1 Introduction
We work over an algebraically closed field k of characteristic zero. Given a contraction
$f:X\rightarrow Z$
, that is, a projective morphism such that
$f_* \mathcal O_X=\mathcal O_Z$
, a fundamental problem is to relate the singularities on X and those on Z. This problem is important as it appears frequently in inductive arguments. Assume that f is a Fano contraction, McKernan conjectured that in this case the singularities on Z are bounded in terms of those on X.
Conjecture 1.1 (McKernan)
Fix a positive integer r and a real number
$0<\epsilon \leq 1$
. There exists
$\delta>0$
depending only on
$r,\epsilon $
and satisfying the following. Assume:
-
•
$f:X\rightarrow Z$
is a contraction of relative dimension r; -
• X is
$\epsilon $
-lc; -
•
$-K_X$
is ample over Z; and -
• Z is
$\mathbb Q$
-Gorenstein.
Then, Z is
$\delta $
-lc.
Recently, this conjecture was solved by Birkar [Reference Birkar9]. Indeed, he proved a more general conjecture—Shokurov conjecture (see Conjecture 1.7 below), which implies McKernan conjecture. Another interesting consequence of Shokurov conjecture is that under the setting of Conjecture 1.1, the multiplicity of the fiber of f over a codimension one point of Z is bounded above. For more historical results on these two conjectures, we refer to [Reference Alexeev and Borisov1], [Reference Birkar5], [Reference Birkar6], [Reference Birkar10], [Reference Birkar and Chen11], [Reference Chen14], [Reference Han, Jiang and Luo19], [Reference Mori and Prokhorov26], [Reference Mori and Prokhorov27].
The next problem is to give an explicit value for
$\delta $
in terms of
$r,\epsilon $
. When
$r=1$
and
$\epsilon =1$
, Han, Jiang, and Luo [Reference Han, Jiang and Luo19] showed that the optimal value of
$\delta $
is 1/2. When
$r=1$
, in [Reference Chen14] the author showed that one can take
$\delta =\epsilon ^2/2$
. The main purpose of this article is to treat the toric case for arbitrary
$r,\epsilon $
. Our main result is the following.
Theorem 1.2. Let r be a positive integer and
$0<\epsilon \leq 1$
be a real number. Let
${f:X\rightarrow Z}$
be a toric contraction of relative dimension r such that
$-K_X$
is ample over Z and X is
$\epsilon $
-lc vertically over Z, that is,
$a(E,X,0)\geq \epsilon $
for any prime divisor E over X with
${f(\operatorname {\mathrm {center}}_X E)\neq Z}$
. Let
$$ \begin{align} \delta=\delta(r,\epsilon)= \frac{\epsilon^{2^r}}{2^{2^r-1}\prod\limits_{i=1}^r i^{2^i}}. \end{align} $$
Then:
(1) if Z is
$\mathbb Q$
-Gorenstein, then Z is
$\delta $
-lc;
(2) for any codimension one point z of Z, the multiplicity of each component of
$f^*z$
is bounded from above by
$1/\delta $
.
Remark 1.3. (1) Comparing with Conjecture 1.1, in Theorem 1.2, we require a weaker condition that X is
$\epsilon $
-lc vertically over Z instead of the original condition that X is
$\epsilon $
-lc. Note that under the original condition the general fiber F of f is an
$\epsilon $
-lc Fano variety, so it belongs to a bounded family by [Reference Birkar7, Reference Birkar8]. However, under the new condition, the general fiber may not belong to a bounded family.
(2) For the first assertion in Theorem 1.2, when
$r=2$
, the order
$O(\epsilon ^4)$
is optimal. Indeed, Alexeev and Borisov [Reference Alexeev and Borisov1, Theorem 1.5] constructed a sequence of toric Fano contractions
$X\to Z$
such that
$\dim X=4$
,
$\dim Z=2$
,
$\operatorname {\mathrm {mld}}(X)\to 0$
and
$\operatorname {\mathrm {mld}}(Z)\approx C\cdot \operatorname {\mathrm {mld}}(X)^4$
.
(3) For the second assertion in Theorem 1.2, the order
$O(\epsilon ^{2^r})$
is optimal by the following example.
Example 1.4. Let
$q,r$
be two positive integers. Let
$u_{i,q}~(i\in \mathbb Z_{>0})$
be a sequence of integers defined recursively as follows:
$$ \begin{align*}u_{1,q}=q,\quad u_{k+1,q}=u_{k,q}(u_{k,q}+1) \text{ for any } k\in \mathbb Z_{>0}.\end{align*} $$
Then,
$u_{r+1,q}\in O(q^{2^r})$
when
$q\to +\infty $
.
Let
$e_1,\ldots ,e_{r+1}$
be the standard basis of
$\mathbb Z^{r+1}$
and denote
$e=\sum _{i=1}^r e_i$
. Let
$$ \begin{align*} v_i&=(1+u_{i,q})e_1-qe \text{ for }1\leq i\leq r,\\ v_{r+1}&=-e, \quad v_{r+2}=(u_{r+1,q}-1)e_{r+1}-qe. \end{align*} $$
Let X be the toric variety associated to the fan in
$\mathbb R^{r+1}$
whose maximal cones are generated by
$v_{r+2}$
and subsets of
$\{v_1,\ldots ,v_{r+1}\}$
of size r. The support of the fan of X is
$\mathbb R^r\times \mathbb R_{\geq 0}$
. The projection
$\mathbb Z^{r+1}\to \mathbb Z$
onto the
$(r+1)$
th coordinate induces a toric morphism
$f:X\to Z$
, where
$Z=\mathbb A^1$
with a distinguished point o. Then,
$f:X\to Z$
is a toric Fano contraction of relative dimension r. Moreover,
$$ \begin{align*}f^*o=(u_{r+1,q}-1)\cdot D,\end{align*} $$
where D is the toric divisor on X corresponding to the ray
$\mathbb R_{\geq 0} \cdot v_{r+2}$
.
Let S be the lattice simplex in
$\mathbb R^{r+1}$
with vertices
$v_1,\ldots ,v_{r+2}$
. Let F be the face of S which is the intersection of S and the subspace spanned by
$e_1,\ldots ,e_r$
. Then, X is
$\frac {1}{q}$
-lc if and only if
$$ \begin{align*}\text{int}(\frac{1}{q} S)\cap \mathbb Z^{r+1}=\emptyset \quad \text{and} \quad \text{relint}(\frac{1}{q} F)\cap \mathbb Z^{r+1}=\{\textbf{0}\}. \end{align*} $$
This condition is satisfied because S is contained in the lattice simplex
$S'$
with vertices
$$ \begin{align*}v_i~(1\leq i\leq r),~u_{r+1,q}e_{r+1}-qe,~-u_{r+1,q}e_{r+1}-qe\end{align*} $$
and
$\text {int}(\frac {1}{q}S')\cap \mathbb Z^{r+1}=\{\textbf {0}\}$
by [Reference Zou30, Theorem 1.3]. Therefore, X is
$\frac {1}{q}$
-lc.
The following is a local and more general version of Theorem 1.2.
Theorem 1.5. Let r be a positive integer,
$0<\epsilon \leq 1$
be a real number and
$\delta =\delta (r,\epsilon )$
as in (1.1). Let
$f:X\rightarrow Z$
be a toric contraction of relative dimension r and let
$z\in Z$
be a codimension
$\geq 1$
point. Suppose there is a pair
$(X,B)$
on X such that
$K_X+B\sim _{\mathbb R} 0/Z$
and
$\operatorname {\mathrm {mld}}(X/Z\ni z, B)\geq \epsilon $
. Then:
(1) if Z is
$\mathbb Q$
-Gorenstein, then
$\operatorname {\mathrm {mld}}(Z\ni z, 0)\geq \delta $
;
(2) if the codimension of z in Z is one, then the multiplicity of each component of
$f^*z$
is bounded from above by
$1/\delta $
.
Here, we denote by
$\operatorname {\mathrm {mld}}(X/Z\ni z,B)$
(resp.
$\operatorname {\mathrm {mld}}(Z\ni z,0)$
) the infimum of the log discrepancy of E with respect to
$(X,B)$
(resp.
$(Z,0)$
), where E runs over all prime divisors over X (resp. Z) whose image on Z is the closure
$\overline {z}$
of z (see Definition 2.3).
Remark 1.6. Notice that the assumption “
$\operatorname {\mathrm {mld}}(X/Z\ni z, B)\geq \epsilon $
” is weaker than “X is
$\epsilon $
-lc over some neighborhood of z” since the former does not put restriction on the log discrepancy of such prime divisor whose image on Z is not
$\overline {z}$
but contains z.
As mentioned earlier, Shokurov proposed a more general conjecture which implies McKernan conjecture. In order to state Shokurov conjecture, we recall some background on adjunction for fibrations (also called the canonical bundle formula). Let
$f:X\rightarrow Z$
be a contraction between normal varieties. Let
$(X,B)$
be a pair which is lc over the generic point of Z and such that
$K_X+B\sim _{\mathbb R} 0/Z$
. By the work of Kawamata [Reference Kawamata22], [Reference Kawamata23] and Ambro [Reference Ambro2], [Reference Ambro3], we may write
$$ \begin{align*}K_X+B\sim_{\mathbb R} f^*(K_Z+B_Z+M_Z),\end{align*} $$
where
$B_Z$
is called the discriminant divisor and
$M_Z$
is called the moduli divisor. The discriminant divisor is defined by lc thresholds, more precisely, the coefficient of a prime divisor D in
$B_Z$
is set to be
$1-t,$
where t is the largest number such that
$(X,B+tf^*D)$
is lc over the generic point of D. The moduli divisor is then automatically determined up to
$\mathbb R$
-linear equivalence.
