Proof. By [Reference Xu37, Lemma 1], we can construct a plt blow-up $\pi \colon Y \rightarrow X$ of $(X,\,\Delta )$ over $x$. Let $\pi _Y \colon W\rightarrow Y$ be a log resolution of $(Y,\,\Delta _Y+E)$, so we can write

\[ K_W+\Delta_W+F_W = \pi_Y^{*}(K_Y+\Delta_Y+E), \]

where $\Delta _W$ is the strict transform of $\Delta _Y$ on $W$, and $F_W$ is an exceptional divisor. We can take $\epsilon$ small enough so that

\[ \textrm{coeff}_D(\Delta_W + F_W - E_W ) < 1-\epsilon \]

for all exceptional prime divisors $D$, where $E_W$ is the strict transform of $E$ on $W$. By [Reference Kollár and Mori25, Lemma 2.29], we conclude that every exceptional divisor over $W$ has log discrepancy either zero or greater than $\epsilon$ with respect to the purely log terminal pair $(Y,\,\Delta _Y+E)$.

Proof. By Lemma 2.7, we can extract an $\epsilon$-plt blow-up over $x\in X$ for some real number $\epsilon$. By [Reference Birkar7, Theorem 1.8] there exists a strong $n$-complement $K_Y+\Delta _Y+\Gamma _Y+E$ for $K_Y+\Delta _Y+E$ over $x$, for $n$ only depending on $d$ and $\mathcal {S}$. Since $\mathcal {S}$ is a fixed set then $n$ only depends on $d$. Since $K_Y+\Delta _Y+\Gamma _Y+E$ is $\mathbb {Q}$-linearly trivial over $x\in X$ then we can write

\[ K_Y+\Delta_Y+\Gamma_Y+E = \pi^{*}(K_X+\Delta+\Gamma) \]

for some boundary $\Gamma$ on $X$ so that $(X,\,\Delta +\Gamma )$ is a log canonical pair. By the exceptionality of $(X,\,\Delta )$ at $x$ we deduce that $(X,\,\Delta +\Gamma )$ has a unique log canonical place and hence $(Y,\,\Delta _Y+\Gamma _Y+E)$ has a unique log canonical place $E$. Moreover, since $n(K_Y+\Delta _Y+\Gamma _Y+E)$ is Cartier over $x$, we conclude that

\[ a_F( Y, \Delta_Y+E) +1 \geq a_F(Y, \Delta_Y+\Gamma_Y +E) + 1 \geq \frac{1}{n} \]

for every divisor $F$ over $Y$ which is not equal to $E$. Thus, it suffices to take $\epsilon _d=\frac {1}{n}$.

Proof. Let $\mathcal {X}\rightarrow T$ be the bounding family. By Notherian induction, we can stratify the base $T$ into finitely many locally closed subvarieties $T_i\subset T$, satisfying the following: For each fibre over $t\in T_i$ of $\mathcal {X}_i\rightarrow T_i$ we have that $K_{\mathcal {X}_i}|_{\mathcal {X}_t}=K_{\mathcal {X}_t}$. We may assume that the $T_i$'s are disjoint by successively replacing $T_i,\,T_j$ with $T_i\setminus \operatorname {supp}(T_i\cap T_j),\, T_j\setminus \operatorname {supp}(T_i\cap T_j)$ and $T_i\cap T_j$. For each $i$, we can consider a very ample effective Cartier divisor $\mathcal {A}_i$ on $\mathcal {X}_i$. For each fibre, we define $\mathcal {A}_{\mathcal {X}_t}:=\mathcal {A}_i|_{\mathcal {X}_t}$. Then, we have that $\mathcal {A}_{\mathcal {X}_t}^{d}(-K_{\mathcal {X}_t}) = \mathcal {A}^{d-1}(-K_{\mathcal {X}})|_{\mathcal {X}_t}$ for each $t\in T_i$. Thus, the intersection $\mathcal {A}_{\mathcal {X}_t}^{d}(-K_{\mathcal {X}_t})$ only takes finitely many values over $T_i$. Analogously, $\mathcal {A}_{\mathcal {X}_t}^{d}$ only takes finitely many values over $T_i$. We define $C$ to be the maximum of all these intersection numbers among all the $T_i$'s. Then, for each $X\in \mathcal {P}$ it suffices to consider $A$ to be the pull-back of $\mathcal {A}_{\mathcal {X}_t}$ to $X$ via the isomorphism $X\rightarrow \mathcal {X}_t$.

