1.2**.**

Let $I$ be a DCC set such that $I=I_{+}$. Assume that $1\in I$ with $1$ the only possible accumulation point, then for any $1\leq k\leq n$, $\lim^{k}(\operatorname{PET}_{n}(I))\subseteq K_{n-k}(I)$.

The definition of the set $K_{n-k}(I)$ is given in Section 3.1 Equation (6). Roughly speaking, $c\in K_{n-k}(I)$ if there exists an $(n-k)$-dimensional Fano variety $X$ of Picard number $1$, such that there exists a numerically trivial generalized klt pair $K_{X}+B+cM\equiv 0$ with either $M\not\equiv 0$ or $B$ has a coefficient $\frac{m-1+f+kc}{m}$ with $f\in I_{+}$ and $m,k\in\mathbb{N}$. Although the last condition sounds cumbersome, in view of Fujita’s log spectrum conjecture 1.1, it is the most simple case. In fact, as $\frac{m-1+f+kc}{m}<1$, we must have $c<1$. Hence, an accumulation point of such $c$ automatically lies in the range predicted by the conjecture when $k\leq n-1$.

The proof of Theorem 1.2 relies heavily on the minimal model program, especially the resent progress on ACC for log canonical thresholds, generalized polarized pairs and BAB conjecture. Part of the argument follows along the same line as that on accumulation points of log canonical thresholds in [HMX14]. The generalized polarized pairs have the advantage over the standard pairs in dealing with such problems. In fact, suppose that $c$ is the pseudo-effective threshold of $M$ with respect to $K_{X}+\Delta$, then $(X,\Delta+cM)$ may not necessarily be a standard part, but it is a generalized polarized pair. This point of view has been used in [HL17] to deal with ACC for pseudo-effective thresholds.

As an application, we give an upper bound for the $k$-th accumulation points when $k$ is large.

Proposition 1.3**.**

Let $I\subseteq[0,1]$ be a DCC set. Suppose that $1$ is the only possible accumulation point of $I$. Then $\lim^{n-1}(\operatorname{PET}_{n}(I))\leq 2$ and $\lim^{n}(\operatorname{PET}_{n}(I))\leq 1$.

Notice that under the above condition, such upper bounds are sharp. Hence at least for surfaces, we have upper bounds for all terms of pseudo-effective accumulation points.

Finally, upper bounds for all the $k$-th accumulation points are obtained in [Li18] by a modified version of Theorem 1.2 together with other techniques.

The paper is organized as follows. In Section 2, we recall the necessary background for the proof. In Section 3, Theorem 1.2 is proved after a series of reductions. An application towards Fujita’s log spectrum conjecture is given at the end of this section.

Acknowledgements. J. H. thanks his advisor Gang Tian for constant supports and encouragement. Z. L. thanks Chen Jiang for extensive discussions on special cases related to the problem. The first part of the paper was completed in the summer of 2017 while Z. L. stayed at the University of Lübeck and he thanks the quiet environment provided by the Zentrale Hochschulbibliothek Lübeck. Both authors thank Chenyang Xu for constant supports. Both authors also thank the anonymous referee for carefully reading the manuscript and providing constructive suggestions. This work is partially supported by NSFC Grant No.11601015.

2. Preliminaries

We work with complex numbers. Throughout the paper, $\mathbb{Z}$ denotes the set of integers and $\mathbb{N}$ denotes the set of positive integers. We collect relevant definitions and results on generalized polarized pairs developed in [BZ16].

2.1. Generalized polarized pairs

Definition 2.1** (Generalized polarized pair, [BZ16, Definition 1.4]).**

A generalized polarized pair consists of a normal variety $X^{\prime}$ equipped with projective morphisms

[TABLE]

where $f$ is birational and $X$ is normal, an $\mathbb{R}$-boundary $B^{\prime}$, and an $\mathbb{R}$-Cartier divisor $M$ on $X$ which is nef$/Z$ such that $K_{X^{\prime}}+B^{\prime}+M^{\prime}$ is $\mathbb{R}$-Cartier, where $M^{\prime}\coloneqq f_{*}M$. We call $B^{\prime}$ the boundary part and $M$ the nef part.

For simplicity, we sometimes also call $M^{\prime}$ the nef part without referring to $M$. When $Z$ is a closed point, we write $X\xrightarrow{f}X^{\prime}$ instead of $X\xrightarrow{f}X^{\prime}\to\{\rm pt\}$. From the definition, we see that $X$ could be replaced by any log resolution over $X$, and $M$ could be replaced by the pullback of $M$ accordingly. To define the generalized log discrepancy of a divisor $E$ over $X^{\prime}$, considering a high enough model $X$ which contains $E$ (say a resolution as above), if

[TABLE]

then the generalized log discrepancy of $E$ is defined as $1-{\rm mult}_{E}B$ (see [BZ16, Definition 4.1]). We say that $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized lc (resp. generalized klt) if the generalized log discrepancy of any prime divisor is $\geq 0$ (resp. $>0$). Just as in the standard setting, one can define generalized non-klt/lc centers and generalized non-klt/lc places. Besides, as $M$ is a nef divisor, if $M^{\prime}$ is $\mathbb{R}$-Cartier, then by the negativity lemma (see [Sho92, (1.1)] or [BCHM10, Lemma 3.6.2]), $f^{*}M^{\prime}=M+E$ with $E\geq 0$ an exceptional divisor. In particular, this implies that if $K_{X^{\prime}}+B^{\prime}$ is $\mathbb{R}$-Cartier, then the log discrepancy of a divisor $F$ with respect to $(X^{\prime},B^{\prime})$ is no less than the generalized log discrepancy of $F$ with respect to $(X^{\prime},B^{\prime}+M^{\prime})$.

2.2. Generalized adjunction

Generalized adjunction for generalized polarized pairs is defined in [BZ16, Definition 4.7], it will be used in the induction argument to lower the dimensions.

Definition 2.2** (Adjunction for generalized polarized pairs).**

Let $(X^{\prime},B^{\prime}+M^{\prime})$ be a generalized polarized pair with data $X\xrightarrow{f}X^{\prime}\to Z$ and $M$. Assume that $S^{\prime}$ is the normalization of a component of $\lfloor B^{\prime}\rfloor$ and $S$ is its birational transform on $X$. Replacing $X$ we may assume that $f$ is a log resolution of $(X^{\prime},B^{\prime}+M^{\prime})$. Write

[TABLE]

and let

[TABLE]

where $B_{S}=(B-S)|_{S}$ and $M_{S}=M|_{S}$. Let $g$ be the induced morphism $S\to S^{\prime}$ and let $B_{S^{\prime}}=g_{*}B_{S}$ and $M_{S^{\prime}}=g_{*}M_{S}$. Then we get the equality

[TABLE]

which is refered as generalized adjunction.

The morphisms $S\xrightarrow{g}S^{\prime}\to Z$ and $M|_{S}$ give a generalized polarized structure on $(S^{\prime},B_{S^{\prime}}+M_{S^{\prime}})$. The singularities and coefficients behave just as in the standard adjunctions (see [Sho92, K*+*92]). To be precise, when $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized lc, $B_{S^{\prime}}$ is a boundary divisor on $S^{\prime}$ (see [BZ16, Remark 4.8]) and $(S^{\prime},B_{S^{\prime}}+M_{S^{\prime}})$ is still generalized lc. Moreover, suppose that $M=\sum\mu_{j}M_{j}$ with $M_{j}$ nef/$Z$ Cartier divisor for each $j$, and $B^{\prime}=\sum b_{i}B_{i}^{\prime}$ is the prime decomposition. Assume that $M_{j}^{\prime}$ are $\mathbb{Q}$-Cartier divisors, then the coefficient of a divisor $V$ in $B_{S^{\prime}}$ is either $1$ or of the form

[TABLE]

with $m,d_{i},e_{j}\in\mathbb{N}$ (see [BZ16, proof of Proposition 4.9]). In fact, when the coefficient of $V$ is less than $1$, $m$ is the Cartier index along $V$ (see [K*+*92, (16.6.3)]). The term $\frac{m-1}{m}+\sum b_{i}\frac{d_{i}}{m}$ in (2) comes from the adjunction $(K_{X^{\prime}}+B^{\prime})|_{S^{\prime}}$. The $M^{\prime}$ and $M$ contribute to the last term $\sum\mu_{j}\frac{e_{j}}{m}$. Indeed, for $M^{\prime}_{j}=f_{*}M_{j}$, we have $f^{*}M^{\prime}_{j}=M_{j}+E_{j}$ with $E_{j}\geq 0$. $mM^{\prime}_{j}$ is Cartier along the image of $V$, and $M_{j}$ is Cartier. Let $\tilde{V}$ be the strict transform of $V$ in $S\subset X$. Then $mE_{j}=mf^{*}M^{\prime}_{j}-mM_{j}$ is a Cartier divisor along $V$. Thus the coefficient of $\tilde{V}$ in $E_{j}|_{S}$ is of the form $\frac{e_{j}}{m}$ with $e_{j}\in\mathbb{N}$, and so is the coefficient of $V$ in $g_{*}(E_{j}|_{S})$.

2.3. MMP for generalized polarized pairs

Although the the minimal model program (MMP) for generalized polarized pairs is not established in the full generality, some of the most important cases could be derived from the standard MMP. The following results are contained in [BZ16, §4] which are elaborated in [HL18, §3].

Assume that $K_{X^{\prime}}+B^{\prime}+M^{\prime}+A^{\prime}$ is nef/$Z$ for some $\mathbb{R}$-Cartier divisor $A^{\prime}\geq 0$ which is big/$Z$. Moreover, assume that

($\star$) for any $s\in(0,1)$ there exists a boundary

[TABLE]

such that $(X^{\prime},\Delta^{\prime}+(1-s)A^{\prime})$ is klt.