For each birational model
$Z'$
over Z, similarly we can define
$B_{Z'},M_{Z'}$
so that their pushdowns on Z coincide with
$B_Z,M_Z$
. This defines a discriminant
$\boldsymbol {b}$
-divisor
$\mathbf {{B}}$
and a moduli
$\boldsymbol {b}$
-divisor
$\mathbf {{M}}$
over Z. It was shown that
$\mathbf {{M}}$
is a
$\boldsymbol {b}$
-nef
$\boldsymbol {b}$
-divisor and we hence obtain a generalized pair
$(Z,B_Z,\mathbf {{M}})$
, which is called the generalized pair given by adjunction for
$f:(X,B)\rightarrow Z$
(see §2.4 for more details). We are now ready to state Shokurov conjecture.
Conjecture 1.7 (Shokurov)
Fix a positive integer r and a real number
$0<\epsilon \leq 1$
. There exists
$\delta>0$
depending only on
$r,\epsilon $
and satisfying the following. Let
$(X,B)$
be a pair and
$f:X\rightarrow Z$
be a contraction such that:
-
•
$\dim X - \dim Z=r$
; -
•
$(X,B)$
is
$\epsilon $
-lc; -
•
$K_X+B\sim _{\mathbb R} 0/Z$
; and -
• X is of Fano type over Z, equivalently,
$-K_X$
is big over Z.
Let
$(Z, B_Z,\mathbf {{M}})$
be the generalized pair given by adjunction for
$f:(X,B)\rightarrow Z$
. Then,
$(Z, B_Z,\mathbf {{M}})$
is generalized
$\delta $
-lc.
As mentioned earlier, Shokurov conjecture was proved by Birkar [Reference Birkar9] recently. Before this celebrated result, in [Reference Birkar and Chen11, Theorem 1.4] Birkar and Chen showed a variant of Shokurov conjecture in the toric setting, which says that Shokurov conjecture holds after taking an average with the toric boundary. This is enough for some interesting applications. Building on ideas from their work and combining the main result in [Reference Chen14], we give an explicit value for
$\delta $
in [Reference Birkar and Chen11, Theorem 1.4] as follows.
Theorem 1.8. Let r be a positive integer and
$0<\epsilon \leq 1$
be a real number. Assume:
-
•
$f\colon X\rightarrow Z$
is a toric contraction of relative dimension r with
$z\in Z$
a codimension
$\geq 1$
point; -
•
$(X,B)$
is a pair
$($
B is not necessarily toric
$)$
such that
$\operatorname {\mathrm {mld}}(X/Z\ni z,B)\geq \epsilon $
; -
•
$K_X+B\sim _{\mathbb R}0/Z$
; and -
•
$\Delta $
is the toric boundary divisor of X.
Let
$$ \begin{align*}\Gamma^{\alpha}=\alpha B+(1-\alpha)\Delta, \quad \text{ where } \alpha=1/r! \end{align*} $$
and let
$(Z,\Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha })$
be the generalized pair given by adjunction for
$f:(X,\Gamma ^{\alpha })\rightarrow Z$
. Then,
$$ \begin{align*}\operatorname{\mathrm{mld}}(Z\ni z, \Gamma_Z^{\alpha},\mathbf{{N}}^{\alpha}) \geq \delta,\end{align*} $$
where
$\delta =\delta (r,\epsilon )$
as in (1.1).
Theorems 1.2 and 1.5 are consequences of Theorem 1.8. Another interesting corollary is the following.
Theorem 1.9. Let r be a positive integer,
$0<\epsilon \leq 1$
be a real number and
$\delta =\delta (r,\epsilon )$
as in (1.1). Let
$f:X\rightarrow Z$
be a toric contraction of relative dimension r with a toric pair
$(X,B)$
and a codimension one point
$z\in Z$
. Suppose there is an
$\mathbb R$
-divisor
$B^+ ($
not necessary toric
$)$
such that
$B^+\geq B$
,
$K_X+B^+\sim _{\mathbb R} 0/Z,$
and
$\operatorname {\mathrm {mld}}(X/Z\ni z, B^+)\geq \epsilon $
. Then,
$(X,B+\delta f^*\overline {z})$
is lc over some neighborhood of
$z\in Z$
.
Remark 1.10. After this work was completed, Ambro informed me that he also got some explicit lower bounds in the toric case. Let
$f:X \to Z$
be a toric Fano contraction of relative dimension r with X being
$\epsilon $
-lc. Let F be the general fiber and let
$\gamma $
be the
$\alpha $
-invariant of F. There exists a sharp lower bound for
$\gamma $
just in terms of r and
$\epsilon $
(cf. [Reference Ambro4]). Ambro got explicit bounds for
$\delta $
in terms of
$\epsilon ,r,$
and
$\gamma $
in Theorems 1.2, 1.5, and 1.9.
1.1 Idea of the proof of Theorem 1.8
The proof is built on ideas from [Reference Birkar and Chen11] with some modifications. In [Reference Birkar and Chen11], by running toric minimal model program (MMP), they reduced the problem to the case for toric Mori fiber spaces. Then, they showed that after taking a finite cover and extracting a toric divisor, a
$\mathbb Q$
-factorial toric Mori fiber space can be factored as the composition of toric contractions of smaller relative dimension. Therefore, they can reduce the problem to the case for contractions of relative dimension one. However, after taking a finite cover and extracting a divisor, the pullback of
$K_X+B$
may be a sub-pair rather than a pair, so it is necessary to take average
$\Gamma ^{\alpha }=\alpha B+(1-\alpha )\Delta $
with the toric boundary to make its pullback a pair. To guarantee that the singularities of
$(X,\Gamma ^{\alpha })$
are not too bad,
$\alpha $
can not be too small and hence it is important to control the order n of the finite cover and the log discrepancy of the extracted divisor. They showed the boundedness of the order n, however, it seems not easy to give an explicit bound for n, as it involves all possibilities of the fans corresponding to
$\epsilon $
-lc toric Fano varieties up to the action of
$GL_r(\mathbb Z)$
. In this article, we make some modifications to their method. We factor a toric Mori fiber space after extracting a toric divisor with log discrepancy
$\leq r$
, without taking a finite cover (see Lemma 3.6). Recall that in relative dimension one, an explicit value for
$\delta $
in Shokurov conjecture was given in [Reference Chen14]. Combining this result, we obtain an explicit value for
$\delta $
in [Reference Birkar and Chen11, Theorem 1.4].
2 Preliminaries
We will freely use the standard notations and definitions in [Reference Birkar, Cascini, Hacon and McKernan12], [Reference Kollár and Mori24]. A contraction
$f:X\rightarrow Z$
is a projective morphism of varieties with
$f_*\mathcal {O}_X=\mathcal {O}_Z$
. An extremal contraction is a contraction
$f:X\rightarrow Z$
with the relative Picard number
$\rho (X/Z)=1$
.
2.1 Fano type varieties
Let
$X\to Z$
be a contraction of normal varieties. We say X is of Fano type over Z if there is a klt pair
$(X,B)$
on X such that
$-(K_X+B)$
is ample over Z.
We say
$X\to Z$
is a Mori fiber space if
$-K_X$
is ample over Z and the relative Picard number
$\rho (X/Z)=1$
.
2.2 b-divisors
Let X be a normal variety. A
$\boldsymbol {b}$
-divisor
$\textbf {D}$
over X is a collection of
$\mathbb R$
-divisors
$\textbf {D}_Y$
for each birational model Y over X, such that
$\sigma _*\textbf {D}_{Y_1}=\textbf {D}_{Y_2}$
for any birational morphism
$\sigma :Y_1\rightarrow Y_2/X$
.
Let
$\mathbf {{D}}$
be a
$\boldsymbol {b}$
-divisor over X and
$Y_0$
be a birational model over X. We say
$\mathbf {{D}}$
descends to
$Y_0$
if
$\textbf {D}_{Y_0}$
is an
$\mathbb R$
-Cartier
$\mathbb R$
-divisor and
$\textbf {D}_{Y}=\sigma ^* \textbf {D}_{Y_0}$
for any birational morphism
$\sigma :Y\rightarrow Y_0/X$
.
Let
$X\rightarrow U$
be a projective morphism. We say that a
$\boldsymbol {b}$
-divisor
$\textbf {D}$
over X is
$\boldsymbol {b}$
-nef
$/U$
(resp.
$\boldsymbol {b}$
-semiample
$/U$
) if
$\mathbf {{D}}$
descends to some birational model
$Y_0$
and
$\textbf {D}_{Y_0}$
is nef
$/U$
(resp. semiample
$/U$
).
We denote by
$\textbf {0}$
the
$\boldsymbol {b}$
-divisor
$\mathbf {{D}}$
such that
$\textbf {D}_Y=0$
for each birational model Y over X.
2.3 Generalized pairs
We will follow the original definitions in [Reference Birkar and Zhang13] and adopt the notations in [Reference Hacon and Liu20].
