Throughout this paper, we work over the complex number field ℂ.
1 Introduction
A family of projective pairs of dimension d over a smooth curve (possibly non-proper) is an object
$$ \begin{align*}f:(X,\Delta)\rightarrow C,\end{align*} $$
consisting of a morphism of schemes
$f:X\rightarrow C$
and an effective
$\mathbb {Q}$
-divisor
$\Delta $
satisfying the following properties,
-
• f is projective, flat, of finite type, of relative dimension d, with reduced fibers,
-
• every irreducible component
$D_i\subset \mathrm {Supp} (\Delta )$
dominates C, and all fibers of
$\mathrm {Supp} (D)$
have pure dimension
$d-1$
, and -
• f is smooth at generic points of
$X_s\cap \mathrm {Supp} (D)$
for every
$s\in C$
, where
$X_s:=f^{-1}(s)$
denotes the fiber over s.
Usually, fibers of a family of projective pairs behave wildly. [Reference Kollár13, 1.43] gives several examples of families of projective surfaces whose special fibers are canonically polarized and general fibers are even not of general type. [Reference Kollár13] also shows that such jumps of Kodaira dimension happen when the canonical class of the total space is not
$\mathbb {Q}$
-Cartier. Thus, it is natural to consider the case when
$K_X+\Delta $
is
$\mathbb {Q}$
-Cartier. And in this case,
$K_{X_s}+\Delta _s=(K_X+\Delta )|_{X_s}$
is also
$\mathbb {Q}$
-Cartier for all closed point
$s\in C$
according to the adjunction formula.
In general, a family of projective pairs
$f:(X,\Delta )\rightarrow C$
over a smooth curve such that
$K_X+\Delta $
is
$\mathbb {Q}$
-Cartier and
$(X_s,\Delta _s)$
is an slc pair for every closed point
$s\in C$
is called locally stable. The notion of locally stable morphisms has been verified to be a very important definition in the moduli of varieties and satisfies many nice properties – for example, the plurigenera are constant; see [Reference Kollár13, Theorem 5.11] (see Definition 2.1 for the definition of slc pairs).
For a family of projective pairs
$f:(X,\Delta )\rightarrow C$
which is locally stable over
$0\in C$
, we call
$(X_0,\Delta _0)$
an slc degeneration of
$\{(X_s,\Delta _s),s\neq 0\}$
. In this paper, we study the birational boundedness of slc degenerations of polarized log Calabi–Yau fibrations. The boundedness of polarized log Calabi–Yau pairs is studied in [Reference Birkar6]. The first result shows that for a family of projective pairs over a curve, suppose the log canonical divisor of the total space is relatively trivial, general fibers are in a fixed bounded family of polarized log Calabi–Yau pairs and the special fiber is an slc degeneration. Then every irreducible component of the slc degeneration is bounded up to birationally equivalence (see Definition 2.4 for the definition of boundedness).
Theorem 1.1. Fix a natural number d and positive rational numbers
$c,v$
. Let X be a quasi-projective normal variety,
$f:(X,\Delta )\rightarrow C$
a family of projective pairs of dimension d over a smooth curve C and
$0\in C$
a closed point. Suppose
-
•
$K_X+\Delta \sim _{\mathbb {Q},C} 0$
, -
•
$(X_0,\Delta _0)$
is an slc pair, and -
• there is a divisor N on X such that a general fiber
$(X_g,\Delta _g),N_g$
is a
$(d,c,v)$
-polarized log Calabi–Yau pair (see Definition 4.1).
Then every irreducible component of
$X_0$
is birationally bounded.
Note
$K_X+\Delta \sim _{\mathbb {Q},C} 0$
implies
$K_X+\Delta $
is
$\mathbb {Q}$
-Cartier. Because the discrepancy is a lower semi-continuous function (see [Reference Kollár12, Corollary 4.10]), then
$(X_0,\Delta _0)$
is an slc pair implies
$(X_s,\Delta _s)$
is an slc pair for every s in an open neighborhood of
$0$
, and it means
$(X,\Delta )\rightarrow C$
is locally stable over an open neighborhood of
$0$
.
Fano varieties naturally have polarized log Calabi–Yau structures. The following corollary is an application of Theorem 1.1 to families of log Fano pairs.