Condition $(\star)$ will be satisfied if $A^{\prime}$ is generalized ample/$Z$ and either

(i) $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized klt, or

(ii) $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized lc and $(X^{\prime},0)$ is klt.

Under the above assumptions, we can run a $(K_{X^{\prime}}+B^{\prime}+M^{\prime})$-MMP with scaling of $A^{\prime}$. Under suitable assumptions, this MMP terminates. This is summarized in the following lemma.

Lemma 2.3** ([BZ16, Lemma 4.4]).**

Let $(X^{\prime},B^{\prime}+M^{\prime})$ be a $\mathbb{Q}$-factorial generalized lc polarized pair with data $X\xrightarrow{f}X^{\prime}\to Z$ and $M$. Assume that $(X^{\prime},B^{\prime}+M^{\prime})$ satisfies (i) or (ii) above. Run an MMP/$Z$ on $(K_{X}^{\prime}+B^{\prime}+M^{\prime})$ with scaling of some general ample/$Z$ $\mathbb{R}$-Cartier divisor $A^{\prime}\geq 0$. Then the following hold:

1. (1)

Assume that $K_{X}^{\prime}+B^{\prime}+M^{\prime}$ is not pseudo-effective/$Z$. Then the MMP terminates with a Mori fibre space. 2. (2)

Assume that

• •

$K_{X^{\prime}}+B^{\prime}+M^{\prime}$* is pseudo-effective/$Z$,*

• •

$(X^{\prime},B^{\prime}+M^{\prime})$* is generalized klt, and that*

• •

$K_{X^{\prime}}+(1+\alpha)B^{\prime}+(1+\beta)M$* is $\mathbb{R}$-Cartier and big/$Z$ for some $\alpha,\beta\geq 0$.*

Then the MMP terminates with a minimal model $X^{\prime\prime}$ and $K_{X^{\prime\prime}}+B^{\prime\prime}+M^{\prime\prime}$ is semi-ample/$Z$, hence it defines a contraction $\phi:X^{\prime\prime}\to T^{\prime\prime}/Z$.

As an application, one can obtain an analogy for dlt modifications. However, we tacitly avoid to introduce the notation of generalized dlt pairs for simplicity.

Proposition 2.4** ([BZ16, Lemma 4.5]).**

Let $(X^{\prime},B^{\prime}+M^{\prime})$ be a generalized lc polarized pair with data $X\xrightarrow{f}X^{\prime}\to Z$ and $M$. Let $S_{1},\ldots,S_{r}$ be prime divisors on birational models of $X^{\prime}$ which are exceptional/$X^{\prime}$ and whose generalized log discrepancies with respect to $(X^{\prime},B^{\prime}+M^{\prime})$ are at most $1$. Then perhaps after replacing $f$ by a high resolution, there exist a $\mathbb{Q}$-factorial generalized lc polarized pair $(X^{\prime\prime},B^{\prime\prime}+M^{\prime\prime})$ with data $X\xrightarrow{g}X^{\prime\prime}\to Z$ and $M$, and a projective birational morphism $\phi:X^{\prime\prime}\to X^{\prime}$ such that

1. (1)

$S_{1},\ldots,S_{r}$* appear as divisors on $X^{\prime\prime}$,* 2. (2)

each exceptional divisor of $\phi$ is one of the $S_{i}$ or is a component of $\lfloor B^{\prime\prime}\rfloor$, 3. (3)

$(X^{\prime\prime},0)$* has klt singularities,* 4. (4)

$K_{X^{\prime\prime}}+B^{\prime\prime}+M^{\prime\prime}=\phi^{*}(K_{X^{\prime}}+B^{\prime}+M^{\prime})$, and 5. (5)

if $\lfloor B^{\prime\prime}\rfloor=0$, then $(X^{\prime\prime},B^{\prime\prime}+M^{\prime\prime})$ is generalized klt.

In particular, the exceptional divisors of $\phi$ are exactly the $S_{i}$ if $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized klt.

Proof.

Each claim except (3) and (5) is explicitly stated in [BZ16, Lemma 4.5]. (3) holds because we obtain $X^{\prime\prime}$ by running an MMP/$X^{\prime}$ with scaling, and in each step of this process, we construct a klt pair. Hence, in each step, the variety itself is klt. In particular, in the last step, $(X^{\prime\prime},0)$ is klt. For (5), when $\lfloor B^{\prime\prime}\rfloor=0$, by the construction of [BZ16, Lemma 4.5], we have $\lfloor B^{\prime}\rfloor=0$, and the generalized log discrepancy of any exceptional divisor in the log resolution $X\to X^{\prime}$ is larger than [math] (otherwise, it would appear in $\lfloor B^{\prime\prime}\rfloor$). The notion of generalized klt singularity is independent of the log resolution, hence the original pair $(X^{\prime},B^{\prime}+M^{\prime})$ is generalized klt. This implies that $(X^{\prime\prime},B^{\prime\prime}+M^{\prime\prime})$ is also generalized klt. ∎

We can extract the divisors which are exactly $S_{1},\ldots,S_{r}$ when $X^{\prime}$ has better singularities.

Proposition 2.5** ([BZ16, Lemma 4.6]).**

Under the notation and assumptions of Proposition 2.4, further assume that $(X^{\prime},C^{\prime})$ is klt for some $C^{\prime}$, and that the generalized log discrepancies of the $S_{i}$ with respect to $(X^{\prime},B^{\prime}+M^{\prime})$ are $<1$. Then we can construct $\phi$ so that in addition it satisfies:

1. (1)

its exceptional divisors are exactly $S_{1},\ldots,S_{r}$, and 2. (2)

if $r=1$ and $X^{\prime}$ is $\mathbb{Q}$-factorial, then $\phi$ is an extremal contraction.

2.4. A collection of relevant results

For reader’s convenience, we collect some relevant results. A set of real numbers is called ACC (ascending chain condition) if there is no strictly increasing infinite sequence. In the same fashion, we define a set to be DCC (descending chain condition). The first two results are called ACC for generalized lc thresholds and global ACC ([BZ16, Theorem 1.5, 1.6]) respectively. They generalize the corresponding results in standard setting ([HMX14]). For the definition of the generalized lc threshold, see [BZ16, Definition 4.3].

Theorem 2.6** (ACC for generalized lc thresholds).**

Let $\Lambda$ be a DCC set of nonnegative real numbers and $n$ a natural number. Suppose that $(X^{\prime},B^{\prime}+M^{\prime})$ is a generalized polarized pair with data $X\xrightarrow{f}X^{\prime}\to Z$ and nef divisor $M$ on $X$. Assume that $D^{\prime}$ on $X^{\prime}$ is an effective $\mathbb{R}$-divisor and that $N$ on $X$ is an $\mathbb{R}$-divisor which is nef$/Z$ and that $D^{\prime}+N^{\prime}$ is $\mathbb{R}$-Cartier with $N^{\prime}=f_{*}N$. Suppose that the following conditions are satisfied.

1. (1)

$(X^{\prime},B^{\prime}+M^{\prime})$* is generalized lc of dimension $n$,* 2. (2)

$M=\sum\mu_{j}M_{j}$* where $M_{j}$ are nef*$/Z$* Cartier divisors and $\mu_{j}\in\Lambda$,* 3. (3)

$N=\sum\nu_{k}N_{k}$* where $N_{k}$ are nef*$/Z$* Cartier divisors and $\nu_{k}\in\Lambda$, and* 4. (4)

the coefficients of $B^{\prime}$ and $D^{\prime}$ belong to $\Lambda$.

Then there is an ACC set $\Theta$ depending only on $\Lambda,n$ such that the generalized lc threshold of $D^{\prime}+N^{\prime}$ with respect to $(X^{\prime},B^{\prime}+M^{\prime})$ belongs to $\Theta$.

Theorem 2.7** (Global ACC).**

Let $\Lambda$ be a DCC set of nonnegative real numbers and $n$ a natural number. Then there is a finite subset $\Lambda^{0}\subseteq\Lambda$ depending only on $\Lambda,d$ such that if $(X^{\prime},B^{\prime}+M^{\prime})$ is a generalized pair with data $X\to X^{\prime}\to Z$ and $M$ satisfying the following conditions,

1. (1)

$(X^{\prime},B^{\prime}+M^{\prime})$* is generalized lc of dimension $n$,* 2. (2)

$Z$* is a point,* 3. (3)

$M=\sum\mu_{j}M_{j}$* where $M_{j}$ are nef Cartier divisors and $\mu_{j}\in\Lambda$,* 4. (4)

$\mu_{j}=0$* if $M_{j}\equiv 0$,* 5. (5)

the coefficients of $B^{\prime}$ belong to $\Lambda$, and 6. (6)

$K_{X^{\prime}}+B^{\prime}+M^{\prime}\equiv 0$,

then the coefficients of $B^{\prime}$ and the $\mu_{j}$ belong to $\Lambda^{0}$.

The next result is [Bir16, Theorem 1.1] which is known as Borisov-Alexeev-Borisov conjecture (BAB conjecture).

Theorem 2.8** (BAB conjecture).**

Let $n$ be a natural number and $\epsilon$ a positive real number. Then the projective varieties $X$ such that

1. (1)

$(X,B)$* is $\epsilon$-lc of dimension $n$ for some boundary $B$, and* 2. (2)

$-(K_{X}+B)$* is nef and big,*

form a bounded family

Finally, for completeness, we mention the ACC for pseudo-effective thresholds (see (3)) of generalized polarized pairs. This will not be used in the rest of the paper and it can be obtained by the same argument as [HL17, Theorem 1.4].

Theorem 2.9**.**

Let $\Lambda$ be a DCC set of nonnegative real numbers and $n$ a natural number. Let $(X^{\prime},B^{\prime}+N^{\prime})$ be a generalized polarized pair satisfying the following conditions.