Definition 2.1. A generalized sub-pair (g-sub-pair for short)
$(X,B,\mathbf {{M}})/U$
consists of a normal variety X associated with a projective morphism
$X\rightarrow U$
, an
$\mathbb R$
-divisor B on X, and a
$\boldsymbol {b}$
-nef
$/U \boldsymbol {b}$
-divisor
$\mathbf {{M}}$
over X.
A g-sub-pair
$(X,B,\mathbf {{M}})/U$
is called a sub-pair if
$\mathbf {{M}}=\textbf {0}$
. In this case, we denote it by
$(X,B)/U$
or
$(X,B)$
.
A g-sub-pair
$(X,B,\mathbf {{M}})/U$
is called a generalized pair (g-pair for short) if
$B\geq 0$
. A sub-pair
$(X,B)$
is called a pair if
$B\geq 0$
.
We may drop U when we emphasize the structures of
$(X,B,\mathbf {{M}})$
that are independent of the choice of U, for example, the singularities of
$(X,B,\mathbf {{M}})$
.
Definition 2.2. Let
$(X,B,\mathbf {{M}})/U$
be a g-(sub-)pair and E be a prime divisor over X, that is, a prime divisor on a normal variety Y with a birational morphism
$\pi :Y\rightarrow X$
. The center of E on X is defined as the image of E on X under the morphism
$\pi $
and it is denoted by
$\operatorname {\mathrm {center}}_X E$
. Write
$$ \begin{align*}K_Y+B_Y+\mathbf{{M}}_Y:=\pi^*(K_X+B+\mathbf{{M}}_X).\end{align*} $$
Then, the log discrepancy of E with respect to
$(X,B,\mathbf {{M}})$
is defined as
$1-\operatorname {\mathrm {mult}}_E B_Y$
and it is denoted by
$a(E,X,B,\mathbf {{M}})$
, where
$\operatorname {\mathrm {mult}}_E B_Y$
is the coefficient of E in
$B_Y$
.
Definition 2.3. Let
$(X,B,\mathbf {{M}})/U$
be a g-(sub-)-pair,
$f:X\rightarrow Z/U$
be a projective morphism and
$z\in Z$
be a
$($
not necessary closed
$)$
point. The minimal log discrepancy of
$(X,B,\mathbf {{M}})$
over z is defined as
$$ \begin{align*} \operatorname{\mathrm{mld}}(X/Z\ni z,B,\mathbf{{M}}):=\inf\{a(E,X,B,\mathbf{{M}}) \mid ~ &E\text{ is a prime divisor over }X\\ &\text{with }f(\operatorname{\mathrm{center}}_X(E))=\overline{z}\}. \end{align*} $$
In the case, that
$Z=X$
,
$z=x$
and f is the identity morphism, we will use
$\operatorname {\mathrm {mld}}(X\ni x,B,\mathbf {{M}})$
instead of
$\operatorname {\mathrm {mld}}(X/Z\ni z,B,\mathbf {{M}})$
.
Definition 2.4. A g-(sub-)pair
$(X,B,\mathbf {{M}})$
is said to be (sub-)
$\epsilon $
-glc (resp. (sub-)
$\epsilon $
-gklt, (sub-)glc, (sub-)gklt) if
$\operatorname {\mathrm {mld}}(X\ni x,B,\mathbf {{M}})\geq \epsilon $
(resp.
$>\epsilon $
,
$\geq 0$
,
$> 0$
) for any codimension
$\geq 1,$
point
$x\in X$
.
If
$\mathbf {{M}}=\textbf {0}$
and
$(X,B,\mathbf {{M}})$
is (sub-)
$\epsilon $
-glc (resp. (sub-)
$\epsilon $
-gklt, (sub-)glc, (sub-)gklt), we say that
$(X,B)$
is (sub-)
$\epsilon $
-lc (resp. (sub-)
$\epsilon $
-klt, (sub-)lc, (sub-)klt). In the case when
$B=0$
, we also say X is
$\epsilon $
-lc (resp.
$\epsilon $
-klt, lc, klt).
Definition 2.5. Let
$(X,B,\mathbf {{M}})/U$
be a g-(sub-)pair and D be an effective
$\mathbb R$
-Cartier
$\mathbb R$
-divisor on X. The lc threshold of D with respect to
$(X,B,\mathbf {{M}})$
is defined to be
$$ \begin{align*}\operatorname{\mathrm{lct}}(X,B,\mathbf{{M}};D):=\sup\{t\geq 0\mid (X,B+tD,\mathbf{{M}}) \text{ is (sub-)glc}\}.\end{align*} $$
Definition 2.6. Let
$(X,B,\mathbf {{M}})/U$
and
$(X,\Gamma ,\mathbf {{N}})/U$
be two g-(sub-)pairs. We say
$(X,B,\mathbf {{M}})$
has better singularities than
$(X,\Gamma ,\mathbf {{N}})$
if
$$ \begin{align*}a(E,X,B,\mathbf{{M}})\geq a(E,X,\Gamma,\mathbf{{N}})\end{align*} $$
for any prime divisor E over X.
Lemma 2.7. Let
$(X,B,\mathbf {{M}})/U$
be a g-
$(sub-)$
-pair,
$f:X\rightarrow Z/U$
be a projective morphism and
$z\in Z$
be a
$($
not necessary closed
$)$
point. Then,
$\operatorname {\mathrm {mld}}(X/Z\ni z,B,\mathbf {{M}})\geq 0$
if and only if
$(X,B,\mathbf {{M}})$
is
$(sub-)$
glc over some neighborhood of
$z\in Z$
.
Proof. This is essentially [Reference Han, Jiang and Luo19, Lemma 2.8] where it was stated only for
$\mathbf {{M}}=0$
. By definition, the “if” part is obvious. Next, we show the “only if” part.
Assume the contrary that
$\operatorname {\mathrm {mld}}(X/Z\ni z,B,\mathbf {{M}})\geq 0$
but
$(X,B,\mathbf {{M}})$
is not (sub-)glc over any neighborhood of
$z\in Z$
. Then, there is a prime divisor E over X such that
$z\in f(\operatorname {\mathrm {center}}_X E)$
and
$a(E,X,B,\mathbf {{M}})< 0$
. Let
$\pi :Y\rightarrow X$
be a resolution with
$K_Y+B_Y+\mathbf {{M}}_Y=\pi ^*(K_X+B+\mathbf {{M}}_X)$
such that:
-
•
$\mathbf {{M}}$
descends to Y; -
• E is a prime divisor on Y;
-
•
$\overline {\pi ^{-1}f^{-1}(z)}$
is a divisor on Y, say F; and -
•
$E+F$
is a simple normal crossing divisor on Y.
We can find an irreducible component D of F such that
$f(\pi (D\cap E))=\overline {z}$
(indeed, since
$z\in f(\pi (E))$
, there is a point
$\eta \in E$
such that
$f(\pi (\eta ))=z$
, then, we take D to be a component of F which contains
$\eta $
).
Denote
$d=\operatorname {\mathrm {mult}}_D B_Y$
and
$e=\operatorname {\mathrm {mult}}_E B_Y>1$
. Blowing up
$D\cap E$
, we get a new resolution
$\pi ':Y'\rightarrow X$
with
$K_{Y'}+B_{Y'}+\mathbf {{M}}_{Y'}=\pi ^{\prime *} (K_X+B+\mathbf {{M}}_X)$
. Denote by
$D'$
the exceptional
$/Y$
divisor on
$Y'$
and by
$E'$
the birational transformation of E on
$Y'$
. By construction, we have
$f(\pi '(D'))=\overline {z}$
,
$D'$
meets
$E'$
transversely,
$f(\pi '(D'\cap E'))=\overline {z}$
and
$\operatorname {\mathrm {mult}}_{D'} B_{Y'}\geq d+e-1>d$
.
So, by successively blowing up, we eventually obtain a prime divisor
$\widetilde {D}$
over X such that
$f(\operatorname {\mathrm {center}}_X \widetilde {D})=\overline {z}$
and
$a(\widetilde {D},X,B,\mathbf {{M}})<0$
, which contradicts that
$\operatorname {\mathrm {mld}}(X/Z\ni z, B,\mathbf {{M}})\geq 0$
.
2.4 Adjunction for generalized fibrations
Let
$f:X\rightarrow Z$
be a contraction between normal varieties over U with
$\dim Z>0$
. Let
$(X,B,\mathbf {{M}})/U$
be a g-pair which is glc over the generic point of Z and such that
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R} 0/Z$
. Then,
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R} f^* L$
for some
$\mathbb R$
-Cartier
$\mathbb R$
-divisor L on Z.
For any prime divisor D on Z, let
$t_D$
be the lc threshold of
$f^*D$
with respect to
$(X,B,\mathbf {{M}})$
over the generic point of D. This make sense even if D is not
$\mathbb Q$
-Cartier because we only need the pullback of D over the generic point of
$D,$
where Z is smooth. We set
${B_Z=\sum _D (1-t_D)D,}$
where D runs over all prime divisors on Z and set
$M_Z=L-K_Z-B_Z$
. The former is called the discriminant divisor and the latter is called the moduli divisor.