Corollary 1.2. Fix a natural number d and positive rational numbers
$c,\epsilon $
. Let X be a normal quasi-projective variety,
$f:(X,\Delta )\rightarrow C$
a family of projective pairs of dimension d over a smooth curve C and
$0\in C$
a closed point. Suppose
-
•
$-(K_X+\Delta )$
is ample over C, -
•
$(X_0,\Delta _0)$
is an slc pair, -
• a general fiber
$(X_g,\Delta _g)$
is
$\epsilon $
-lc, and -
•
$\mathrm {coeff}(\Delta )\subset c \mathbb {N}$
.
Then every irreducible component of
$X_0$
is birationally bounded.
Note that in Theorem 1.1, we only assume the existence of polarization on general fibers, the slc degeneration has no assumption on positivity, and hence, it does not have a polarized log Calabi–Yau pair structure. Example 2.5 shows that boundedness up to birational equivalence is the best result one can hope for. Example 2.6 shows that the number of irreducible components of the slc degeneration cannot be bounded either.
After the paper has been completed, Birkar informed the author that he and Santai Qu [Reference Birkar and Qu8] obtained Theorem 1.1 and Corollary 1.2 independently.
Sketch of the proof of Theorem 1.1
. The main tools used in this paper are the moduli space of polarized log Calabi–Yau pairs [Reference Birkar6] and the weak semistable reduction [Reference Abramovich and Karu1]; see also [Reference Adiprasito, Liu and Temkin2]. With the same notation as in Theorem 1.1, because a general fiber
$(X_g,\Delta _g), N_g$
is a
$(d,c,v)$
-polarized log Calabi–Yau pair, there exists a moduli map
$C\setminus 0\rightarrow \mathcal {S}$
, where
$\mathcal {S}$
is the moduli space of
$(d,c,v)$
-polarized log Calabi–Yau pairs. Because
$\mathcal {S}$
is proper, the moduli map extends to a morphism
$C\rightarrow \mathcal {S}$
, and after a finite cover, it will define a new fibration
$(X',\Delta '), N'\rightarrow C$
whose fibers are
$(d,c,v)$
-polarized log Calabi–Yau pairs. In particular,
$(X^{\prime }_0,\Delta ^{\prime }_0)$
is log bounded. Because
$(X,\Delta )\rightarrow C$
and
$(X',\Delta ')\rightarrow C$
are both log Calabi–Yau fibrations over C with the same generic fiber, then they are crepant birationally equivalent over C. Therefore, any irreducible component of
$X_0$
is an lc place of
$(X',\Delta ')$
. Note that an lc center of
$(X',\Delta ')$
contained in
$X^{\prime }_0$
is also an lc center of
$(X^{\prime }_0,\Delta ^{\prime }_0)$
by adjunction which is in a bounded family. The main difficulty is to use the boundedness of lc centers to prove the birational boundedness of lc places since the contraction from an exceptional divisor to its image can not be controlled. We use weak semistable reduction to make singularities toric, and for toric cases, such contraction is well understood according to [Reference Hu, Liu and Yau10].
2 Preliminary
2.1 Notations and basic definition
We will use the same notation as in [Reference Kollár and Mori15] and [Reference Lazarsfeld17].
A sub-log pair
$(X,\Delta )$
consists of a normal quasi-projective variety X and a
$\mathbb {Q}$
-divisor
$\Delta $
such that
$K_X+\Delta $
is
$\mathbb {Q}$
-Cartier. We call
$(X,\Delta )$
a log pair if in addition,
$\Delta $
is effective. If
$g: Y\rightarrow X$
is a birational morphism and E is a divisor on Y, the discrepancy
$a(E,X,\Delta )$
is
$-\mathrm {coeff}_{E}(\Delta _Y)$
, where
$K_Y+\Delta _Y :=g^*(K_X+\Delta ) $
. A sub-log pair
$(X,\Delta )$
is called sub-klt (respectively sub-lc) if for every birational morphism
$Y\rightarrow X$
as above,
$a(E,X,\Delta )>-1$
(respectively
$\geq -1$
) for every divisor E on Y. A log pair
$(X,\Delta )$
is called klt (respectively lc) if
$(X,\Delta )$
is sub-klt (respectively sub-lc) and
$(X,\Delta )$
is a log pair.