1. (1)

$\dim X=n$, coefficients of $B^{\prime}$ and $N^{\prime}$ are in $\Lambda$, 2. (2)

$(X^{\prime},B^{\prime}+N^{\prime})$* is generalized lc such that $B^{\prime}$ is the boundary part and $N^{\prime}$ the nef part, and* 3. (3)

$M^{\prime}$* is a nef and big Cartier divisor.*

Then the set of pseudo-effective thresholds of $M^{\prime}$ with respect to $K_{X^{\prime}}+B^{\prime}+N^{\prime}$ is an ACC set.

3. Accumulation points of pseudo-effective thresholds

3.1. A characterization of $k$-th accumulation points

Let $M$ be a big $\mathbb{Q}$-Cartier divisor. Define the pseudo-effective threshold of $M$ with respect to $D$ to be

[TABLE]

For a log pair $(X,\Delta)$, set the pseudo-effective threshold of $M$ with respect to $K_{X}+\Delta$ to be

[TABLE]

The following lemma gives extra flexibility on singularities when working with the accumulation points of pseudo-effective thresholds.

Lemma 3.1**.**

Let $D,B$ be $\mathbb{Q}$-Cartier divisors, and $M$ be a big $\mathbb{Q}$-Cartier divisor. Then

[TABLE]

Proof.

By considering $-B$, we can assume that $\epsilon\to 0^{+}$. Because $M$ is big, there is an $a\in\mathbb{R}_{>0}$ such that $M-aB\equiv E\geq 0$. By

[TABLE]

we have

[TABLE]

On the other hand, because there exists $b>0$ such that $bM+B\equiv F>0$,

[TABLE]

By taking $\epsilon\to 0^{+}$, we get the desired result. ∎

We use the following notation and conventions. Assume that $I\subseteq[0,1]$, and $c\in\mathbb{R}_{\geq 0}$. Recall that $I_{+}=\{\sum n_{i}a_{i}\leq 1\mid a_{i}\in I,n_{i}\in\mathbb{N}\}\cup\{0\}$. Set

[TABLE]

By direct computations (for example, see [MP04, Lemma 4.4]), we have

[TABLE]

Moreover, when $c>0$,

[TABLE]

For a divisor $\Delta$, we write $\Delta\in I$ if the coefficients of $\Delta$ lie in $I$. We are interested in the set of pseudo-effective thresholds

[TABLE]

Notice that the $\operatorname{PET}_{n}$ in Fujita’s log spectrum conjecture 1.1 is contained in $\operatorname{PET}_{n}(\{1\})$.

Suppose that $X\to X^{\prime}$ is a generalized polarized pair with $M^{\prime}$ the nef part ($M^{\prime}$ may not be effective) and $f_{*}M=M^{\prime}$. For $c\in\mathbb{R}_{\geq 0}$, a generalized polarized pair $(X^{\prime},B^{\prime}+cM^{\prime})$ is said to satisfy condition ($\dagger$) if the following conditions hold:

($\dagger$)

1. (1)

$(X^{\prime},B^{\prime}+cM^{\prime})$ is generalized lc, 2. (2)

$K_{X^{\prime}}+B^{\prime}+cM^{\prime}\equiv 0$, and $B^{\prime}\in D(I)\cup D_{c}(I)$, 3. (3)

$M^{\prime}=f_{*}M$ with $M$ a nef Cartier divisor, and $M^{\prime}$ a $\mathbb{Q}$-Cartier divisor, 4. (4)

if $M\equiv 0$, then at least one coefficient of $B^{\prime}$ lies in $D_{c}(I)$.

The following sets will be considered in the sequel. First, for $c\in\mathbb{R}_{\geq 0}$, let

[TABLE]

Notice that when $M\equiv 0$ (or equivalently $M^{\prime}\equiv 0$), this generalized polarized pair belongs to $\mathfrak{N}_{n}(I,c)$ defined in [HMX14] Page 559. It is crucial to require $M$ to be Cartier. This property will be preserved under the generalized MMP and all the actions performed below. For $n\in\mathbb{N}$, set

[TABLE]

Notice that in $N_{n}(I)$ and $K_{n}(I)$, we also consider varieties of dimensions less than $n$. Besides, set

[TABLE]

Notice that $N_{0}(I)\subseteq K_{1}(I)\subseteq K_{n}(I)$. In fact, by $\frac{1}{2}\in D(I)$, for any $f\in I_{+},f\leq 1$, let $X^{\prime}=\mathbb{P}^{1},B^{\prime}=\frac{1}{2}(p_{1}+p_{2})+fq$ and $M^{\prime}=mz$, where $p_{1},p_{2},q,z\in X^{\prime}$ are distinct closed points. Then $K_{X^{\prime}}+B^{\prime}+cM^{\prime}\equiv 0$ for $c=\frac{1-f}{m}$.

Lemma 3.2**.**

Let $I\subseteq[0,1],c\in\mathbb{R}_{\geq 0}$ and $(X^{\prime},B^{\prime}+cM^{\prime})$ be a generalized lc pair with data $f:X\to X^{\prime}$ and $cM$. Suppose that $B^{\prime},M\in I$, $M^{\prime}$ is $\mathbb{Q}$-Cartier, and $K_{X^{\prime}}+B^{\prime}+cM^{\prime}\equiv 0$. Let $S^{\prime}$ be the normalization of an irreducible component of $\lfloor B^{\prime}\rfloor$. Let $S$ be the strict transform of $S^{\prime}$ on $X$, and

[TABLE]

be the generalized adjunction. If $M^{\prime}|_{S^{\prime}}\not\equiv 0$, but $M|_{S}\equiv 0$, then $B_{S^{\prime}}\in D(I)\cup D_{c}(I)$, and at least one coefficient is $\frac{m-1+f+kc}{m}$ with $m,k\in\mathbb{N},f\in I_{+}$. Moreover, in this case, $(S^{\prime},B_{S^{\prime}})$ is lc and $K_{S^{\prime}}+B_{S^{\prime}}\equiv 0$.

Proof.

By the generalized adjunction formula, we have $B_{S^{\prime}}\in D(I)\cup D_{c}(I)$, and $(S^{\prime},B_{S^{\prime}})$ is lc because $({S^{\prime}},B_{S^{\prime}}+cM_{S^{\prime}})$ is generalized lc. Moreover, as $M|_{S}\equiv 0$, $M_{S^{\prime}}\equiv 0$, we have $K_{S^{\prime}}+B_{S^{\prime}}\equiv 0$. We claim that there exists a coefficient $\frac{m-1+f+kc}{m}$ of $B_{S^{\prime}}$ with $k>0$.

By the negativity lemma, we have

[TABLE]

with $E$ an effective exceptional divisor. Let $g$ be the morphism $S\to S^{\prime}$, then $(f^{*}M^{\prime})|_{S}=g^{*}(M^{\prime}|_{S^{\prime}})$. Hence $g^{*}(M^{\prime}|_{S^{\prime}})=(f^{*}M^{\prime})|_{S}=M|_{S}+E|_{S}$. As $M^{\prime}|_{S^{\prime}}\not\equiv 0$ and $M|_{S}\equiv 0$,

[TABLE]

Thus $g_{*}(E|_{S})\equiv M^{\prime}|_{S^{\prime}}\not\equiv 0$. Because $B_{S^{\prime}}=g_{*}((B-S)|_{S}+cE|_{S})$, we show the claim (see the last paragraph of Section 2.2). ∎

The following lemma is in the same spirit of [HMX14, Lemma 11.4].

Lemma 3.3**.**

Suppose that $1\in I$, then $N_{n}(I)=K_{n}(I)$.

Proof.

The “$\supseteq$” is by definition, we only need to show the inclusion “$\subseteq$”. We do induction on $n$.

When $n=1$, for $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{N}_{1}(I,c)$, $X^{\prime}$ is an elliptic curve or $\mathbb{P}^{1}$. In the former case, $B^{\prime}=0,M^{\prime}\equiv 0$, $(X^{\prime},B^{\prime}+cM^{\prime})$ is generalized klt, and $\rho(X^{\prime})=1$. Thus $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{K}_{1}(I,c)$ and $c\in K_{1}(I)$. When $X^{\prime}=\mathbb{P}^{1}$, for $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{N}_{1}(I,c)$, if $(X^{\prime},B^{\prime}+cM^{\prime})$ is generalized lc but not generalized klt, then at least one coefficient of $B^{\prime}$ is $1$. By $0\in I_{+}$, $\frac{1}{2}\in D(I)$. If $p_{1},p_{2},p_{3},p_{4},q\in X^{\prime}$ are distinct closed points, then for any $c\in\mathbb{R}_{\geq 0}$, $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{K}_{1}(I,c)$ for $B^{\prime}=\frac{1}{2}(p_{1}+p_{2}+p_{3}+p_{4})$ and $M^{\prime}=cq$. In particular, $c=0\in K_{1}(I)$. We can assume that $c>0$.

First, we consider the case $M\not\equiv 0$. Write $B^{\prime}=p+B^{\prime\prime}$, where $p$ has coefficient $1$. Choose $p_{1},p_{2}\in X^{\prime}$ to be two closed points different from $\operatorname{Supp}B^{\prime\prime}$, then

[TABLE]

We can continue this process if there exists a component of $B^{\prime\prime}$ whose coefficient is $1$. By doing this, we get $\tilde{B}^{\prime}\in D(I)\cup D_{c}(I)$ such that $B^{\prime}\equiv\tilde{B}^{\prime}$ and $(X^{\prime},\tilde{B}^{\prime}+cM^{\prime})$ is generalized klt. Thus $c\in K_{1}(I)$.