Let
$\sigma : Z'\rightarrow Z$
be a birational morphism from a normal variety
$Z'$
and let
$X'$
be the resolution of the main component of
$X\times _Z Z'$
with induced morphism
$\tau :X'\rightarrow X$
and
$f':X'\rightarrow Z'$
. Write
$K_{X'}+B'+\mathbf {{M}}_{X'}=\tau ^*(K_X+B+\mathbf {{M}}_X)$
, then
$K_{X'}+B'+\mathbf {{M}}_{X'}\sim _{\mathbb R} f^{\prime *} \sigma ^* L$
. Similarly, we can define the discriminant divisor
$B_{Z'}$
and the moduli divisor
$M_{Z'}$
for the contraction
$(X',B',\mathbf {{M}})\rightarrow Z'$
. One can check that
$\sigma _* B_{Z'}=B_Z$
and
$\sigma _* M_{Z'}=M_Z$
. Hence, there exist
$\boldsymbol {b}$
-divisors
$\mathbf {{B}}^Z,\textbf {M}^Z$
such that
$\mathbf {{B}}^Z_{Z'}=B_{Z'}$
and
$\textbf {M}^Z_{Z'}=M_{Z'}$
for any birational model
$Z'$
over Z, which are called the discriminant
$\boldsymbol {b}$
-divisor and the moduli
$\boldsymbol {b}$
-divisor, respectively. By construction, we have
$$ \begin{align*}K_X+B+\mathbf{{M}}_X\sim_{\mathbb R} f^*(K_Z+B_Z+\mathbf{{M}}^Z_Z).\end{align*} $$
It was shown that
$\mathbf {{M}}^Z$
is a
$\boldsymbol {b}$
-nef
$/U \boldsymbol {b}$
-divisor over Z (see [Reference Chen, Han, Liu and Xie15, Theorem 11.4.4]). Hence, we can regard
$(Z,B_Z,\mathbf {{M}}^Z)/U$
as a g-pair. We call
$(Z,B_Z,\mathbf {{M}}^Z)/U$
the g-pair given by adjunction for
$f:(X,B,\mathbf {{M}})\rightarrow Z$
. In the case, that
$(X,B,\mathbf {{M}})$
is glc,
$(Z,B_Z,\mathbf {{M}}^Z)$
is also a glc g-pair.
For more details about adjunction for generalized fibrations, we refer the readers to [Reference Filipazzi17], [Reference Jiao, Liu and Xie21] and [Reference Chen, Han, Liu and Xie15, Section 11.4].
Lemma 2.8 [Reference Birkar and Chen11, Lemma 2.1]
Assume that:
-
•
$(X,B,\mathbf {{M}})/U$
is a g-pair which is glc over the generic point of Z; -
•
$X \xrightarrow {g} Y\xrightarrow {h} Z$
are contractions of normal varieties
$/U$
with
$\dim Z>0$
; and -
•
$K_X+B+\mathbf {{M}}_X \sim _{\mathbb {R}}0/Z$
.
Let
$(Y,B_Y,\mathbf {{M}}^Y)/U$
be the g-pair given by adjunction for
$g:(X,B,\mathbf {{M}})\rightarrow Y$
and let
$(Z,B_Z,\mathbf {{M}}^Z)/U$
be the g-pair given by adjunction for
$h\circ g:(X,B,\mathbf {{M}})\rightarrow Z$
. Then,
$(Z,B_Z,\mathbf {{M}}^Z)/U$
is also the g-pair given by adjunction for
$h:(Y,B_Y,\mathbf {{M}}^Y)\rightarrow Z$
.
Lemma 2.9. Let
$f:X\rightarrow Z$
be a contraction between normal varieties over U. Let
$(X,B,\mathbf {{M}})/U$
and
$(X,\Gamma ,\mathbf {{N}})/U$
be two g-pairs on X which are glc over the generic point of Z. Assume that
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R } 0/Z$
and
$K_X+\Gamma +\mathbf {{N}}_X\sim _{\mathbb R } 0/Z$
. Let
$(Z,B_Z,\mathbf {{M}}^Z)/U$
and
$(Z,\Gamma _Z,\mathbf {{N}}^Z)/U$
be the g-pairs given by adjunction for
$(X,B,\mathbf {{M}})$
and
$(X,\Gamma ,\mathbf {{N}})$
over
$Z,$
respectively. Suppose that
$(X,B,\mathbf {{M}})$
has better singularities than
$(X,\Gamma ,\mathbf {{N}})$
, then
$(Z,B_Z,\mathbf {{M}}^Z)$
has better singularities than
$(Z,\Gamma _Z,\mathbf {{N}}^Z)$
(see Definition 2.6 for “better singularities”).
Proof. Let D be a prime divisor on some high resolution
$Z'\rightarrow Z$
. Let
$\pi :X'\rightarrow X$
be a high enough resolution such that the induced map
$f':X'\dashrightarrow Z'$
is a morphism. Write
$K_{X'}+B'+\mathbf {{M}}_{X'}$
(resp.
$K_{X'}+\Gamma '+\mathbf {{N}}_{X'}$
) for the pullback of
$K_X+B+\mathbf {{M}}_{X}$
(resp.
$K_X+\Gamma +\mathbf {{N}}_{X}$
). Denote by t (resp. s) the lc threshold of
$f^{\prime *} D$
with respect to
$(X',B',\mathbf {{M}})$
(resp.
$(X',\Gamma ',\mathbf {{N}})$
) over the generic point of D. It suffices to show
$t\geq s$
.
By construction,
$(X',\Gamma '+sf^{\prime *} D, \mathbf {{N}})$
is sub-glc over the generic point of D. Since
$(X,B,\mathbf {{M}})$
has better singularities than
$(X,\Gamma ,\mathbf {{N}})$
,
$(X',B'+sf^{\prime *} D,\mathbf {{M}})$
also has better singularities than
$(X',\Gamma '+sf^{\prime *} D, \mathbf {{N}})$
and it hence is sub-glc over the generic point of D. Therefore,
$t\geq s$
.
Lemma 2.10. Let
$f:X\rightarrow Z$
be a contraction of normal varieties over U. Let
$(X,B,\mathbf {{M}})/U$
and
$(X,\Gamma ,\mathbf {{N}})/U$
be two g-pairs on X which are glc over the generic point of Z. Assume that
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R } 0/Z$
and
$K_X+\Gamma +\mathbf {{N}}_X\sim _{\mathbb R } 0/Z$
. Let
$0\leq \alpha \leq 1$
be a real number and let
$$ \begin{align*}\Omega= \alpha B+(1-\alpha) \Gamma ~\text{ and }~ \mathbf{{L}}=\alpha \mathbf{{M}}+(1-\alpha) \mathbf{{N}}.\end{align*} $$
Let
$(Z,B_Z,\mathbf {{M}}^Z)/U$
,
$(Z,\Gamma _Z,\mathbf {{N}}^Z)/U,$
and
$(Z,\Omega _Z,\mathbf {{L}}^Z)/U$
be the g-pairs given by adjunction for
$(X,B,\mathbf {{M}})$
,
$(X,\Gamma ,\mathbf {{N}}),$
and
$(X,\Omega ,\mathbf {{L}})$
over
$Z,$
respectively. Then,
$(Z,\Omega _Z,\mathbf {{L}}^Z)$
has better singularities than
$$ \begin{align} (Z,\alpha B_Z+(1-\alpha)\Gamma_Z, \alpha \mathbf{{M}}^Z+(1-\alpha)\mathbf{{N}}^Z). \end{align} $$
See Definition 2.6 for “better singularities”.
Proof. Let
$Z'\rightarrow Z$
be any resolution and D be a prime divisor on
$Z'$
. Take a high enough resolution
$X'\rightarrow X$
such that the induced map
$h':X'\dashrightarrow Z'$
is a morphism. Let t (resp. s) be the lc threshold of
$h^{\prime *} D$
with respect to
$(X',B',\mathbf {{M}})$
(resp.
$(X',\Gamma ',\mathbf {{N}})$
) over the generic point of
$D,$
where
$K_{X'}+B'+\mathbf {{M}}_{X'}$
(resp.
$K_{X'}+\Gamma '+\mathbf {{N}}_{X'}$
) is the pullback of
$K_X+B+\mathbf {{M}}_X$
(resp.
$K_{X}+\Gamma +\mathbf {{N}}_{X}$
). By definition, the coefficient of D in
$B_{Z'}$
(resp.
$\Gamma _{Z'}$
) is
$1-t$
(resp.
$1-s$
), where
$K_{Z'}+B_{Z'}+\mathbf {{M}}^{Z}_{Z'}$
(resp.
$K_{Z'}+\Gamma _{Z'}+\mathbf {{N}}_{Z'}^Z$
) is the pullback of
$K_Z+B_Z+\mathbf {{M}}_Z^Z$
(resp.
$K_{Z}+\Gamma _Z+\mathbf {{M}}_{Z}^Z$
). Hence,
$a(D,Z,B_Z,\mathbf {{M}}^Z)=t$
and
$a(D,Z,\Gamma _Z,\mathbf {{N}}^Z)=s$
. So the log discrepancy of D with respect to the g-pair (2.1) is
$\alpha t+(1-\alpha )s$
.