Proof. By Lemma 2.17, we can find a positive constant $C$ so that for each $X\in \mathcal {P}$ there is a very ample Cartier divisor $A$ with $A^{d}\leq C$ and $A^{d-1}(-K_X) \leq C$. The set $\textrm {coeff}(\Delta )$ satisfies the descending chain condition, so there exists $\delta >0$ small enough so that $\delta < \textrm {coeff}(\Delta )$ for every boundary $\Delta$ of a pair $(X,\,\Delta )$ on $\mathcal {Q}$. Since $-(K_X+\Delta )$ is pseudo-effective we have that $A^{d-1}\cdot (-K_X-\Delta ) \geq 0$, hence we conclude that

\[ A^{d-1}\cdot \left( {\delta} \Delta_\textrm{red} \right) \leq A^{d-1}\cdot \Delta \leq A^{d-1} \cdot \left({-}K_X \right) \leq C, \]

where $\Delta _\textrm {red}$ is $\Delta$ with the reduced structure. Thus, we get that $A^{d-1}\cdot \Delta _\textrm {red}\leq \delta ^{-1} C$, so by [Reference Alexeev2, Lemma 3.7.(2)], we conclude that the set of pairs $\mathcal {Q}$ is log bounded.

Proof. Note that the dimension of the varieties of $\mathcal {P}$ is bounded. Let $(\mathcal {X},\,\mathcal {B})\rightarrow T$ be a log bounding family for $\mathcal {P}$. We can stratify the base $T$ into finitely many locally closed subvarieties $T_i\subset T$, satisfying the following: For each fibre over $t\in T_i$ of $\mathcal {X}_i\rightarrow T_i$, we have that $(K_{\mathcal {X}_i}+\mathcal {B}_i)|_{\mathcal {X}_t}=K_{\mathcal {X}_t}+\mathcal {B}_t$. In particular, we conclude that there exists a constant $c$, only depending on $\mathcal {P}$, so that $c(K_X+\Delta )$ is Cartier for every $(X,\,\Delta )\in \mathcal {P}$. Then, applying Kollár's effective base point freeness [Reference Kollár21, Theorem 1.1] with $a=1$ and $L=-c(K_X+\Delta )$, we conclude that there exists a constant $m$, only depending on $\mathcal {P}$, so that $|-m(K_X+\Delta )|$ is base point free.

Proof. Since ampleness is open in families, we may restrict to an open set $U\subset T$ so that $-(K_{\mathcal {X}_t} +\mathcal {B}_t)$ is ample for every $t\in U$. Furthermore, by Noetherian induction, we may assume that the induced morphism $\mathcal {X}_U \rightarrow U$ is a projective smooth morphism of relative dimension $d$ with normal fibres. In this case, we have that

\[ \operatorname{vol}( -(K_{\mathcal{X}_t}+\mathcal{B}_t)) = ( -(K_{\mathcal{X}_t}+\mathcal{B}_t))^{d} \]

is an upper-semicontinuous function on $t\in U$. Hence, it takes finitely many values on $U$.

Proof. By Lemma 2.19, we can find a positive natural number $m$, only depending on $\mathcal {P}$, so that $|-m(K_X+\Delta )|$ is a base point free linear system for any $(X,\,\Delta ) \in \mathcal {P}$. Hence, a general curve $C$ on the rational equivalence class of $(-m(K_X+\Delta ))^{d-1}$ will be an ample curve. Moreover, by Lemma 2.21, we have that

\[ -(K_X+\Delta) \cdot C = m^{d-1} (-(K_X+\Delta))^{d} = m^{d-1} \operatorname{vol}(-(K_X+\Delta)) < m^{d-1}v_2. \]

Thus, it suffices to take $M=m^{d-1}v_2$.