Next, we consider the case $M\equiv 0$. Then $B^{\prime}$ has a coefficient $\frac{m-1+f+kc}{m}$ with $k\in\mathbb{N}$. If $\frac{m-1+f+kc}{m}<1$, then we can do the same thing as (9) and obtain a $\tilde{B}^{\prime}$ so that $(X^{\prime},\tilde{B}^{\prime}+cM^{\prime})$ is generalized klt with $\frac{m-1+f+kc}{m}$ to be a coefficient of $\tilde{B}^{\prime}$. Thus $c\in K_{1}(I)$. If $\frac{m-1+f+kc}{m}=1$, then $c=\frac{1-f}{k}>0$. Consider $B^{\prime}=\frac{1}{2}p_{1}+\frac{1}{2}p_{2}+fp_{3}$ and $M^{\prime}=kq$, then $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{K}_{1}(I,c)$ for $c=\frac{1-f}{k}$. Thus $c=\frac{1-f}{k}\in K_{1}(I)$. This completes the $n=1$ case.

From now on, assume that the inclusion holds for any dimension less than $n$.

Let $(X^{\prime},B^{\prime}+cM^{\prime})\in\mathfrak{N}_{n}(I,c)$. If $M^{\prime}\equiv 0$, then this is just [HMX14, Lemma 11.4], hence we can assume that $M^{\prime}\not\equiv 0$.

By Proposition 2.4, there exists a generalized lc pair $(X^{\prime\prime},B^{\prime\prime}+cM^{\prime\prime})$ such that

[TABLE]

with $(X^{\prime\prime},0)$ has klt singularities, hence it satisfies property (ii) in §2.3. Let $T^{\prime\prime}=\lfloor B^{\prime\prime}\rfloor$. If $T^{\prime\prime}=0$, then $(X^{\prime\prime},B^{\prime\prime}+cM^{\prime\prime})$ is generalized klt by Proposition 2.4 (5). Moreover, $M^{\prime\prime}$ can be obtained as a push-forward of a Cartier divisor from a common resolution. Thus $(X^{\prime\prime},B^{\prime\prime}+cM^{\prime\prime})\in\mathfrak{N}_{n}(I,c)$.

By Lemma 2.3 (or just [BCHM10]), we can run a $(K_{X^{\prime\prime}}+B^{\prime\prime})$-MMP with scaling, which is the same as a $(-cM^{\prime\prime})$-MMP. As $-cM^{\prime\prime}$ is not pseudo-effective, we can obtain a Mori fibre space $Y\to Z$, and $-M^{\prime\prime}$ is not contracted in this MMP. Let $(Y,B_{Y}+cM_{Y})$ be the log pair obtained above. There are three possibilities.

Case (1). When $T^{\prime\prime}=0$, then $(Y,B_{Y}+cM_{Y})$ is a generalized pair with generalized klt singularities. Taking a general fibre $F$ and restricting to $F$, we have

[TABLE]

with $M_{Y}|_{F}\not\equiv 0$ (because $M_{Y}$ is positively intersects with curves contracted to $Z$). $(F,B_{Y}|_{F}+cM_{Y}|_{F})$ is still generalized klt with $M_{Y}|_{F}$ a push-forward of a Cartier divisor. When $\dim Z>0$, then $\dim F<\dim X$, and we get the result by induction hypothesis. Otherwise, $F=Y$ has Picard number $1$, and thus $(F,B_{Y}|_{F}+cM_{Y}|_{F})\in\mathfrak{K}_{n}(I)$. Hence $c\in K_{n}(I)$.

Case (2). When $T^{\prime\prime}\neq 0$, and $T^{\prime\prime}$ is not contracted in $X^{\prime\prime}\to Y$. We do the same thing as Case (1) when $\dim Z>0$, and obtain $(F,B_{Y}|_{F}+cM_{Y}|_{F})\in\mathfrak{N}_{\dim F}(I)$, hence the result holds by induction hypothesis. When $\dim Z=0$, then $\rho(Y)=1$, and as $M_{Y}\not\equiv 0$, $M_{Y}|_{S_{Y}}\not\equiv 0$, where $S_{Y}$ is an irreducible component of the push-forward of $T^{\prime\prime}$. Hence we do the generalized adjunction on the normalization $S$ of $S_{Y}$,

[TABLE]

By generalized adjunction and (4), $B_{S}\in D(D(I)\cup D_{c}(I))\subseteq D(I)\cup D_{c}(I)$. Either $M_{S}\not\equiv 0$, or $M_{S}\equiv 0$ but $M_{Y}|_{S}\neq 0$. In the later case, by Lemma 3.2, we have at least one component of $B_{S}$ has coefficient in $D_{c}(D(I)\cup D_{c}(I))\subseteq D_{c}(I)$. Hence, for both cases, we are done by induction hypothesis.

Case (3). When $T^{\prime\prime}$ is contracted in some step of the above MMP. Without loss of generality, we can assume that a component $S^{\prime\prime}$ is contracted in $X^{\prime\prime}\to Y^{\prime}$. Thus, by the negativity lemma, $S^{\prime\prime}$ is a covering family of curves $\{C\}$ such that $S^{\prime\prime}\cdot C<0$. But $M^{\prime\prime}\cdot C>0$ as we run $(-cM^{\prime\prime})$-MMP, and thus $M^{\prime\prime}|_{S^{\prime\prime}}\not\equiv 0$. Hence as Case (2), we do adjunction on the normalization of $S^{\prime\prime}$ and complete the argument by induction hypothesis. ∎

Proposition 3.4**.**

Assume $1\in I$, then $\lim^{1}\operatorname{PET}_{n}(I)\subseteq\lim^{1}N_{n}(I)$.

Proof.

We proof the claim by induction on $n$. For $n=1$, $\operatorname{PET}_{1}(I)\subseteq N_{1}(I)$ by definition, hence the claim holds. We can assume that the claim holds for any dimension less than $n$.

Step 1. Let $c_{i}=\operatorname{pet}(X_{i},\Delta_{i};M_{i})$. Suppose that $\{c_{i}\}_{i\in\mathbb{N}}$ has an accumulation point $c$. We can assume that $c_{i}>0$. Taking a dlt modification $f_{i}$ of $(X_{i},\Delta_{i})$

[TABLE]

where $T_{i}$ is a reduced divisor, $W_{i}$ is $\mathbb{Q}$-factorial and $({W_{i}},\Delta_{W_{i}}+T_{i})$ is dlt. Moreover, $c_{i}$ is the same as the pseudo-effective threshold of $f_{i}^{*}M_{i}$ with respect to $K_{W_{i}}+\Delta_{W_{i}}+T_{i}$. Replacing by $({W_{i}},\Delta_{W_{i}}+T_{i})$, we can assume that $X_{i}$ is $\mathbb{Q}$-factorial and $(X_{i},\Delta_{i})$ is dlt.

Step 2. Choose an ample divisor $A_{i}$ and $1\gg\epsilon_{i}>0$ such that $(X_{i},\Delta_{i}+\epsilon_{i}A_{i})$ is lc. Then there exists a $\tilde{\Delta}_{i}$ satisfying

[TABLE]

such that $(X_{i},\tilde{\Delta}_{i})$ is klt. By Lemma 3.1, for any $1\gg\delta_{i}>0$, we can find a sufficiently small $\epsilon_{i}>0$, such that

[TABLE]

Because $M_{i}$ is nef, $(X_{i},\tilde{\Delta}_{i}+tM_{i})$ is generalized klt for any $t\geq 0$ with data $X_{i}\xrightarrow{\rm id}X_{i}$ and $M_{i}$. In the same way, $(X_{i},\Delta_{i}+\epsilon A_{i}+tM_{i})$ is generalized lc. By Lemma 2.3 (2), we can run a $(K_{X_{i}}+\tilde{\Delta}_{i}+c^{\prime}_{i}M_{i})$-MMP, $X_{i}\dashrightarrow Y_{i}$. As $K_{X_{i}}+\tilde{\Delta}_{i}+c^{\prime}_{i}M_{i}$ is pseudo-effective, $Y_{i}$ is birational to $X_{i}$. Moreover, $K_{Y_{i}}+\tilde{\Delta}_{Y_{i}}+c^{\prime}_{i}M_{Y_{i}}$ is nef but not big as $c^{\prime}_{i}$ is the pseudo-effective threshold. Then by Lemma 2.3 (2) again, $K_{Y_{i}}+\tilde{\Delta}_{Y_{i}}+c^{\prime}_{i}M_{Y_{i}}$ is semi-ample, and thus it defines a fibration $g_{i}:Y_{i}\to Z_{i}$. Taking a general fibre $F_{i}$ (if $Z_{i}$ is a point, then $F_{i}=Y_{i}$), and restricting to $F_{i}$, we get

[TABLE]

Notice that because $M_{i}$ is big (by the definition of $\operatorname{PET}_{n}(I)$), $M_{Y_{i}}|_{F_{i}}$ is big. Moreover, as $(X_{i},\Delta_{i}+\epsilon A_{i}+c^{\prime}_{i}M_{i})$ is generalized lc, $(Y_{i},\Delta_{Y_{i}}+\epsilon A_{Y_{i}}+c^{\prime}_{i}M_{Y_{i}})$ is also generalized lc, where $\Delta_{Y_{i}},A_{Y_{i}}$ are push-forwards of $\Delta_{i},A_{i}$. In fact, the MMP above is also an MMP on $K_{X_{i}}+\Delta_{i}+\epsilon A_{i}+c^{\prime}_{i}M_{i}$. The restriction to a general fibre preserves the generalized lc property. Moreover, as $K_{Y_{i}}+\Delta_{Y_{i}}+c_{i}M_{Y_{i}}$ is the push-forward of $K_{X_{i}}+\Delta_{i}+c_{i}M_{i}$, it is pseudo-effective and so is $K_{F_{i}}+\Delta_{Y_{i}}|_{F_{i}}+c_{i}M_{Y_{i}}|_{F_{i}}$. By (10),

[TABLE]

In particular, $\lim_{i\to\infty}\tau_{i}=\lim_{i\to\infty}c_{i}$. Moreover, $(F_{i},\Delta_{Y_{i}}|_{F_{i}}+c_{i}^{\prime}M_{Y_{i}}|_{F_{i}})$ is generalized lc.