Let
$(Y,\Delta _Y),(X,\Delta )$
be two sub-log pairs and
$h:Y\rightarrow X$
a projective birational morphism. We say
$(Y,\Delta _Y)\rightarrow (X,\Delta )$
is a crepant birational morphism if
$K_Y+\Delta _Y\sim _{\mathbb {Q}}h^*(K_X+\Delta )$
, two sub-log pairs
$(X_i,\Delta _i),i=1,2$
are crepant birationally equivalent if there is a sub-log pair
$(Y,\Delta _Y)$
and two crepant birational morphisms
$(Y,\Delta _Y)\rightarrow (X_i,\Delta _i),i=1,2$
.
Let
$(X,\Delta )$
be a sub-log pair. We say a divisor P over X is a log place of
$(X,\Delta )$
if the discrepancy
$a(P,X,\Delta )<0$
. A closed subvariety
$W\subset X$
is called a log center of
$(X,\Delta )$
if W is the image of a log place of
$(X,\Delta )$
on X. In particular, a log place P of a sub-log pair
$(X,\Delta )$
such that
$a(P,X,\Delta )\leq -1$
is called a nklt place, respectively, a nklt center is the image of a nklt place. When
$(X,\Delta )$
is sub-lc, a nklt place (respectively, a nklt place) is also called an lc place (respectively, an lc center).
A contraction is a projective morphism
$f:X\rightarrow Z$
of quasi-projective varieties with
$f_*\mathcal {O}_X=\mathcal {O}_Z$
. If X is normal, then so is Z, and the fibers of f are connected. A fibration is a contraction
$f:X\rightarrow Z$
of normal quasi-projective varieties with
$\mathrm {dim}X>\mathrm {dim}Z$
.
For a scheme X, a stratification of X is a disjoint union
$\coprod _i X_i$
of finitely many locally closed subschemes
$X_i\hookrightarrow X$
such that the corresponding morphism
$\coprod _i X_i\rightarrow X$
is both monomorphism and surjective.
Definition 2.1. A semi-pair
$(X,\Delta )$
consists of a reduced quasi-projective scheme of pure dimension and a
$\mathbb {Q}$
-divisor
$\Delta \geq 0$
on X satisfying the following conditions:
-
• X is
$S_2$
with nodal codimension one singularities, -
• no component of
$\mathrm {Supp} (\Delta )$
is contained in the singular locus of X, and -
•
$K_{X}+\Delta $
is
$\mathbb {Q}$
-Cartier.
We say that
$(X,\Delta )$
is semi-log canonical (slc) if, in addition, we have
-
• if
$\pi :X^\nu \rightarrow X$
is the normalization of X and
$\Delta ^\nu $
is the sum of the birational transform of
$\Delta $
and the conductor divisor of
$\pi $
, then every irreducible component of
$(X^\nu ,\Delta ^\nu )$
is lc. We call
$(X^\nu ,\Delta ^\nu )$
the normalization of
$(X,\Delta )$
.
For an slc pair
$(X,\Delta )$
with normalization
$(X^\nu ,\Delta ^\nu )$
, we say a divisor P over X is a log place (respectively, an lc place) of
$(X,\Delta )$
if P is a log place (respectively, an lc place) of an irreducible component of
$(X^\nu ,\Delta ^\nu )$
, and the image of P on X is called a log center (respectively, an lc center) of
$(X,\Delta )$
.
The following is the definition of locally stable morphisms defined in [Reference Kollár13, Chapter 4]. In this paper, we only work on the case when the base is smooth, and in this case, the definition of locally stable morphisms is much more simple; see Lemma 2.3.
Definition 2.2. Let S be a reduced scheme and n a natural number. A projective family of pairs of dimension n over S is an object
$$ \begin{align*}f:(X,\Delta)\rightarrow S,\end{align*} $$
consisting of a morphism of schemes
$f:X\rightarrow S$
and an effective
$\mathbb {Q}$
-divisor
$\Delta $
satisfying the following properties:
-
• f is projective, flat, of finite type, of pure relative dimension n, with geometrically reduced fibers,
-
• every irreducible component
$D_i\subset \mathrm {Supp}(\Delta )$
dominates an irreducible component of S, and all nonempty fibers of
$\mathrm {Supp}(\Delta )\rightarrow S$
have pure dimension
$n-1$
. In particular,
$\mathrm {Supp}(\Delta )$
does not contain any irreducible component of any fiber of f, and -
• the morphism f is smooth at generic points of
$X_s\cap \mathrm {Supp}(D)$
for every
$s\in S$
.