Proof. Let $\pi \colon Y \rightarrow X$ be the $\epsilon$-plt blow-up at $x\in X$ and write

\[ K_Y-a_E(X,0)E = \pi^{*}K_X. \]

By assumption of $X$ being klt and $K_X$ being Cartier at $x$ we know that $a_E(X,\,0)$ is a non-negative integer. We write

\[ K_E+\Delta_E = ( K_Y+E)|_E. \]

Thus, the pair $(E,\,\Delta _E)$ is $\epsilon$-lc and $-(K_E+\Delta _E)$ is ample. By boundedness of Fano varieties [Reference Birkar6, Theorem 1.1], we conclude that the projective varieties $E$ belong to a bounded family. By [Reference Hacon and Kovács15, Theorem 3.34], we know that the coefficients of $\Delta _E$ are standard, therefore the pairs $(E,\,\Delta _E)$ are log bounded by Lemma 2.18. Moreover, since $(E,\,\Delta _E)$ is $\epsilon$-log canonical then the coefficients of $\Delta _E$ are at most $1-\epsilon$, hence belong to a finite set of rational numbers. By Lemma 2.16, conclude that the pairs $(E,\,\Delta _E)$ belong to a strictly log bounded family $\mathcal {P}$, which only depends on the dimension $d$ and the positive real number $\epsilon$.

By [Reference Shokurov33, Proposition 3.9], we know that at codimension $2$ points of $Y$, every Weil divisor has Cartier index bounded by $p$, for some constant $p$ which only depends on $\epsilon$. Hence, there exists a closed subset $Z$ on $Y$ of codimension at least $3$, so that the Cartier index of $E$ is a divisor of $p$ outside $Z$. By Lemma 2.22, we may find an ample curve $C$ so that

\[ -(K_E+\Delta_E) \cdot C \leq M, \]

for some constant $M$ which only depends on $\mathcal {P}$. Thus, up to replacing $C$ with a rationally equivalent curve, we may assume that $C$ does not intersect $Z$, so we have $p E \cdot C$ is a negative integer, or equivalently,

\[ - E \cdot C \in \mathbb{Z}_{{>}0}\left[ \frac{1}{p} \right]. \]

Moreover, since $(E,\,\Delta _E)$ belongs to a strictly log bounded family, we conclude that the Cartier index of $-(K_E+\Delta _E)$ is bounded by a constant which only depends on $\mathcal {P}$ [Reference Birkar7, Lemma 2.25]. In particular,

\[ -(K_Y+E) \cdot C ={-}(K_E+\Delta_E)\cdot C \]

belongs to a finite set $\mathcal {F}$ of positive rational numbers, which only depends on $\mathcal {P}$. Finally, observe that from the relation

\[ (K_Y+E -(a_E(X,0)+1) E ) \cdot C = \pi^{*}(K_X)\cdot C =0, \]

we conclude that

\[ \frac{(a_E(X,0)+1)}{p} \leq (a_E(X,0)+1) (- E\cdot C) \in \mathcal{F}, \]

so

\[ a_E(X,0) \leq p \max\{\mathcal{F}\}-1, \]

where the right-hand side only depends on $\epsilon$ and $\mathcal {F}$. Since $\mathcal {F}$ only depends on $\mathcal {P}$, and $\mathcal {P}$ only depends on $d$ and $\epsilon$, we deduce that $l=p \max \{ \mathcal {F}\}-1$ only depends on $d$ and $\epsilon$. This proves the first statement. Since $a_E(X,\,0)$ is a non-negative integer, we conclude that there are finitely many possible values for it, proving the second statement.

Proof. Let $\mathcal {X}\rightarrow T$ be a bounding family. Let $\mathcal {B}\subset \mathcal {X}$ be an effective divisor so that $(\mathcal {X},\,\mathcal {B})$ strictly log bound the family $\mathcal {P}$. By Noetherian induction, we can stratify the base $T$ into finitely many locally closed subvarieties $T_i\subset T$, satisfying the following: For each induced family $\mathcal {X}_i\rightarrow T_i$, we can find a log resolution of $(\mathcal {X}_i,\,\mathcal {B}_i)$ so that admits a log resolution over $T_i$ and each blow-up is horizontal over $T_i$. By [Reference Kollár and Mori25, Corollary 2.32.(2)], we conclude that the total discrepancy of each fibre of $(\mathcal {X}_i,\,\mathcal {B}_i)\rightarrow T_i$ belongs to a finite set. Hence, the total discrepancy of each fibre of $(\mathcal {X},\,\mathcal {B})\rightarrow T$ belongs to a finite set. Then, the same holds for pairs $(X,\,\Delta )\in \mathcal {P}$.