Step 3. First suppose that $\dim F_{i}<\dim Y_{i}$. If $(F_{i},\Delta_{Y_{i}}|_{F_{i}}+\tau_{i}M_{Y_{i}}|_{F_{i}})$ is generalized lc, then the result follows from the induction hypothesis. If $(F_{i},\Delta_{Y_{i}}|_{F_{i}}+\tau_{i}M_{Y_{i}}|_{F_{i}})$ is not generalized lc, let $\tau_{i}^{\prime}$ be the generalized log canonical threshold of $M_{Y_{i}}|_{F_{i}}$ with respect to $(F_{i},\Delta_{Y_{i}}|_{F_{i}})$, then $\tau_{i}>\tau_{i}^{\prime}\geq c_{i}^{\prime}$ ($\tau_{i}^{\prime}\geq c_{i}^{\prime}$ because $(F_{i},\Delta_{Y_{i}}|_{F_{i}}+c_{i}^{\prime}M_{Y_{i}}|_{F_{i}})$ is generalized lc). By Proposition 2.5, and notice that $(F_{i},0)$ is klt, we can extract a generalized lc place $S_{i}$ such that

[TABLE]

where $\Delta_{F^{\prime\prime}_{i}}$ is the strict transform of $\Delta_{Y_{i}}|_{F_{i}}$. Moreover, the relative Picard number $\rho(F_{i}^{\prime\prime}/F_{i}^{\prime})=1$ and $S_{i}$ is anti-ample/$F_{i}^{\prime}$. Suppose that $V_{i}$ is the generalized lc center of $S_{i}$ with $\tilde{V}_{i}$ its normalization. Let $S_{i}\to\tilde{V}_{i}$ be the corresponding morphism, and $\Upxi_{i}$ be a general fibre. Then we have

[TABLE]

For $(\Upxi_{i},\Delta_{\Upxi_{i}}+\tau_{i}^{\prime}M_{\Upxi_{i}})$, it could happen that $M_{\Upxi_{i}}\equiv 0$. But if this is the case, then at least one component of $\Delta_{\Upxi_{i}}$ lies in $D_{\tau^{\prime}_{i}}(I)$. The verification of the claim is identical to that in Step 5 of Proposition 3.5 (from (14) to the end, especially Case (b1)), and thus we leave the details to that argument. Now $({\Upxi_{i}},\Delta_{\Upxi_{i}}+\tau_{i}^{\prime}M_{\Upxi_{i}})$ is a generalized lc pair satisfying condition ($\dagger$) such that $\lim_{i\to\infty}\tau_{i}^{\prime}=c$. Then by the induction hypothesis, we are done.

Step 4. Now, suppose that $F_{i}=Y_{i}$, then $\rho(Y_{i})=1$, and $K_{Y_{i}}+\Delta_{Y_{i}}+\tau_{i}M_{Y_{i}}\equiv 0$. If $({Y_{i}},\Delta_{Y_{i}}+\tau_{i}M_{Y_{i}})$ is generalized lc, then $\tau_{i}\in N_{n}(I)$. Otherwise, let $\tau_{i}^{\prime}$ be the generalized lc threshold of $M_{Y_{i}}$ with respect to $(Y_{i},\Delta_{Y_{i}})$. Again, $\tau_{i}>\tau_{i}^{\prime}\geq c_{i}^{\prime}$. Then the same argument as Step 3 gives the desired result. ∎

Proposition 3.5**.**

Let $I$ be a DCC set such that $I=I_{+}$. Assume that $1\in I$ with $1$ the only possible accumulation point. Let $n\in\mathbb{N}$ be a fixed integer, and $\{c_{i}\}_{i\in\mathbb{N}}$ be a strictly decreasing sequence with limit $c,c>0$. Suppose that $(X^{\prime}_{i},\Delta^{\prime}_{i})$ satisfies the following conditions.

1. (1)

$\dim X_{i}^{\prime}=n$, 2. (2)

$K_{X^{\prime}_{i}}+\Delta^{\prime}_{i}\equiv 0$* with $\Delta^{\prime}_{i}=A^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i}$,* 3. (3)

$(X^{\prime}_{i},A^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$* is generalized lc with $A^{\prime}_{i}+B^{\prime}_{i}$ the boundary part and $c_{i}M^{\prime}_{i}$ the nef part, $M_{i}^{\prime}$ is a $\mathbb{Q}$-Cartier divisor,* 4. (4)

the coefficients of $A^{\prime}_{i}$ are either [math] or approaching $1$, 5. (5)

$B^{\prime}_{i}\in D(I)\cup D_{c_{i}}(I)$, and 6. (6)

$M^{\prime}_{i}$* is a push-forward of a nef Cartier divisor $M_{i}$, such that either $M_{i}^{\prime}\not\equiv 0$, or when $M_{i}^{\prime}\equiv 0$, at least one component of $B^{\prime}_{i}$ lies in $D_{c_{i}}(I)$.*

Then $c\in N_{n-1}(I)$.

Proof.

We do induction on $n$.

First, for the base case $n=1$, by $c>0$, we have $X_{i}^{\prime}=\mathbb{P}^{1}$. Let $a_{i}=\deg A_{i}^{\prime},l_{i}=\deg M_{i}^{\prime}$ and $\frac{m_{i}^{(j)}-1+f_{i}^{(j)}+k_{i}^{(j)}c_{i}}{m_{i}^{(j)}}$ be a coefficient of $B_{i}^{\prime}$, then

[TABLE]

Notice that $\lim_{i\to+\infty}a_{i}=0,1$ by assumption (4) and $c>0$, $l_{i}\in\mathbb{Z}_{\geq 0}$ and if $l_{i}=0$, then there exists $k_{i}^{(j)}>0$ by assumption (6). By $c>0$, $k_{i}^{(j)},l_{i}$ are bounded above, and thus by passing to a subsequence, we can assume that they are equal to $k$ and $l$ respectively. Because $c$ is an accumulation point, by passing to a subsequence, we can assume that $c_{i}$ are distinct. Hence if $\lim_{i\to+\infty}a_{i}=0$ (i.e. $\deg A_{i}=0$), then $m_{i}^{(j)}$ or $f_{i}^{(j)}$ does not belong to a finite set. Otherwise, $c_{i}$ also belongs to a finite set. Because $1$ is the only possible accumulation point of $I=I_{+}$, if $f_{i}^{(j)}$ does not belong to a finite set, then by passing to a subsequence, $\lim_{i\to+\infty}f_{i}^{(j)}=1$. Thus one coefficient $\frac{m_{i}^{(j)}-1+f_{i}^{(j)}+k_{i}^{(j)}c_{i}}{m_{i}^{(j)}}$ of $B_{i}^{\prime}$ has limit $1$. Taking $i\to+\infty$, we have

[TABLE]

where $f\in I_{+}=I$, $m\in\mathbb{N}$, $k,l\in\mathbb{Z}_{\geq 0}$ and at least one of $k,l$ is non-zero. Thus $c=\frac{1}{l}$ or $\frac{1-f}{k+ml}$, which belongs to $N_{0}(I)$ (see (7)). This shows $n=1$ case. From now on, we assume that the result holds for any dimension less than $n$.

Next, the problem can be simplified by the following. The $M_{i}^{\prime}\equiv 0$ case is just [HMX14, Proposition 11.7]. In fact, $M_{i}=f_{i}^{*}M_{i}^{\prime}-E\equiv-E$ with $E\geq 0$ (see (8)). If $E>0$, then $M_{i}$ cannot be nef. Thus $M_{i}\equiv 0$. Besides, according to (6), at least a component of $B_{i}^{\prime}$ lies in $D_{c_{i}}(I)$, by $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}\leq 1$, we have $c_{i}\leq 1$. Notice that the $B_{i}+C_{i}$ in [HMX14, Proposition 11.7] is $B_{i}$ in our case. Hence, we assume $M_{i}^{\prime}\not\equiv 0$ in the following.

We can assume that $A^{\prime}_{i},B^{\prime}_{i}$ do not have common components. Indeed, if there is a component of $B^{\prime}_{i}$ which approaching $1$, then we add it to $A^{\prime}_{i}$. Besides, we can assume that in the coefficients $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}$ of $B^{\prime}_{i}$, $m_{i}$ are bounded. Because $1$ is the only possible accumulation point of $I=I_{+}$, if $f_{i}<1$ is strictly increasing with limit $f$, then $f=1$, and thus $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}$ approaching $1$. Hence, by passing to a subsequence, we can assume that those $f_{i}$ appearing in the coefficients of $B_{i}^{\prime}$ are chosen from a finite set.

Step 1. In this step, we reduce the problem to $\mathbb{Q}$-factorial Fano and Picard number $1$ case.

By Proposition 2.4, there exists $f:X_{i}^{\prime\prime}\to X_{i}^{\prime}$ such that

[TABLE]

and $(X_{i}^{\prime\prime},A_{i}^{\prime\prime}+B_{i}^{\prime\prime}+c_{i}M_{i}^{\prime\prime})$ is still generalized lc, $X_{i}^{\prime\prime}$ is $\mathbb{Q}$-factorial, and $(X_{i}^{\prime\prime},0)$ is klt with $A_{i}^{\prime\prime},B_{i}^{\prime\prime}\in I$. We can run a $(K_{X_{i}^{\prime\prime}}+A_{i}^{\prime\prime}+B_{i}^{\prime\prime})$-MMP by Lemma 2.3 (1), $X_{i}\dashrightarrow Y_{i}$, which is the same as a $(-M_{i}^{\prime\prime})$-MMP and thus we can assume that there exists a Mori fibre space $Y_{i}\to Z_{i}$. When $\dim Z_{i}>0$ for infinitely many $i$, we take a general fibre and restricting everything to this fibre. Then we complete the proof by induction hypothesis. Otherwise, we can assume $\dim Z_{i}=0$ for each $i$, and thus $\rho(Y_{i})=1$. Replacing $(X^{\prime}_{i},\Delta^{\prime}_{i})$ by $(Y_{i},{f_{i}}_{*}\Delta^{\prime}_{i})$, we can assume that $\rho(X^{\prime}_{i})=1$. In particular, we can assume that $X^{\prime}_{i}$ is a $\mathbb{Q}$-factorial Fano variety and $M^{\prime}_{i}$ is big.