We say a projective family of pairs
$f:(X,\Delta )\rightarrow S$
is well-defined if further,
-
• there exists an open subset
$U\subset X$
such that-
– codimension of
$X_s\setminus U_s$
is
$\geq 2$
for every
$s\in S$
, and -
–
$\Delta |_U$
is
$\mathbb {Q}$
-Cartier.
-
Let
$f:(X,\Delta )\rightarrow S$
be a well-defined projective family of pairs over a reduced scheme S. We say f is locally stable if it satisfies the following conditions:
-
•
$K_{X/S}+\Delta $
is
$\mathbb {Q}$
-Cartier, and -
•
$(X_s,\Delta _s)$
is an slc pair for every
$s\in S$
.
We say f is stable if further,
-
•
$K_{X/S}+\Delta $
is ample over S.
According to [Reference Kollár13, Theorem-Definition 4.3], when S is normal, a family of projective family of pairs is naturally well-defined.
Lemma 2.3 [Reference Kollár13, Corollary 4.55].
Let S be a smooth scheme and
$f:(X,\Delta )\rightarrow S$
a morphism. Then f is locally stable if and only if the pair
$(X,\Delta +f^*D)$
is slc for every snc divisor
$D\subset S$
.
Definition 2.4. We say that a set
$\mathscr {X}$
of varieties is bounded (respectively, birationally bounded) if there is a projective morphism
$\mathcal {W}\rightarrow \mathcal {T}$
, where
$\mathcal {T}$
is of finite type, such that for every
$X\in \mathscr {X}$
, there is a closed point
$t\in \mathcal {T}$
and an isomorphism
$\mathcal {W}_t\rightarrow X$
(respectively, a birational map
$\mathcal {W}_t\dashrightarrow X$
).
Example 2.5. [Reference Hacking and Prokhorov11, Corollary 1.2] shows that for any
$(\alpha ,\beta ,\gamma )\in \mathbb {N}^3$
such that
$\alpha ^2+\beta ^2+\gamma ^2=3\alpha \beta \gamma $
, there exists a morphism
$X\rightarrow C$
over a smooth curve germ
$0\in C$
, such that
$K_X\sim _{\mathbb {Q},T}0$
,
$X_t$
is isomorphic to
$ \mathbb {P}^2$
for
$t\neq 0$
and
$X_0$
is isomorphic to the weighted projective space
$\mathbb {P}(\alpha ,\beta ,\gamma )$
. Write
$U:=C\setminus 0$
and
$X_U:=X\times _C U$
. Because
$-K_{X_U}$
is very ample over U, let
$\Delta_U,N_U\in |-K_{X_U/U}|$
be two general elements. Then a general fiber of
$(X_U,\Delta _U),N_U\rightarrow U$
is a
$(2,1,9)$
-polarized log Calabi–Yau pair. Since the function
$\alpha ^2+\beta ^2+\gamma ^2=3\alpha \beta \gamma $
has infinitely many positive integer solutions, the special fiber is not bounded, while they are all birationally equivalent to
$\mathbb {P}^2$
.
Example 2.6. Let
$f:(X,\Delta )\rightarrow C$
be a family of pairs over a smooth curve satisfying the assumptions in Theorem 1.1. Suppose
$X_0$
has more than two irreducible components. Note that every irreducible component of
$X_0$
is an lc center of
$(X,\Delta +X_0)$
. Then
$(X,\Delta +X_0)$
has infinitely many lc places over
$0$
.
Fix a positive integer
$m\gg 0$
. Suppose
$(Y,\Delta _Y)\rightarrow (X,\Delta +X_0)$
is a crepant birational morphism that only extracts at least m lc places of
$(X,\Delta +X_0)$
over c. Denote the morphism
$Y\rightarrow C$
by
$f_Y$
. Then
$\Delta _Y$
is equal to the strict transform of
$\Delta $
plus
$\mathrm {red}(f_Y^*0)$
. Suppose l is the least common multiple of the set of coefficients of
$f_Y^*0$
. Let
$\pi :C'\rightarrow C$
be a ramified cover whose ramified index along
$0'$
is l, where
$0'$
is a closed point of
$\pi ^{-1}(0)$
. Let
$Y'$
be the normalization of
$Y\times _C C'$
. Then by [Reference Kollár13, Lemma 2.53],
$f_{Y'}:Y'\rightarrow C'$
has reduced fibers.