Now,
$$ \begin{align*}(X',\alpha B'+(1-\alpha)\Gamma'+\alpha t h^{\prime *} D +(1-\alpha) s h^{\prime *} D, \alpha \mathbf{{M}}+(1-\alpha)\mathbf{{N}})\end{align*} $$
is glc over the generic point of D. Assuming that
$K_{X'}+\Omega '+\mathbf {{L}}_{X'}$
is the pullback of
${K_X+\Omega +\mathbf {{L}}_X}$
, we have
$\Omega '=\alpha B'+(1-\alpha )\Gamma '$
and
$\mathbf {{L}}=\alpha \mathbf {{M}}+(1-\alpha ) \mathbf {{N}}$
. Hence, the lc threshold of
$h^{\prime *} D$
with respect to
$(X',\Omega ',\mathbf {{L}})$
over the generic point of D is as least
$\alpha t+(1-\alpha )s$
. By definition, the coefficient of D in
$\Omega _{Z'}$
is at most
$1-\alpha t-(1-\alpha )s$
, where
$K_{Z'}+\Omega _{Z'}+\mathbf {{L}}^{Z}_{Z'}$
is the pullback of
$K_Z+\Omega _Z+\mathbf {{L}}^Z_Z$
. So
$$ \begin{align*}a(D,Z,\Omega_Z,\mathbf{{L}}^Z)\geq \alpha t+(1-\alpha)s\end{align*} $$
and the proof is completed.
2.5 Toric varieties and toric morphisms
We refer to [Reference Fulton18], [Reference Oda28] or [Reference Cox, Little and Schenck16] for the general theory of toric varieties. Here, we collect some definitions and facts on toric varieties. All toric varieties in this article are assumed to be normal.
A toric variety X is given by a pair
$(N_X,\Sigma _X)$
, where
$N_X$
is a lattice and
$\Sigma _X$
is a rational polyhedral fan in
$N_X\otimes \mathbb {R}$
. A toric morphism between toric varieties X and Y is given by a
$\mathbb Z$
-linear map
$\phi :N_X\rightarrow N_Y$
which is compatible with the fan
$\Sigma _X$
and
$\Sigma _Y$
, that is to say, for any cone
$\sigma _1\in \Sigma _X$
, there is a cone
$\sigma _2 \in \Sigma _Y$
such that
$\phi _{\mathbb R}(\sigma _1)\subset \sigma _2$
, where
$\phi _{\mathbb {R}}:N_X\otimes \mathbb {R}\rightarrow N_Y\otimes \mathbb {R}$
is the extension of
$\phi $
.
A toric divisor on a toric variety X is a divisor which is invariant under the torus action. We say a pair
$(X,B)$
is a toric pair if X is a toric variety and B is a toric
$\mathbb R$
-divisor.
There is a one-to-one correspondence between the cones
$\sigma $
in
$\Sigma _X$
and the torus orbits
$O(\sigma )$
in X. The dimension of the cone
$\sigma $
is equal to the codimension of the orbit
$O(\sigma )$
in X. In particular, a one-dimensional-cone
$\sigma $
, called a ray, corresponds to a toric prime divisor
$\overline {O(\sigma )}$
.
If
$\Delta $
is the toric boundary divisor of a toric variety X, that is,
$\Delta $
is the sum of all the toric prime divisors on X, then
$(X,\Delta )$
is lc and
$K_X+\Delta \sim 0$
. Moreover,
$a(D,X,\Delta )=0$
for any toric prime divisor D over X.
A toric variety X is
$\mathbb {Q}$
-factorial if and only if the fan
$\Sigma _X$
is simplicial, that is, every cone in
$\Sigma _X$
is generated by a set of
$\mathbb R$
-linear independent vectors.
If a toric morphism
$f:X\rightarrow Y$
given by
$\phi :N_X\rightarrow N_Y$
is a contraction, then
$\phi $
is surjective.
If
$f:X\rightarrow Z$
is a toric contraction, then X is of Fano type over Z.
Every toric varieties is a Mori dream space, that is to say, if
$X\to Z$
is a toric contraction, then we can run a MMP on any
$\mathbb R$
-Cartier
$\mathbb R$
-divisor D relatively over Z and it terminates with either a D-negative fibre space or a D-minimal model. Moreover, all the steps of the MMP are toric (see [Reference Matsuki25, Chapter 14] for the
$\mathbb Q$
-factorial case.
Lemma 2.11 [Reference Cox, Little and Schenck16, p. 133]
Let
$X,Z$
be two toric varieties given by
$(N_X,\Sigma _X)$
,
$(N_Z,\Sigma _Z),$
respectively, and
$f:X\to Z$
be a toric morphism given by a surjective
$\mathbb Z$
-linear map
${\phi :N_X\to N_Z}$
. Let F be a toric varieties given by
$(N_0,\Sigma _0),$
where
$N_0=\ker (\phi )$
and
$$ \begin{align*}\Sigma_0=\{\sigma\in \Sigma_X\mid \sigma\subset (N_0)_{\mathbb R}\}\end{align*} $$
is a sub-fan of
$\Sigma _X$
. Then,
$f^{-1}(T_Z)\simeq F\times T_Z$
, where
$T_Z$
is the torus in Z.
We also need the following lemma in [Reference Birkar and Chen11] regarding adjunction for toric pairs.
Lemma 2.12 [Reference Birkar and Chen11, Lemma 2.11]
Let
$f:X\rightarrow Z$
be a toric contraction between toric varieties and
$\Delta ,\Delta _Z$
be the toric boundary divisors of
$X,Z$
respectively. Then,
$(Z,\Delta _Z,\mathbf {0})$
is the g-pair given by adjunction for
$f:(X,\Delta )\rightarrow Z$
.
3 Proofs of main results
In this section, we will prove a more general form of Theorem 1.8 for generalized pairs as follows.
Theorem 3.1 (cf. [Reference Birkar and Chen11, Theorem 1.7])
Let r be a positive integer and
$0<\epsilon \leq 1$
be a real number. Assume:
-
(a)
$f\colon X\rightarrow Z$
is a toric contraction of relative dimension r between toric varieties over U with a codimension
$\geq 1$
point
$z\in Z$
; -
(b)
$(X,B,\mathbf {{M}})/U$
is a g-pair
$($
not necessarily toric
$)$
with
$\operatorname {\mathrm {mld}}(X/Z\ni z,B,\mathbf {{M}}) \geq \epsilon $
; -
(c)
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R}0/Z$
; and -
(d)
$\Delta $
is the toric boundary divisor of X.
Let
$$ \begin{align*}\Gamma^{\alpha}=\alpha B+(1-\alpha)\Delta ~ \text{ and } ~\mathbf{{N}}^{\alpha}=\alpha \mathbf{{M,}} \quad \text{ where } \alpha=1/r! \end{align*} $$
and let
$(Z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})/U$
be the g-pair on Z given by adjunction for
$f:(X,\Gamma ^{\alpha },\mathbf {{N}}^{\alpha })\rightarrow Z$
. Then,
$$ \begin{align*}\operatorname{\mathrm{mld}}(Z\ni z, \Gamma^{\alpha}_Z,\mathbf{{N}}^{\alpha,Z}) \geq \delta,\end{align*} $$
where
$\delta =\delta (r,\epsilon )$
as in (1.1).
We start with showing a generalized version of [Reference Chen14, Theorem 1.4], that is, showing that one can take
$\delta =\epsilon ^2/2$
in a generalized version of Shokurov conjecture in relative dimension one. Its proof is similar to that of [Reference Birkar and Chen11, Lemma 3.1].
Lemma 3.2 (cf. [Reference Birkar and Chen11, Lemma 3.1])
Let
$f:X\rightarrow Z$
be a contraction between normal varieties over U,
$(X,B,\mathbf {{M}})/U$
be a g-pair and
$z\in Z$
be a codimension
$\geq 1$
point such that:
-
•
$\dim X - \dim Z=1$
; -
•
$K_X+B+\mathbf {{M}}_X\sim _{\mathbb R} 0/Z$
; -
•
$\operatorname {\mathrm {mld}}(X/Z\ni z,B,\mathbf {{M}})\geq \epsilon $
, where
$0<\epsilon \leq 1$
; and -
• X is of Fano type over Z.
Let
$(Z,B_Z,\mathbf {{M}}^Z)/U$
be the g-pair given by adjunction for
$f:(X,B,\mathbf {{M}})\rightarrow Z$
. Then,
$$ \begin{align*} \operatorname{\mathrm{mld}}(Z\ni z, B_Z,\mathbf{{M}}^Z) \geq \delta(1,\epsilon)=\epsilon^2/2. \end{align*} $$
Proof. Since the singularities of
$(Z, B_Z,\mathbf {{M}}^Z)/U$
are independent of the choice of U, we may assume that
$U=Z$
. Shrinking Z around z, by Lemma 2.7, we may assume that
$(X,B,\mathbf {{M}})$
is glc. Let D be a prime divisor on some high resolution
$Z'\rightarrow Z$
with
${\operatorname {\mathrm {center}}_Z D=\overline {z}}$
. Let
$\pi :X'\rightarrow X$
be a high enough resolution such that
$\mathbf {{M}}$
descends to
$X'$
and the induced map
$f':X'\dashrightarrow Z'$
is a morphism. Write
$K_{X'}+B'+\mathbf {{M}}_{X'}$
for the pullback of
$K_X+B+\mathbf {{M}}_{X}$
. Then,
$(X',B')$
is sub-lc and
$\operatorname {\mathrm {mld}}(X'/Z\ni z, B')\geq \epsilon $
. Denote by t the lc threshold of
$f^{\prime *} D$
with respect to
$(X',B')$
over the generic point of D. It suffices to show that t is bounded from below by
$\epsilon ^2/2$
.