Step 2. In this step, we show that there is no $\epsilon>0$ such that $(X^{\prime}_{i},\Delta^{\prime}_{i})$ is generalized $\epsilon$-lc for each $i$.

Suppose that there exists $\epsilon>0$ such that for infinitely many $i$, the total generalized log discrepancy of $(X^{\prime}_{i},\Delta^{\prime}_{i})$ is greater than $\epsilon$. As the total generalized log discrepancy of $(X^{\prime}_{i},A^{\prime}_{i}+B^{\prime}_{i})$ is no less than that of $(X^{\prime}_{i},\Delta^{\prime}_{i})$, $(X^{\prime}_{i},A^{\prime}_{i}+B^{\prime}_{i})$ is $\epsilon$-lc. By Theorem 2.8, such $X^{\prime}_{i}$ forms a bounded family. Moreover, as coefficient of $A^{\prime}_{i}$ are either [math] or approaching $1$, by passing to a tail, we can assume $A^{\prime}_{i}=0$ (otherwise the total generalized log discrepancies will $<\epsilon$). The coefficients of $B^{\prime}_{i}$ are of the form $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}$ with $f_{i}\in I_{+}$, by passing to a subsequence, we can assume that $m_{i}$ is fixed. As above, $f_{i}$ is chosen from a finite set. By considering a very ample divisor on the bounded family and by passing to a subsequence again, we see that $(K_{X^{\prime}_{i}}+\Delta^{\prime}_{i})\cdot H^{n-1}=0$. Moreover, $K_{X^{\prime}_{i}}\cdot H^{n-1},B^{\prime}_{i}\cdot H^{n-1}$ are bounded. $M^{\prime}_{i}\cdot H^{n-1}$ is also bounded because $\lim c_{i}=c\neq 0$ and $M^{\prime}_{i}$ is a prime divisor as it is a push-forward of a Cartier divisor. Notice that if $D\subseteq X_{i}^{\prime}$ is a Weil divisor and $H$ is a very ample divisor, then $D\cdot H^{n-1}\in\mathbb{Z}$. For $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}$, those $m_{i},k_{i},f_{i}$ are bounded and thus are fixed by passing to a subsequence. This contradicts to the strictly decreasing of $c_{i}$ with limit $c\neq 0$. Thus we can assume that for any $\epsilon>0$, there exists $(X^{\prime}_{i},\Delta^{\prime}_{i})$ whose total generalized log discrepancy is less than $\epsilon$.

Step 3. In this step, we show that $A_{i}^{\prime}$ can be assumed to be non-zero.

If $A^{\prime}_{i}=0$, then by Proposition 2.4, the definition of generalized log discrepancy and the convention before Step 1, there exists $g_{i}:Y_{i}\to X_{i}^{\prime}$ such that

[TABLE]

where $E_{i}$ is a reduced divisor (possibly be [math]), and $1\geq a_{i}\geq 1-\epsilon$. By putting $A_{i}=E_{i}+a_{i}T_{i}$, we can assume that $A_{i}\neq 0$. Moreover, when $\lfloor A_{i}\rfloor=0$, $({Y_{i}},B_{Y_{i}}+E_{i}+a_{i}T_{i}+c_{i}M_{Y_{i}})$ is generalized klt. Replacing $(X_{i}^{\prime},\Delta_{i}^{\prime})$ by $(Y_{i},B_{Y_{i}}+E_{i}+a_{i}T_{i}+c_{i}M_{Y_{i}})$, we can assume that $A_{i}\neq 0$, and when $\lfloor A_{i}\rfloor=0$, $(X^{\prime}_{i},\Delta^{\prime}_{i})$ is generalized klt. Notice that in this case, $\rho(X_{i}^{\prime})$ may not be $1$, but $M_{i}^{\prime}$ is still big.

Run a $(K_{X^{\prime}_{i}}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$-MMP, $f_{i}:X^{\prime}_{i}\dashrightarrow Y_{i}$, and as $K_{X^{\prime}_{i}}+B^{\prime}_{i}+c_{i}M^{\prime}_{i}\equiv-A^{\prime}_{i}$ is not pseudo-effective, we can assume that the MMP terminates with a Mori fibre space $Y_{i}\to Z_{i}$. Let $F_{i}$ be a general fibre, because $M^{\prime}_{i}$ is big, ${f_{i}}_{*}M^{\prime}_{i},{f_{i}}_{*}M^{\prime}_{i}|_{F_{i}}\neq 0$. If $\dim Z_{i}>0$, then we are done. Thus we can assume that $\dim Z_{i}=0$, and thus $\rho(Y_{i})=1$. Replacing by $Y_{i},{f_{i}}_{*}A^{\prime}_{i},{f_{i}}_{*}B^{\prime}_{i},{f_{i}}_{*}M^{\prime}_{i}$ , we can assume that $\rho(X_{i}^{\prime})=1$. Moreover, $A^{\prime}_{i}$ is not contracted by $f_{i}$ because we run a $(-A^{\prime}_{i})$-MMP.

Step 4. In this step, we show the claim when $\lfloor A^{\prime}_{i}\rfloor\neq 0$.

Suppose $\lfloor A^{\prime}_{i}\rfloor\neq 0$, let $S^{\prime}_{i}$ be the normalization of an irreducible component of $\lfloor A^{\prime}_{i}\rfloor$. Because $\rho(X^{\prime}_{i})=1$, $M^{\prime}_{i}|_{S_{i}^{\prime}}\not\equiv 0$. By generalized adjunction (see Definition 2.2)

[TABLE]

To be precise, suppose that $h_{i}:X_{i}\to X_{i}^{\prime}$ is a log resolution, then

[TABLE]

with $A_{i}$ the strict transform of $A_{i}^{\prime}$. Let $g_{i}$ denote the restriction of $h_{i}$ to $S_{i}\to S_{i}^{\prime}$, then

[TABLE]

Hence, $M_{S^{\prime}_{i}}$ is the push-forward of the nef and Cartier divisor $M_{i}|_{S_{i}}$. $A_{S^{\prime}_{i}}$ is either [math] or approaching $1$. $B_{S_{i}^{\prime}}\in D(I)\cup D_{c_{i}}(I)$, and by Lemma 3.2, when $M_{i}|_{S_{i}}\equiv 0$, at least one coefficients of $B_{S^{\prime}_{i}}$ is in $D_{c_{i}}(I)$. The $(S_{i}^{\prime},A_{S^{\prime}_{i}}+B_{S^{\prime}_{i}}+c_{i}M_{S^{\prime}_{i}})$ is still generalized lc, but $M_{S^{\prime}_{i}}$ may not necessarily be $\mathbb{Q}$-Cartier. By Proposition 2.4, there exists a $\mathbb{Q}$-factorial variety $\tilde{S}_{i}$ with a birational morphism $\pi_{i}:\tilde{S}_{i}\to S_{i}^{\prime}$ such that

[TABLE]

The generalized polarized pair $(\tilde{S}_{i},A_{\tilde{S}_{i}}+B_{\tilde{S}_{i}}+c_{i}M_{\tilde{S}_{i}})$ satisfies all the assumptions of the proposition with $\dim\tilde{S}_{i}=n-1$. Hence by the induction hypothesis, we have $c\in N_{n-2}(I)\subseteq N_{n-1}(I)$.

Step 5. In this step, we deal with $\lfloor A^{\prime}_{i}\rfloor=0$ case. We create a reduced divisor on $X_{i}^{\prime}$ and discuss some of its properties.

If $\lfloor A^{\prime}_{i}\rfloor=0$, then $(X^{\prime}_{i},\Delta^{\prime}_{i})$ is generalized klt by Step 3. As $c<c_{i}$, $({X^{\prime}_{i}},A^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i})$ is also generalized klt. Let $T^{\prime}_{i}$ be the reduced divisor which is the support of $A^{\prime}_{i}$. We claim that by passing to a subsequence, $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i})$ is still generalized lc. Otherwise, let $t_{i}$ be the generalized lc threshold of $T^{\prime}_{i}$ with respect to $(X^{\prime}_{i},B^{\prime}_{i}+cM^{\prime}_{i})$. Then $t_{i}<1$. But $t_{i}$ is greater or equal to the minimal coefficients of $A_{i}$. As the coefficients of $A_{i}$ is approaching $1$, $t_{i}$ is approaching $1$. This contradicts to the ACC for generalized lc thresholds (Theorem 2.6). By passing to a subsequence, we can assume that $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i})$ is generalized lc for each $i$.

Now we consider the generalized polarized pair $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$. Until the end of this step, we show that when $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is not generalized lc for infinite $i$, the proposition holds. The remaining case when $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is generalized lc will be discussed in Step 5.