Proof. We proceed by contradiction. Recall that

\begin{align*} \mathcal{M}\mathcal{C}_x(X,\Delta) & := \sup \{ a(E,\Delta_E)+1 \mid (E,\Delta_E) \text{ is a Koll}\unicode{x00E1}\text{r component of}\\ & \qquad\qquad(X,\Delta)\text{ over }x\in X \}. \end{align*}

We want to prove that the supremum on the right is actually a maximum. Assume it is not. Then, we can find a sequence of Kollár components $\pi _i \colon Y_i \rightarrow X$ over $x\in X$ for the pair $(X,\,\Delta )$ so that the total discrepancies of the pairs $(E_i,\, \Delta _{E_i})$ are in an infinite increasing sequence. Since each $(E_i,\,\Delta _{E_i})$ is klt, its total log discrepancy is positive. We are assuming that the total discrepancy of the sequence $(E_i,\,\Delta _{E_i})$ is infinite and strictly increasing. Hence, there exists $\epsilon >0$ so that each $(E_i,\,\Delta _{E_i})$ is $\epsilon$-log canonical. Thus, by [Reference Birkar7, Theorem 1.1], we conclude that the varieties $E_i$ belong to a bounded family.

Since the coefficients of $\Delta _{Y_i}$ are fixed by $\textrm {coeff}(\Delta )$, we conclude that the coefficients of $\Delta _{E_i}$ belong to a finite set of rational numbers which only depend on $\textrm {coeff}(\Delta )$ [Reference Filipazzi and Moraga11, Lemma 5.3]. By Lemma 2.18, we conclude that the pairs $(E_i,\,\Delta _{E_i})$ are log bounded. By Lemma 2.16, we conclude that the pairs $(E_i,\,\Delta _{E_i})$ are strictly log bounded. Since the pairs $(E_i,\,\Delta _{E_i})$ belong to a strictly log bounded family, then their total discrepancies can take only finitely many values (see Lemma 2.24). This leads to a contradiction. We conclude that there is no such infinite sequence, hence the supremum is a maximum.

Proof. Let $\pi \colon Y \rightarrow X$ be a projective birational morphism and $Y'\rightarrow Y\times _{X} X'$ the normalization of the main component of the fibre product, then we have a commutative diagram

(2.1)

and we can write

\[ {\pi'}^{*}(K_{X'}+\Delta')=K_{Y'}+\Delta'_{Y'}+E' \text{ and } \pi^{*}(K_X+\Delta)=K_Y+\Delta_Y+E. \]

Here $\Delta '_{Y'}$ is the strict transform of $\Delta '$ on $Y'$. By Lemma 2.27, we know that $(X',\,\Delta ')$ is a log pair. Thus, we have the relation

\[ \phi_Y^{*}(K_Y+\Delta_Y+E) = K_{Y'}+\Delta'_{Y'}+E'. \]

Therefore, by the pull-back formula for quasi-finite morphisms we have that

\[ a_{E}(X',\Delta')+1 = r_E(a_E(X,\Delta)+1), \]

where $r_E$ is the ramification index of $\phi _Y$ at the generic point of $E$. By Theorem 2.28, we know that for every divisorial valuation on $X'$ we can find $\pi \colon Y \rightarrow X$ so that the centre of this valuation on $Y'$ is a divisor, where $Y'$ is as in the commutative diagram (

). This proves the first statement.