We may assume that X is
$\mathbb Q$
-factorial. Since X is of Fano type over Z, X is klt and
$-K_X$
is big over Z. So we can write
$$ \begin{align*}\pi^*(-K_X)\sim_{\mathbb Q} H'+C'/Z,\end{align*} $$
where
$H'$
is ample over Z and
$C'\geq 0$
. We can also write
$\pi ^*K_X=K_{X'}+E'$
. Then,
$E'\leq B'$
and
$(X',E')$
is sub-klt. For each sufficiently small real number
$u>0$
, let
$$ \begin{align*}B^{\prime}_u=(1-u)B'+uE',\end{align*} $$
then we have
$(X',B^{\prime }_u)$
is sub-klt and
$\operatorname {\mathrm {mld}}(X'/Z\ni z, B^{\prime }_u)\geq \epsilon $
. So we can find a general
$$ \begin{align*}0\leq L'\sim_{\mathbb R} (1-u)\mathbf{{M}}_{X'}+uH'/Z\end{align*} $$
(note that
$H'$
is ample
$/Z$
and
$\mathbf {{M}}_{X'}$
is nef
$/Z$
) such that letting
$$ \begin{align*}\Delta'=B^{\prime}_u+uC'+L',\end{align*} $$
we have
$\operatorname {\mathrm {mld}}(X'/Z\in z,\Delta ')\geq \epsilon '$
where
$\epsilon -\epsilon '>0$
is sufficiently small. Moreover, over Z we have
$$ \begin{align*} K_{X'}+\Delta'&\sim_{\mathbb R} K_{X'}+(1-u)B'+uE'+uC'+(1-u)\mathbf{{M}}_{X'}+uH'\\ &=(1-u)(K_{X'}+B'+\mathbf{{M}}_{X'})+u(K_{X'}+E')+u(H'+C')\\ &\sim_{\mathbb R} (1-u)(K_{X'}+B'+\mathbf{{M}}_{X'}) \sim_{\mathbb R} 0. \end{align*} $$
Therefore, letting
$\Delta =\pi _*\Delta '$
, we deduce that
$K_{X'}+\Delta '$
is the pullback of
$K_X+\Delta $
.
Now, if we choose
$u>0$
to be sufficiently small, the lc threshold s of
$f^{\prime *} D$
with respect to
$(X',\Delta ')$
over the generic point of D is sufficiently close to t. Applying [Reference Chen14, Theorem 1.4] to
$(X,\Delta )\to Z$
, we deduce that
$s\geq \epsilon ^{\prime 2} /2$
, where
$\epsilon -\epsilon '>0$
is sufficiently small. Hence,
$t\geq \epsilon ^2/2$
.
To prove Theorem 3.1, we need a couple of lemmas.
Lemma 3.3 (cf. [Reference Birkar and Chen11, Lemma 3.2])
Let
$0<\epsilon \leq 1$
be a real number and
$r,s,t$
be positive integers such that
$r=s+t$
. Suppose Theorem 3.1 holds in relative dimension s and in relative dimension t. Keep the assumptions
$(a)$
,
$(b)$
,
$(c),$
and
$(d)$
in Theorem 3.1. Additionally assume that
$f:X\rightarrow Z$
can be factored as
$X\xrightarrow {g} Y \xrightarrow {h} Z,$
where h and g are toric contractions of relative dimension s and
$t,$
respectively. Let
$$ \begin{align*}\Gamma^{\beta}=\beta B+(1-\beta)\Delta ~ \text{ and } ~\mathbf{{N}}^{\beta}=\beta \mathbf{{M,}} \quad \text{ where } \beta=1/(s!t!) \end{align*} $$
and let
$(Z, \Gamma ^{\beta }_Z,\mathbf {{N}}^{\beta ,Z})/U$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\beta },\mathbf {{N}}^{\beta })\rightarrow Z$
. Then,
$$ \begin{align*}\operatorname{\mathrm{mld}}(Z\ni z, \Gamma^{\beta}_Z,\mathbf{{N}}^{\beta,Z}) \geq \delta\big(t,\delta(s,\epsilon)\big)= \frac{\epsilon^{2^{s+t}}}{2^{2^{s+t}-1}\prod\limits_{i=1}^s i^{2^{i+t}}\cdot\prod\limits_{i=1}^t i^{2^i}}.\end{align*} $$
Proof. By assumption, Theorem 3.1 holds for both h and g in the following sense. Let
$$ \begin{align*}\Gamma^{\lambda}=\lambda B+(1-\lambda) \Delta ~\text{ and } ~\mathbf{{N}}^{\lambda}=\lambda \mathbf{{M,}} \quad \text{ where } \lambda=1/s!\end{align*} $$
and let
$(Y,\Gamma ^{\lambda }_Y,\mathbf {{N}}^{\lambda ,Y})/U$
be the g-pair given by adjunction for
$g:(X,\Gamma ^{\lambda },\mathbf {{N}}^{\lambda })\rightarrow Y$
. Then,
$$ \begin{align*}\operatorname{\mathrm{mld}}(Y/Z\ni z,\Gamma^{\lambda}_Y,\mathbf{{N}}^{\lambda,Y})\geq \delta(s,\epsilon).\end{align*} $$
On the other hand, let
$$ \begin{align*}\Omega_{Y}^\gamma=\gamma \Gamma^{\lambda}_Y+(1-\gamma)\Delta_Y ~\text{and}~ \mathbf{{L}}^{\gamma,Y}=\gamma \mathbf{{N}}^{\lambda,Y},\quad \text{where }\gamma=1/t!\end{align*} $$
and
$\Delta _Y$
is the toric boundary divisor of Y. Let
$(Z,\Omega _{Z}^\gamma ,\mathbf {{L}}^{\gamma ,Z})/U$
be the g-pair given by the adjunction for
$h:(Y,\Omega _Y^{\gamma },\mathbf {{L}}^{\gamma ,Y})\rightarrow Z$
. Then,
$$ \begin{align} \operatorname{\mathrm{mld}}(Z\ni z,\Omega_{Z}^\gamma,\mathbf{{L}}^{\gamma,Z})\geq \delta\big(t,\delta(s,\epsilon)\big). \end{align} $$
Now, let
$$ \begin{align*}\Gamma^{\beta}=\beta B+(1-\beta)\Delta ~ \text{ and } ~\mathbf{{N}}^{\beta}=\beta \mathbf{{M,}} \quad \text{ where } \beta=\lambda\gamma=1/(s!t!). \end{align*} $$
By construction, we have
$$ \begin{align*}\Gamma^{\beta}=\gamma\Gamma^{\lambda}+(1-\gamma)\Delta ~ \text{ and } ~ \mathbf{{N}}^{\beta}=\gamma \mathbf{{N}}^{\lambda}.\end{align*} $$
Let
$(Y, \Gamma ^{\beta }_Y,\mathbf {{N}}^{\beta ,Y})/U$
be the g-pair given by adjunction for
$g:(X,\Gamma ^{\beta },\mathbf {{N}}^{\beta })\rightarrow Y$
. and let
$(Z, \Gamma ^{\beta }_Z,\mathbf {{N}}^{\beta ,Z})$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\beta }+\mathbf {{N}}^{\beta })\rightarrow Z$
. By Lemma 2.8,
$(Z, \Gamma ^{\beta }_Z,\mathbf {{N}}^{\beta ,Z})$
is also the g-pair given by adjunction for
$h:(Y, \Gamma ^{\beta }_Y,\mathbf {{N}}^{\beta ,Y})\rightarrow Z$
.
Since
$$ \begin{align*}\Gamma^{\beta}=\gamma\Gamma^{\lambda}+(1-\gamma)\Delta,~\mathbf{{N}}^{\beta}=\gamma \mathbf{{N}}^{\lambda}\end{align*} $$
and
$$ \begin{align*}\Omega_{Y}^\gamma=\gamma \Gamma^{\lambda}_Y+(1-\gamma)\Delta_Y, ~ \mathbf{{L}}^{\gamma,Y}=\gamma \mathbf{{N}}^{\lambda,Y},\end{align*} $$
by Lemmas 2.10 and 2.12, the g-pair
$(Y, \Gamma ^{\beta }_Y,\mathbf {{N}}^{\beta ,Y})$
has better singularities than
$(Y,\Omega _Y^{\gamma },\mathbf {{L}}^{\gamma ,Y})$
, which implies that
$(Z, \Gamma ^{\beta }_Z,\mathbf {{N}}^{\beta ,Z})$
has better singularities than
$(Z,\Omega _{Z}^\gamma ,\mathbf {{L}}^{\gamma ,Z})$
by Lemma 2.9. So by (3.1), we have
$$ \begin{align*}\operatorname{\mathrm{mld}}(Z\ni z, \Gamma^{\beta}_Z,\mathbf{{N}}^{\beta,Z})\geq \delta\big(t,\delta(s,\epsilon)\big).\\[-37pt] \end{align*} $$
Lemma 3.4. Let F be a
$\mathbb Q$
-factorial complete toric variety given by
$(N,\Sigma )$
with
$\rho (F)=1$
and
$\dim F=r\geq 2$
. Then:
$(1)$
its fan
$\Sigma $
has exactly
$r+1$
rays generated by primitive elements
$v_i$
,
$i=1,\ldots ,r+1$
, and there exist positive integers
$q_i$
such that
$\sum _{i=1}^{r+1} q_iv_i=0$
;
$(2)$
let
$E_i$
,
$i=1,\ldots ,r+1$
, be the prime divisor over F corresponding to
$-v_i$
, then
$$ \begin{align*}a(E_i,F,0)=\frac{q_1+\cdots+\widehat{q_i}+\cdots +q_{r+1}}{q_i},\end{align*} $$
where the hat indicates that we omit that term;
$(3)$
let
$\pi :F'\to F$
be an extremal toric divisorial contraction with the exceptional divisor
$E_i$
for some i, then there exists a toric contraction
$g:F'\to G$
such that
$\dim G=r-1$
.