Because $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is not generalized lc, $M_{i}^{\prime}\not\equiv 0$, and let

[TABLE]

Thus $c\leq c_{i}^{\prime}<c_{i}$. Moreover, there exists a generalized lc center $V_{i}$ which is strictly contained in $M^{\prime}_{i}$. By this we mean the following. Suppose that $h_{i}:X_{i}\to X_{i}^{\prime}$ is a log resolution, then

[TABLE]

with $T_{i}$ the strict transform of $T_{i}^{\prime}$. When $t=c_{i}^{\prime}$, there exists at least one exceptional divisor whose coefficient in $B_{i}(c_{i}^{\prime})$ is one and it is a generalized lc place over $V_{i}$. In particular, they cannot be components of $M_{i}^{\prime}$. Besides, in $h_{i}^{*}M_{i}^{\prime}=M_{i}+E_{i}$, we must have $E_{i}>0$. In fact, $E_{i}\geq 0$ by the negativity lemma, and if $E_{i}=0$, then $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is already generalized lc. By the negativity lemma again, there exists a component $S_{i}$ of $E_{i}$ which is a covering family of curves $\{C\}$, such that ${h_{i}}_{*}C=0$ and $E_{i}\cdot C<0$. Thus $M_{i}\cdot C>0$ when $\operatorname{Supp}E_{i}=S_{i}$.

By Proposition 2.5, and notice that $(X^{\prime},0)$ is klt, we can just extract $S_{i}$. That is,

[TABLE]

such that the push-forward of $S_{i}$ on $X_{i}^{\prime\prime}$, $S^{\prime\prime}_{i}$, is a component of $\lfloor B_{i}^{\prime\prime}\rfloor$. Moreover, the relative Picard number $\rho(X_{i}^{\prime\prime}/X_{i}^{\prime})=1$ and $S^{\prime\prime}_{i}$ is anti-ample/$X_{i}^{\prime}$. On the other hand, we can assume that $X_{i}\xrightarrow{g_{i}}X_{i}^{\prime\prime}\xrightarrow{\phi_{i}}X_{i}^{\prime}$ and $h_{i}=\phi_{i}\circ g_{i}$. Thus $M_{i}^{\prime\prime}={g_{i}}_{*}M_{i}$ satisfies

[TABLE]

for some $a_{i}>0$. We see that $M^{\prime\prime}_{i}$ is ample/$X_{i}^{\prime}$. In particular, for a general fibre $F^{\prime\prime}_{i}$ of $S^{\prime\prime}_{i}\to V_{i}$, $M^{\prime\prime}_{i}|_{F^{\prime\prime}_{i}}$ is ample.

We have

[TABLE]

By generalized adjunction, we have

[TABLE]

where $M_{S_{i}^{\prime\prime}}$ is the push-forward of $M_{i}|_{S_{i}}$. To remedy the problem that $M_{S_{i}^{\prime\prime}}$ may not be $\mathbb{Q}$-Cartier, as Step 4, we can pass to a $\mathbb{Q}$-factorial model $\tilde{S}_{i}$ over $S_{i}^{\prime\prime}$ and take the corresponding general fibre. Hence, without loss of generality, we can assume that $M_{S_{i}^{\prime\prime}}$ is $\mathbb{Q}$-Cartier. After doing this, $M^{\prime\prime}_{i}|_{F^{\prime\prime}_{i}}$ is nef and big. Let $K_{F^{\prime\prime}_{i}},T_{F^{\prime\prime}_{i}},B_{F^{\prime\prime}_{i}},M_{F^{\prime\prime}_{i}}$ be the restrictions of $K_{S^{\prime\prime}_{i}},T_{S^{\prime\prime}_{i}},B_{S^{\prime\prime}_{i}},M_{S^{\prime\prime}_{i}}$ to $F_{i}^{\prime\prime}$ respectively. Then by (14), we have

[TABLE]

We need a detailed analysis on $B_{F^{\prime\prime}_{i}}$ and $M_{F^{\prime\prime}_{i}}$. Recall that $g_{i}:X\to X_{i}^{\prime\prime}$, and we let $q_{i}:S_{i}\to S_{i}^{\prime\prime},\psi_{i}:F_{i}\to F_{i}^{\prime\prime}$ be the corresponding restrictions of $g_{i}$, where $F_{i}$ is the preimage of the general fibre $F_{i}^{\prime\prime}$.

[TABLE]

First, we claim that

[TABLE]

In fact, by definition ${q_{i}}_{*}(M_{i}|_{S_{i}})=M_{S_{i}^{\prime\prime}}$, thus $q_{i}^{*}(M_{S_{i}^{\prime\prime}})=M_{i}|_{S_{i}}+\Theta_{i}$, where $\Theta_{i}$ is $q_{i}$-exceptional. We have

[TABLE]

where $\Theta_{i}|_{F_{i}}$ is $\psi_{i}$-exceptional, and hence the claim.

Let $g_{i}^{*}M_{i}^{\prime\prime}=M_{i}+R_{i}$ with $R_{i}\geq 0$ a $g_{i}$-exceptional divisor. Then

[TABLE]

(a). When $R_{i}|_{S_{i}}$ is an exceptional divisor for $q_{i}:S_{i}\to S_{i}^{\prime\prime}$, then $R_{i}|_{F_{i}}=(R_{i}|_{S_{i}})|_{F_{i}}$ is $\psi_{i}$-exceptional as $F_{i}$ is a general fibre. Restricting (16) to $F_{i}$, we have

[TABLE]

As $M_{i}^{\prime\prime}|_{F_{i}^{\prime\prime}}$ is big, by (15), $M_{F^{\prime\prime}_{i}}={\psi_{i}}_{*}(M_{i}|_{F_{i}})$ is big.

(b). When $R_{i}|_{S_{i}}$ is not an exceptional divisor for $q_{i}:S_{i}\to S_{i}^{\prime\prime}$, write

[TABLE]

where $E_{S_{i}}$ is the summation of exceptional divisors in $R_{i}|_{S_{i}}$ and $T_{S_{i}}$ is the summation of the non-exceptional divisors in $R_{i}|_{S_{i}}$. In particular, we have $T_{S_{i}}\neq 0$. Then, there are two cases to consider:

(b1). Suppose that $T_{S_{i}}$ is a horizontal divisor over $V_{i}$, that is, ${\rm Supp}(T_{S_{i}})$ maps surjectively to $V_{i}$. Now as $T_{S_{i}}$ is not an exceptional divisor for $q_{i}:S_{i}\to S_{i}^{\prime\prime}$, there are summands of $B_{S_{i}^{\prime\prime}}$ whose coefficients are of the form $\frac{m-1+f+kc_{i}^{\prime}}{m}$ with $k\in\mathbb{N}$ (see discussion after equation (2)). Moreover, as $T_{S_{i}}$ is a horizontal divisor, there is a component of ${q_{i}}_{*}(T_{S_{i}})$ whose restriction to $F^{\prime\prime}_{i}$ is non-zero. Thus, at least one coefficients of $B_{F_{i}^{\prime\prime}}$ is in $D_{c_{i}^{\prime}}(I)$.

(b2). Suppose that $T_{S_{i}}$ is a vertical divisor over $V_{i}$, that is, the image of ${\rm Supp}(T_{S_{i}})$ is not equal to $V_{i}$. Then $(T_{S_{i}})|_{F_{i}}=0$. Because $E_{S_{i}}$ is an exceptional divisor for $q_{i}$, $(E_{S_{i}})|_{F_{i}}$ is $\psi_{i}$-exceptional. Put them together, we have $\psi_{i*}(R_{i}|_{S_{i}})|_{F_{i}}=0$. By (17) and the bigness of $M_{i}^{\prime\prime}|_{F_{i}^{\prime\prime}}$, $M_{F^{\prime\prime}_{i}}={\psi_{i}}_{*}(M_{i}|_{F_{i}})$ is big.

In summary, the above shows that: either $M_{F^{\prime\prime}_{i}}$ is big (in particular, non-zero), or at least one coefficient of $B_{F_{i}^{\prime\prime}}$ is in $D_{c_{i}^{\prime}}(I)$. The generalized polarized pair $({F^{\prime\prime}_{i}},T_{F^{\prime\prime}_{i}}+B_{F^{\prime\prime}_{i}}+c_{i}^{\prime}M_{F^{\prime\prime}_{i}})$ satisfies all the assumptions of Proposition 3.5 with $\dim F_{i}^{\prime\prime}<n$ except that $\{c_{i}^{\prime}\}_{i\in\mathbb{N}}$ may not be strictly decreasing.

Now, if $c=c_{i}^{\prime}$ for some $i$, then

[TABLE]

Because $T_{F^{\prime\prime}_{i}}$ is a reduced divisor and $1\in I$, we have

[TABLE]

thus $c\in N_{\dim F^{\prime\prime}_{i}}(I)\subseteq N_{n-1}(I)$.

We can assume that $c\neq c_{i}^{\prime}$ for infinitely many $i$. By passing to a subsequence, $c_{i}^{\prime}$ is strictly decreasing to $c$, and we obtain the result by applying the induction hypothesis to $({F^{\prime\prime}_{i}},T_{F^{\prime\prime}_{i}}+B_{F^{\prime\prime}_{i}}+c_{i}^{\prime}M_{F^{\prime\prime}_{i}})$.

Step 6. In this step, we show the remaining case of Step 5 when $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is generalized lc (see the third paragraph of Step 5) and finish the proof of the whole proposition.

Suppose that $({X^{\prime}_{i}},T^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i})$ is generalized lc for infinite $i$, and $\rho(X_{i}^{\prime})=1$ (see the end of Step 3).

When $K_{X^{\prime}_{i}}+T^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i}\equiv 0$, then let $S^{\prime}_{i}$ be the normalization of an irreducible component of $T^{\prime}_{i}$. Because $\rho(X^{\prime}_{i})=1$, ${M_{i}^{\prime}}|_{S^{\prime}_{i}}\not\equiv 0$ (recall that we assume $M_{i}^{\prime}\not\equiv 0$ at the beginning of the proof). By the generalized adjunction on $S^{\prime}_{i}$, we have

[TABLE]

After possibly passing to a $\mathbb{Q}$-factorial model as before, $(S_{i}^{\prime},\Theta_{S_{i}^{\prime}}+cM_{S_{i}^{\prime}})\in\mathfrak{N}_{n-1}(I,c)$. Thus $c\in N_{n-1}(I)$.