Now we turn to prove the second statement by contradiction. By the proof of the first statement, we know that the log canonical pair $(X',\,\Delta ')$ has at least one log canonical place, and all its log canonical places map onto $x'\in X'$. Assume that $(X',\,\Delta ')$ has more than one log canonical place. Let $\pi \colon Y'\rightarrow X'$ be a dlt modification of $(X',\,\Delta ')$ and write

\[ {\pi'}^{*}(K_{X'}+\Delta')= K_{Y'}+\Delta'_{Y'}. \]

Applying [Reference Kollár and Mori25, Theorem 5.48] to a log resolution of $(Y',\,\Delta '_Y)$, we deduce that $\lfloor \Delta '_{Y'} \rfloor$ is connected. Moreover, since $(X',\,\Delta ')$ has more than one log canonical place, the divisor $\lfloor \Delta '_{Y'}\rfloor$ is connected and has at least two irreducible components. Since the intersection of two log canonical centres is a union of log canonical centres [Reference Ambro5, Theorem 1.1], we deduce that $(Y',\,\Delta '_{Y'})$ has infinitely many log canonical places. Thus, $(X',\,\Delta ')$ has infinitely many log canonical places. By Theorem 2.28, we conclude that each of those log canonical places appears in a commutative diagram as in (2.1). Moreover, at least one log canonical place of $(X',\,\Delta ')$ is exceptional over the dlt model $(Y,\,\Delta _Y)$ of $(X,\,\Delta )$. Therefore, $(X,\,\Delta )$ has at least two log canonical places. This provides the needed contradiction and the claim follows.

Proof. We will construct a strong $(0,\,n)$-complement for the divisor $-(K_Y+\Delta _Y+E)$ with respect to the morphism $\pi \colon Y \rightarrow X$ around $x\in X$. This complement will push-forward to a $(0,\,n)$-complement for $(X,\,\Delta )$ locally around $x\in X$. In order to do so, we will do adjunction to $E$, produce a global complement on $E$ and then pull-back to a neighbourhood of $E$.

By adjunction, we can write

\[ (K_Y+\Delta_Y+E)|_E = K_E+\Delta_E, \]

where $(E,\,\Delta _E)$ is an $\epsilon$-log canonical pair and the coefficients of $\Delta _E$ belong to a set of rational numbers satisfying the descending chain condition with rational accumulation points (see, e.g. [Reference Filipazzi and Moraga11, Lemma 5.3]). Hence, by [Reference Filipazzi and Moraga11, Theorem 1.3.2], we can find $n$ only depending on $d$ and $\epsilon$, and a global strong $(\epsilon,\,n)$-complement $\Gamma _E$ for the pair $(E,\,\Delta _E)$. From the proof of [Reference Filipazzi and Moraga11, Theorem 1.3], we may assume that $\Gamma _E$ does not share prime components with $\Delta _E$. Without loss of generality, we may assume that $n\Delta _Y$ is a Weil divisor. Indeed, since $(Y,\,\Delta _Y+E)$ is $\epsilon$-plt and the coefficients of $\Delta _Y$ are standard, then they belong to a finite set.

Let $\pi _Y \colon W \rightarrow Y$ be a log resolution of $(Y,\,\Delta _Y+E)$, and write

\[{-}N_W := \pi_Y^{*}(K_Y+\Delta_Y+E) = K_W+\Delta_W +E_W, \]

where $E_W$ is the strict transform of $E$ on $W$. We define

\[ L_W:={-}nK_W - nE_W - \lfloor (n+1)\Delta_W \rfloor. \]

Let $P_W$ be the unique integral effective divisor on $W$ so that

\[ \Lambda_W := E_W+(n+1)\Delta_W - \lfloor (n+1)\Delta_W \rfloor + P_W \]

is a boundary on $W$ so that $(W,\,\Lambda _W)$ is plt and $\lfloor \Lambda _W \rfloor =E_W$. We claim that $P_W$ is an exceptional divisor over $Y$. Assume it is not. Indeed, if $D$ is a prime divisor on $W$ which is not contracted on $Y$ then, we have that

\[ \textrm{coeff}_D(\lfloor (n+1)\Delta_W \rfloor) = \textrm{coeff}_D( n\Delta_W), \]

because $n\Delta _Y$ is integral. Therefore

\[ \textrm{coeff}_D( (n+1)\Delta_W - \lfloor (n+1)\Delta_W \rfloor ) = \textrm{coeff}_D( \Delta_W )= \textrm{coeff}_{\pi_Y(D)}( \Delta_Y) \in (0,1). \]

Hence, if $\textrm {coeff}_D(P_W)>0$, then $\textrm {coeff}_P(P_W)\geq 1$, so $\textrm {coeff}_P(\Lambda _W)>1$. In particular, $(W,\,\Lambda _W)$ is not plt. This leads to a contradiction. We conclude that $P_W$ is exceptional over $Y$.