Remark 3.5. Since
$\rho (F')=2$
,
$\overline {\text {NE}}(F')$
has exactly two extremal rays. One corresponds to
$F'\to F$
and the other corresponds to
$F'\to G$
.
Proof. (1) This assertion was showed in the proof of [Reference Birkar and Chen11, Lemma 3.3]. We give another proof here. By [Reference Cox, Little and Schenck16, Proposition 6.4.1], for a
$\mathbb Q$
-factorial complete toric variety, the number of rays in its fan is equal to the sum of its Picard number and its dimension. As
$\rho (F)=1$
and
$\dim F=r$
,
$\Sigma $
has
$r+1$
rays, say
$R_1,\ldots R_{r+1}$
. Let
$v_i$
be the primitive element of the ray
$R_i$
for
$i=1,\ldots ,r+1$
.
As F is complete, the support of
$\Sigma $
(denoted by
$|\Sigma |$
) is
$N_{\mathbb R}$
, so
$v_1,\ldots ,v_{r+1}$
span
$N_{\mathbb R}$
. We may assume that
$v_1,\ldots ,v_r$
form a basis of
$N_{\mathbb Q}$
. Then, there exist rational numbers
$c_1,\ldots ,c_r$
such that
$v_{r+1}=\sum _{i=1}^r c_iv_i$
. We claim that all
$c_i$
are negative. Indeed, if one of them (say
$c_1$
) is non-negative, the support
$|\Sigma |$
is contained in the half space
$\{\sum _{i=1}^r a_iv_i\mid a_1\geq 0\}$
, which leads to a contradiction. We can find a positive integer q such that all
$qc_i$
are integers. Let
$q_i=-qc_i$
for
$i=1,\ldots r$
and let
$q_{i+1}=q$
. Then,
$\sum _{i=1}^{r+1} q_iv_i=0$
.
(2) Without loss of generality, we may suppose that
$i=r+1$
. Let
$\Delta _F$
be the toric boundary divisor of F, then
$a(D,F,\Delta _F)=0$
for any toric prime divisor D over F, which implies that
$a(D,F,0)$
is equal to the coefficient of D in the pullback of
$\Delta _F$
(denoted by
$\operatorname {\mathrm {mult}}_{D} \Delta _F$
).
Since
$-v_{r+1}$
is in the interior of the cone
$\sigma $
generated by
$v_1,\ldots ,v_{r}$
, the center of
$E_{r+1}$
is contained in the affine chart
$U_{\sigma }$
. On the chart
$U_{\sigma }$
,
$\Delta _F$
is determined by
$m\in N^*\otimes \mathbb Q$
with
$\langle m,v_i\rangle =1$
for
$i=1,\ldots ,r$
. Then,
$$ \begin{align*}\operatorname{\mathrm{mult}}_{E_{r+1}} \Delta_F=\langle m,-v_{r+1}\rangle =\frac{\langle m,q_1v_1+\cdots +q_rv_r\rangle}{q_{r+1}}=\frac{q_1+\cdots +q_r}{q_{r+1}}.\end{align*} $$
(3) Without loss of generality, we may suppose that
$i=r+1$
. The toric variety
$F'$
is given by
$(N,\Sigma ')$
, where
$\Sigma '$
is the star subdivision of
$\Sigma $
along
$-v_{r+1}$
, more precisely,
$$ \begin{align*}\Sigma'=(\Sigma\setminus \{\sigma\})\cup \Sigma^*(\sigma),\end{align*} $$
where
$\sigma $
is the cone generated by
$v_1,\ldots ,v_r$
and
$\Sigma ^*(\sigma )$
is the set of all cones generated by subsets of
$\{-v_{r+1},v_1,\ldots ,v_r\}$
not containing
$\{v_1,\ldots ,v_r\}$
.
Let
$\phi :N\to N/(\mathbb Z v_{r+1}):=N_G$
be the quotient map and let
$$ \begin{align*}\Sigma_{G}=\{\phi_{\mathbb R}(\tau)\subset (N_G)_{\mathbb R}\mid \tau \in \Sigma \text{ and } -v_{r+1}\in \tau\}.\end{align*} $$
Then,
$\Sigma _G$
is a fan in
$(N_{G})_{\mathbb R}$
( [Reference Cox, Little and Schenck16, Exercise 3.2.7]). Let G be the toric variety given by
$(N_G,\Sigma _G)$
. We claim that
$\phi $
is compatible with
$\Sigma $
and
$\Sigma _G$
, that is, for any
$\tau \in \Sigma $
,
$\phi _{\mathbb R}(\tau )$
is contained in some cone in
$\Sigma _G$
. Indeed, the claim holds obviously when
$-v_{r+1}\in \tau $
, so we may assume that
$-v_{r+1}\notin \tau $
. Then,
$\tau $
is generated by a subset S of
$\{v_1,\ldots ,v_{r+1}\}$
not containing
$\{v_1,\ldots ,v_r\}$
. Let
$\tau '$
be the cone generated by
$S',$
where
$$ \begin{align*}S' = \begin{cases} S\cup\{-v_{r+1}\}, & \text{if }v_{r+1}\notin S, \\ (S\setminus\{v_{r+1}\})\cup\{-v_{r+1}\}, & \text{if }v_{r+1}\in S. \end{cases} \end{align*} $$
Then,
$\phi _{\mathbb R}(\tau ')\in \Sigma _G$
and
$\phi _{\mathbb R}(\tau )=\phi _{\mathbb R}(\tau ')$
. Therefore, the claim holds and then
$\phi :N\to N_G$
determines a toric contraction from F to G.
Lemma 3.6 (cf. [Reference Birkar and Chen11, Lemma 3.4])
Let
$f:X\rightarrow Z$
be a toric Mori fiber space of relative dimension
$r\geq 2,$
where X is
$\mathbb Q$
-factorial. Then, there is a commutative diagram
such that:
-
•
$\pi ,h,g$
are toric contractions; -
•
$\pi :W\rightarrow X$
is an extremal toric divisorial contraction with the exceptional divisor E satisfying
$a(E,X,0)\leq r$
; and -
•
$\dim W-1=\dim Y>\dim Z$
.
Proof. By Lemma 2.11, over the torus
$T_Z$
in Z,
$f^{-1}(T_Z)$
is isomorphic to
$F\times T_Z$
, where F is a general fiber of f. Since
$f:X\to Z$
is a Mori fiber space, F is a Fano variety with
$\rho (F)=1$
. Moreover, F is
$\mathbb Q$
-factorial, as by Lemma 2.11 its fan
$\Sigma _F$
is a sub-fan of the fan
$\Sigma _X$
of X which is simplicial. By Lemma 3.4 (1), the fan
$\Sigma _F$
has exactly
$r+1$
rays generated by primitive elements
$v_i$
,
$i=1,\ldots ,r+1$
, and there exist positive integers
$q_i$
such that
$\sum _{i=1}^{r+1} q_iv_i=0$
. Pick e such that
$q_e\geq q_i$
for any
$i=1,\ldots ,r+1$
and denote by
$E_F$
the toric prime divisor over F corresponding to
$-v_e$
. Extracting
$E_F$
gives an extremal contraction
$F'\to F$
. By Lemma 3.4 (2), there is a toric contraction
$F'\to G$
with
$\dim G=r-1$
.
The closure E of the exceptional divisor
$E_F\times T_Z$
of
$F'\times T_Z\to F\times T_Z$
is a toric divisor over X, so it determines an extremal toric divisorial contraction
$\pi :W\rightarrow X$
with the exceptional divisor E. Then,
$\rho (W/Z)=2$
. Over
$T_Z$
, the two contractions
$W\to X$
and
$F'\times T_Z\to F\times T_Z$
coincides. By Lemma 3.4 (2), we have
$$ \begin{align*}a(E,X,0)=a(E_F\times T_Z,F\times T_Z,0)\leq r.\end{align*} $$
Let
$g:W\to Y$
be a
$(-E)$
-negative toric extremal contraction over Z. Then,
$\rho (Y/Z)=1$
. Over
$T_Z$
, the restriction
$g|_{T_Z}:F'\times T_Z\to Y|_{T_Z}$
is either an isomorphism or a
$(-E_F\times T_Z)$
-negative toric extremal contraction over
$T_Z$
. But the former case is impossible because
$\rho (F')=2$
and
$\rho (Y/Z)=1$
. Note that
$\overline {\text {NE}}(F'\times T_Z/T_Z)$
has exactly two extremal rays. One corresponds to
$F'\times T_Z\to F\times T_Z,$
and the other corresponds to
$F'\times T_Z\to G\times T_Z$
. So
$g|_{T_Z}$
coincides with one of them. It is impossible that
$g|_{T_Z}$
coincides with
$F'\times T_Z\to F\times T_Z$
because
$-E_F\times T_Z$
is ample over
$F\times T_Z$
. So
$g|_{T_Z}$
coincides with
$F'\times T_Z\to G\times T_Z$
, which implies that
$\dim Y=\dim W-1$
.