When $K_{X^{\prime}_{i}}+T^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i}$ is anti-ample, then there exists $c_{i}^{\prime\prime}$ such that $K_{X^{\prime}_{i}}+T^{\prime}_{i}+B^{\prime}_{i}+c_{i}^{\prime\prime}M^{\prime}_{i}\equiv 0$. By $K_{X^{\prime}_{i}}+A^{\prime}_{i}+B^{\prime}_{i}+c_{i}M^{\prime}_{i}\equiv 0$ and $T_{i}^{\prime}\geq A_{i}^{\prime}$, we see that $c<c_{i}^{\prime\prime}\leq c_{i}$. Hence, by passing to a subsequence, $\{c_{i}^{\prime\prime}\}_{i\in\mathbb{N}}$ is strictly decreases to $c$. Then by the generalized adjunction on $S_{i}^{\prime}$, we obtain the result by the induction hypothesis just as the case $\lfloor A_{i}\rfloor\neq 0$ in Step 4.

Finally, suppose that $K_{X^{\prime}_{i}}+T^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i}$ is ample. Because $c<c_{i}$, $K_{X^{\prime}_{i}}+A^{\prime}_{i}+B^{\prime}_{i}+cM^{\prime}_{i}$ is anti-ample, and thus $T_{i}^{\prime}\neq A_{i}^{\prime}$. Since $\rho(X_{i}^{\prime})$=1, there exists $r_{i}\in(0,1)$, such that

[TABLE]

Because the coefficients of $A_{i}$ is approaching $1$ and at least one coefficient is not $1$ (because $T_{i}^{\prime}\neq A_{i}^{\prime}$), the coefficients of $r_{i}T^{\prime}_{i}+(1-r_{i})A_{i}^{\prime}$ is approaching $1$ and not all of them equal to $1$. Moreover, $({X^{\prime}_{i}},r_{i}T^{\prime}_{i}+(1-r_{i})A_{i}^{\prime}+B^{\prime}_{i}+cM^{\prime}_{i})$ is generalized lc, and by passing to a subsequence, we can assume that the coefficients of $r_{i}T^{\prime}_{i}+(1-r_{i})A_{i}^{\prime}+B^{\prime}_{i}$ lie in a DCC set. But the coefficients must lie in a finite set by the global ACC of generalized polarized pairs (Theorem 2.7). This is a contradiction. ∎

Lemma 3.6**.**

Suppose that $I\subseteq[0,1]$ is a DCC set, then $N_{n}(I)$ is an ACC set.

Proof.

Suppose that there exists a strictly increasing sequence $\{c_{i}\mid c_{i}\in N_{n}(I)\}_{i\in\mathbb{N}}$. We claim that the set

[TABLE]

is a DCC set. Otherwise, we can assume that $\{\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}\}_{i\in\mathbb{N}}$ is a strictly decreasing sequence. By passing to a subsequence, we can assume that $\{m_{i}\}_{i\in\mathbb{N}},\{f_{i}\}_{i\in\mathbb{N}}$ and $\{k_{i}\}_{i\in\mathbb{N}}$ are non-decreasing sequences as $\mathbb{N},\mathbb{Z}_{\geq 0}$ and $I_{+}$ are all DCC sets. We can also assume that $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}<1$, and thus $\{m_{i}\}_{i\in\mathbb{N}}$ is bounded. By passing to a subsequence again, we can assume that $\{m_{i}\}_{i\in\mathbb{N}}$ is a constant sequence and thus $\{f_{i}+k_{i}c_{i}\}_{i\in\mathbb{N}}$ is strictly decreasing. But this leads to a contradiction as $\{f_{i}\}_{i\in\mathbb{N}}$ is non-decreasing and $\{c_{i}\}_{i\in\mathbb{N}}$ is strictly increasing (notice that $k_{i}$ could be [math]).

The set of varieties in $\cup_{i\in\mathbb{N}}\mathfrak{N}(I,{c_{i}})$ have coefficients in a DCC set, and thus by global ACC (Theorem 2.7), they must lie in a finite set. This contradicts to the strictly increasing assumption on $\{c_{i}\}_{i\in\mathbb{N}}$. In fact, either there are infinitely $i$ such that $M^{\prime}_{i}\not\equiv 0$ and we are done by $K_{X^{\prime}_{i}}+B^{\prime}_{i}+c_{i}M^{\prime}_{i}\equiv 0$, or $M^{\prime}_{i}\equiv 0$, with $\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}},k_{i}>0$ as a coefficient of $B^{\prime}_{i}$ . In the later case, $\{\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}\}_{i\in\mathbb{N}}$ is an infinite set. Indeed, by passing to a subsequence, we can assume that $\{f_{i}+k_{i}c_{i}\}_{i\in\mathbb{N}}$ is strictly increasing. In particular, they are not $1$ by passing to a tail. Then no matter $\{m_{i}\}_{i\in\mathbb{N}}$ is bounded or unbounded, $\{\frac{m_{i}-1+f_{i}+k_{i}c_{i}}{m_{i}}\}_{i\in\mathbb{N}}$ is an infinite set. ∎

Corollary 3.7**.**

Let $I$ be a DCC set such that $I=I_{+}$. Assume that $1\in I$ with $1$ the only possible accumulation point, then for any $1\leq k\leq n$, $\lim^{k}N_{n}(I)\subseteq N_{n-k}(I)$.

Proof.

By induction, it is enough to show that $\lim^{1}N_{n}(I)\subseteq N_{n-1}(I)$. By Lemma 3.6, if $\{c_{i}\}_{i\in\mathbb{N}}$ has an accumulation point $c$, then there is a strictly increasing sequence converging to $c$. Moreover, as $0\in N_{n-1}(I)$, we can assume that $c>0$. Then the claim follows from Proposition 3.5 by taking $A_{i}=0$. ∎

Proof of Theorem 1.2.

This follows from Proposition 3.4, Corollary 3.7 and Lemma 3.3. ∎

3.2. An application

We demonstrate an application of Theorem 1.2 towards Fujita’s spectrum conjecture for large $k$. In practice, as long as $I$ is DCC with $1$ to be the only possible accumulation point, we can always enlarge $I$ so that it satisfies all the assumptions in Theorem 1.2. Notice that this could only enlarge the accumulation points.

Lemma 3.8**.**

If $I\subseteq[0,1]$ is a DCC set with $1$ to be the only possible accumulation point, then $J\coloneqq(I\cup\{1\})_{+}$ is a DCC set such that $J_{+}=J$ with $1\in J$ to be the only possible accumulation point.

Proof.

If $I$ is DCC, then $J$ is DCC and $J_{+}=J$ by definition. It is enough to show that $1$ is also the only possible accumulation point of $J$. Otherwise, there exists a sequence $\{c_{+}^{i}\}_{i\in\mathbb{N}}$ of $J$ approaching $c_{+}<1$. Each $c_{+}^{i}=\sum_{j}^{n_{i}}a_{ij}$, where $a_{ij}\in I$ (repetition is allowed). We claim that $n_{i}$ is bounded above. Otherwise there exists a subsequence sequence $\{a_{k_{i}j_{k_{i}}}\}$ decreasing to [math] which contradicts to that $I$ is DCC. By passing to a subsequence, we can assume that $n_{i}=n$ is a fixed number. For each $c_{+}^{i}$, there is an $n$-tuple $(a_{i1},\ldots,a_{in})$ (the order does not matter). By passing to a subsequence again, we can assume that for each $k$, $\{a_{ik}\}_{i\in\mathbb{N}}$ is an increasing sequence. Hence there exists $\lim_{i}a_{ik}=a_{k}$, and $0<a_{k}<1$. This is a contradiction. ∎

Proof of Proposition 1.3.

By Lemma 3.8, replacing $I$ by $(I\cup\{1\})_{+}$, we can assume that $I$ satisfies all the assumptions of Theorem 1.2.

For $k=n-1$, $\lim^{n-1}(\operatorname{PET}_{n}(I))\subseteq K_{1}(I)$ by Theorem 1.2. It suffices to give an upper bound for $K_{1}(I)$. Let $c\in K_{1}(I)$, by definition in Section 3, there exists a smooth curve $X^{\prime}$, and $B^{\prime},M^{\prime}$ such that $K_{X^{\prime}}+B^{\prime}+cM^{\prime}\equiv 0$. There are two cases to consider. If $M^{\prime}\equiv 0$, then some coefficient of $B^{\prime}$ lies in $D_{c}(I)$. This coefficient is of the form

[TABLE]

By generalized klt assumption, all the coefficients of $B^{\prime}$ are less than $1$, hence $kc<1$, and thus $c<1$. If $M^{\prime}\not\equiv 0$, then $M^{\prime}$ is an ample Cartier divisor, hence $c\leq 2$.

For $k=n$, by Theorem 1.2 and (7),

[TABLE]

Thus $c\leq 1$ for any $c\in\lim^{n}(\operatorname{PET}_{n}(I))$. ∎

Proposition 1.3 gives upper bounds for the first and the second accumulation points of surfaces and they are sharp under our conditions.

Remark 3.9**.**

Corollary 3.7 shows that a $k$-th accumulation point $c$ lies in $N_{n-k}(I)$. When $n-k>1$, a similar argument as Proposition 1.3 has the following difficulty: when $M^{\prime}\not\equiv 0$, $M^{\prime}$ is known to be an ample Weil divisor which may not necessarily be Cartier. Thus, there is no upper bound for $c$. If one works with $M$ instead of $M^{\prime}$, then one gains the nef and Cartier property but loses the bigness. However, we are able to overcome such difficulty when $M$ is assumed to be ample and Cartier in $\operatorname{PET}_{n}(I)$ (1) (see [Li18]).