By definition, we have that

\[ L_W+P_W-E_W =K_W+\Lambda_W + (n+1)N_W, \]

is the sum of the klt pair $(W,\,\Lambda _W-E_W)$ and the nef and big divisor $(n+1)N_W$ over a neighbourhood of $x\in X$. Note that $(W,\,\Lambda _W-E_W)$ is klt given that $(W,\,\Lambda _W)$ is plt with $\lfloor \Lambda _W\rfloor =E_W$. Shrinking around $x\in X$ we may assume that $X$ is affine, then $(n+1)N_W$ is nef and big over $X$. By the relative version of Kawamata–Viehweg theorem [Reference Kawamata, Matsuda and Matsuki20, Theorem 1-2-5], we have a surjection

(2.2)\begin{equation} H^{0}(L_W+P_W) \rightarrow H^{0}((L_W+P_W)|_{E_W}). \end{equation}

We denote by $\Gamma _{E_W}$ the pull-back of $\Gamma _E$ to $E_W$. Observe that we have

\[ (L_W+P_W)|_{E_W} \sim G_{E_W} := n\Gamma_{E_W}+n\Delta_{E_W}- \lfloor (n+1)\Delta_{E_W} \rfloor +P_{E_W}, \]

where $P_{E_W}:=P_W|_{E_W}$ and $\Delta _{E_W}:=\Delta _W|_{E_W}$. The divisor $G_{E_W}$ is integral and its coefficients are strictly greater than $-1$, therefore it is indeed effective. By the surjectivity of (2.2), there exists $0\leq G_W \sim L_W+P_W$ which restricts to $G_{E_W}$. We denote by $G_Y$ the push-forward of $G_W$ to $Y$. By pushing-forward the linear equivalence $L_W+P_W\sim G_W$ to $Y$, and using the fact that $P_W$ is $Y$-exceptional, we get that

\[ 0\leq G_Y \sim{-}n(K_Y+\Delta_Y+E). \]

We define $\Gamma _Y := \frac {G_Y}{n}$. Observe that by construction, we have

\[ n(K_Y+\Delta_Y+\Gamma_Y+E)\sim 0 \]

on a neighbourhood of $x\in X$. We claim that $\Gamma _Y|_{E}= \Gamma _E$. Indeed, observe that we can define

\[ n \Gamma_W := G_W -P_W + \lfloor (n+1)\Delta_W \rfloor -n \Delta_W \sim nN_W \sim_{\mathbb{Q},Y} 0, \]

and $n\Gamma _W$ pushes-forward to $G_Y$ on $Y$, hence $\Gamma _W = \pi _Y^{*}(\Gamma _Y)$. On the other hand, we have $n\Gamma _{E_W}= n \Gamma _W|_{E_W}$, which means that $\Gamma _{E_W}=\Gamma _W|_{E_W}$. Thus, we have $\Gamma _Y|_E=\Gamma _E$ as claimed.

Finally, observe that

\[ (K_Y+\Delta_Y+\Gamma_Y +E)|_E = K_W+\Delta_E+\Gamma_E, \]

so by inversion of adjunction, we conclude that $(Y,\,\Delta _Y+\Gamma _Y+E)$ is $\epsilon$-plt. Moreover, since $\Gamma _E$ and $\Delta _E$ prime components, then $\Delta _Y$ and $\Gamma _Y$ do not share prime components as well. Define $\Gamma =\pi _*(\Gamma _Y)$, and observe that

\[ \pi^{*}(K_X+\Delta+\Gamma)= K_Y+\Delta_Y+\Gamma_Y+E. \]

Thus, $(X,\,\Delta +\Gamma )$ is a log canonical pair with a unique log canonical place so that $n(K_X+\Delta +\Gamma ) \sim 0$ on a neighbourhood of $x\in X$. Moreover, $\Delta$ and $\Gamma$ do not share prime components.