Lemma 3.7 (cf. [Reference Birkar and Chen11, Lemma 3.4])
Assume that Theorem 3.1 holds in relative dimension
$\leq r-1$
. Then, Theorem 3.1 holds in relative dimension r when
$f:X\rightarrow Z$
is a toric Mori fiber space and X is
$\mathbb Q$
-factorial.
Proof. By Lemma 3.2, we may suppose that the relative dimension
$r\geq 2$
. By taking a toric
$\mathbb Q$
-factorialisation, we can assume X is
$\mathbb Q$
-factorial. By Lemma 3.6, there is a commutative diagram
satisfying the properties listed in that lemma. Let
$\Delta _W$
be the the toric boundary divisor of W, then
$K_W+\Delta _W=\pi ^*(K_X+\Delta )$
. Write
$K_W+B_W+\mathbf {{M}}_W=\pi ^*(K_X+B+\mathbf {{M}}_X)$
. Let
$$ \begin{align*}\Gamma^{\theta}_W=\theta B_W+(1-\theta) \Delta_W ~\text{ and }~ \mathbf{{N}}^{\theta}=\theta \mathbf{{M}}, \quad \text{ where } \theta=1/r.\end{align*} $$
Since
$a(E,X,0)\leq r$
, the coefficient of E in
$B_W$
is bounded below by
$1-r$
. Then,
$\Gamma ^{\theta }_W\geq 0$
since the coefficient of E in
$\Delta _W$
is 1.
By construction,
$\operatorname {\mathrm {mld}}(W/Z\ni z,\Gamma ^{\theta }_W, \mathbf {{N}}^{\theta })\geq \frac {\epsilon }{r}$
. Applying Lemma 3.3 to
$(W,\Gamma ^{\theta }_W, \mathbf {{N}}^{\theta })$
over Z (taking
$s=1$
and
$t=r-1$
in the lemma), we deduce that if we let
$$ \begin{align*}\Omega_W^{\beta}=\beta \Gamma^{\theta}_W +(1-\beta)\Delta_W ~\text{ and }~ \mathbf{{L}}^{\beta}=\beta \mathbf{{N}}^{\theta}, \quad \text{ where } \beta=1/(r-1)!,\end{align*} $$
and
$(Z,\Omega _Z^{\beta },\mathbf {{L}}^{\beta ,Z})/U$
be the g-pair given by adjunction for
$h\circ g: (W,\Omega _W^{\beta },\mathbf {{L}}^{\beta })\rightarrow Z$
, then
$$ \begin{align*}\operatorname{\mathrm{mld}}(Z\ni z,\Omega_Z^{\beta},\mathbf{{L}}^{\beta,Z})\geq \frac{(\epsilon/r)^{2^r}}{2^{2^{r}-1}\cdot\prod\limits_{i=1}^{r-1} i^{2^i}}=\delta(r,\epsilon).\end{align*} $$
It is easy check that
$$ \begin{align*}\Omega_W^{\beta}=\alpha B_W +(1-\alpha)\Delta_W ~\text{ and }~ \mathbf{{L}}^{\beta}=\alpha \mathbf{{M,}} \quad \text{ where } \alpha=\theta\beta=1/r!.\end{align*} $$
Hence,
$$ \begin{align*}K_W+\Omega_W^{\beta}+\mathbf{{L}}^{\beta}_W=\pi^*(K_X+\Gamma^{\alpha}+\mathbf{{N}}^{\alpha}_X),\end{align*} $$
where
$$ \begin{align*}\Gamma^{\alpha}=\alpha B+(1-\alpha)\Delta ~ \text{ and } ~\mathbf{{N}}^{\alpha}=\alpha \mathbf{{M}}.\end{align*} $$
Therefore,
$(Z,\Omega _Z^{\beta },\mathbf {{L}}^{\beta ,Z})/U$
is also the g-pair given by adjunction for
$f:(X,\Gamma ^{\alpha },\mathbf {{N}}^{\alpha })\rightarrow Z$
. This finishes the proof of the lemma.
Proof of Theorem 3.1
By induction on relative dimension, we may assume that the theorem holds in relative dimension
$\leq r-1$
. Taking a toric
$\mathbb Q$
-factorization of X and running an MMP on
$K_X$
over Z, we may assume that X is
$\mathbb Q$
-factorial and it has a toric Mori fiber space structure
$X\rightarrow Y/Z$
.
If
$Y\rightarrow Z$
is birational, we can replace Z by Y, then we are done by Lemma 3.7. Otherwise
$\dim Y>\dim Z$
. Denote
$s=\dim X-\dim Y$
and
$t=\dim Y-\dim Z$
, then
$r=s+t$
. Applying Lemma 3.3, we deduce that if we let
$$ \begin{align*}\Gamma^{\beta}=\beta B+(1-\beta)\Delta ~ \text{ and } ~\mathbf{{N}}^{\beta}=\beta \mathbf{{M,}} \quad \text{ where } \beta=1/(s!t!) \end{align*} $$
and
$(Z, \Gamma ^{\beta }_Z,\mathbf {{N}}^{\beta ,Z})/U$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\beta },\mathbf {{N}}^{\beta })\rightarrow Z$
, then
$$ \begin{align} \operatorname{\mathrm{mld}}(Z\ni z, \Gamma^{\beta}_Z,\mathbf{{N}}^{\beta,Z}) \geq \frac{\epsilon^{2^{r}}}{2^{2^{r}-1}\prod\limits_{i=1}^s i^{2^{i+t}}\cdot\prod\limits_{i=1}^t i^{2^i}}. \end{align} $$
Let
$$ \begin{align*}\Gamma^{\alpha}=\alpha B+(1-\alpha)\Delta ~ \text{ and } ~\mathbf{{N}}^{\alpha}=\alpha \mathbf{{M,}} \quad \text{ where } \alpha=1/r!. \end{align*} $$
Then, we have
$$ \begin{align*}\Gamma^{\alpha}=\theta \Gamma^{\beta}+(1-\theta)\Delta ~ \text{ and } ~\mathbf{{N}}^{\alpha}=\theta \mathbf{{N}}^{\beta}, \quad \text{ where } \theta=\alpha/\beta=(s!t!)/r!. \end{align*} $$
Let
$(Z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\alpha },\mathbf {{N}}^{\alpha })\rightarrow Z$
. By Lemmas 2.10 and 2.12,
$(Z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})$
has better singularities than
$$ \begin{align*}(Z, \theta \Gamma^{\beta}_Z +(1-\theta) \Delta_Z,\theta\mathbf{{N}}^{\beta,Z}).\end{align*} $$
Hence, by (3.2), we have
$$ \begin{align*} \operatorname{\mathrm{mld}}(Z\ni z, \Gamma^{\alpha}_Z,\mathbf{{N}}^{\alpha,Z})&\geq \frac{s!t!}{r!}\cdot\frac{\epsilon^{2^{r}}}{2^{2^{r}-1}\prod\limits_{i=1}^s i^{2^{i+t}}\cdot\prod\limits_{i=1}^t i^{2^i}}\\ & \geq \frac{\epsilon^{2^r}}{2^{2^r-1}\prod\limits_{i=1}^r i^{2^i}}=\delta(r,\epsilon).\\[-57pt] \end{align*} $$
Proof of Theorem 1.9
By Lemma 2.7, shrinking Z around z, we may suppose that
$(X,B)$
is lc. Since
$(X,B)$
is a toric lc pair, we have
$B\leq \Delta $
, where
$\Delta $
is the toric boundary divisor of X. Let
$$ \begin{align*}\Gamma^{\alpha}=\alpha B^++(1-\alpha)\Delta, \quad \text{ where } \alpha=1/r!. \end{align*} $$
Then,
$\Gamma ^{\alpha }\geq B$
. Let
$(Z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\alpha })\rightarrow Z$
. By Theorem 1.8,
$\operatorname {\mathrm {mld}}(Z\ni z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})\geq \delta =\delta (r,\epsilon )$
. Denote the prime divisor
$\overline {z}$
by D. Then, the coefficient of D in
$\Gamma _Z^{\alpha }$
is bounded from above by
$1-\delta $
. This means that
$(X,\Gamma ^{\alpha }+\delta f^*D)$
is lc over the generic point of D. Since
$\Gamma ^{\alpha }\geq B$
, we deduce that
$(X,B+\delta f^*D)$
is lc over the generic point of D.
Proof of Theorem 1.5
Let
$$ \begin{align*}\Gamma^{\alpha}=\alpha B+(1-\alpha)\Delta, \quad \text{ where } \alpha=1/r! \end{align*} $$
and let
$(Z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})$
be the g-pair given by adjunction for
$f:(X,\Gamma ^{\alpha })\rightarrow Z$
. By Theorem 1.8,
$\operatorname {\mathrm {mld}}(Z\ni z, \Gamma ^{\alpha }_Z,\mathbf {{N}}^{\alpha ,Z})\geq \delta (r,\epsilon )$
. Hence,
$\operatorname {\mathrm {mld}}(Z\ni z, 0)\geq \delta (r,\epsilon )$
and the first assertion holds.
The second assertion is an immediate consequence of Theorem 1.9 (taking
$B=0$
in Theorem 1.9).
Acknowledgements
I would like to thank Caucher Birkar for his valuable comments and constant support. I would also like to thank Yifei Chen and Yu Zou for their helpful comments. I am grateful to Florin Ambro for sharing with me his result (Remark 1.10) and for a lot of useful discussions.
Funding statement
I am supported by the start-up fund from Sun Yat-sen University.