Denote the morphism
$Y'\rightarrow Y$
by
$\pi _Y$
. By the Hurwitz’s formula, there is a
$\mathbb {Q}$
-divisor
$\Delta ^{\prime }_Y$
on
$Y'$
such that
$K_{Y'}+\Delta ^{\prime }_Y\sim _{\mathbb {Q}} \pi _Y^*(K_Y+\Delta _Y)$
. Because
$\Delta _Y\geq \mathrm {red}(f_Y^*0)$
, by [Reference Kollár12, 2.42],
$\Delta ^{\prime }_Y\geq \mathrm {red}(f_{Y'}^*0')=f_{Y'}^*0'$
. Then by the definition,
$(Y',\Delta ^{\prime }_Y-f_{Y'}^*0')\rightarrow C'$
is a locally stable morphism satisfying the assumptions in Theorem 1.1, and
$Y^{\prime }_0$
has at least m irreducible components.
3 Almost semistable reduction and toroidal embedding
3.1 Toric varieties
Let
$N',N\cong \mathbb {Z}^n$
be lattices,
$\Sigma ',\Sigma $
be fans in
$N'$
, N respectively. A map between fans, in notation
$\psi :\Sigma '\rightarrow \Sigma $
, is a homomorphism
$\psi :N'\rightarrow N$
of lattices that satisfies the condition: For each
$\sigma '\in \Sigma '$
, there exists a
$\sigma \in \Sigma $
such that
$\psi (\sigma ')\subset \sigma $
. Such
$\psi $
determines a morphism
$\tilde {\psi }:X_{\Sigma '}\rightarrow X_\Sigma $
. A morphism between toric varieties that arises in this way is called a toric morphism.
Let
$\Sigma ^{\prime }_\sigma $
be the set of cones in
$\Sigma '$
whose interior is mapped to the interior of
$\sigma \in \Sigma $
. Pick
$\sigma '\in \Sigma ^{\prime }_\sigma $
. The image
$\psi (N'/ N^{\prime }_{\sigma '})$
in
$N/N_\sigma $
is independent of the choice of
$\sigma '$
in
$\Sigma ^{\prime }_\sigma $
. We define the index
$[N/N_\sigma :\psi (N'/N^{\prime }_\sigma )]$
to be the index of
$\tilde {\psi }$
over
$O_\sigma $
, and denote it by
$Ind(\sigma )$
.
Let
$\tau '\in \Sigma ^{\prime }_\sigma $
and
$\{\sigma ^{\prime }_1,\sigma ^{\prime }_2,...\}$
be the set of cones in
$\Sigma ^{\prime }_\sigma $
that contains
$\tau '$
as a face. Then each
$\sigma ^{\prime }_i$
determines a cone
$\bar {\sigma ^{\prime }_i}$
in
$\psi ^{-1}((N_\sigma )_{\mathbb {R}})/(N^{\prime }_{\tau '})_{\mathbb {R}}$
, defined by
$$ \begin{align*}\bar{\sigma^{\prime}_i}=(\sigma^{\prime}_i+(N^{\prime}_{\tau'})_{\mathbb{R}})/(N^{\prime}_{\tau'})_{\mathbb{R}}.\end{align*} $$
Note that
$\sigma ^{\prime }_i+(N^{\prime }_{\tau '})_{\mathbb {R}}$
is contained in
$\psi ^{-1}(N_{\sigma })_{\mathbb {R}}$
since
$\tau ',\sigma ^{\prime }_i \in \Sigma ^{\prime }_\sigma $
. Thus,
$\{\bar {\sigma ^{\prime }_1},\bar {\sigma ^{\prime }_2},...\}$
defines a fan in
$\psi ^{-1}((N_\sigma )_{\mathbb {R}})/(N^{\prime }_{\tau '})_{\mathbb {R}}$
. The fan in
$\psi ^{-1}((N_\sigma )_{\mathbb {R}})/(N^{\prime }_{\tau '})_{\mathbb {R}}$
constructed above will be called the relative star of
$\tau '$
over
$\sigma $
and will be denoted by
$Star_\sigma (\tau ')$
A cone
$\tau '\in \Sigma ^{\prime }_\sigma $
is called primitive with respect to
$\psi $
if none of the faces of
$\tau '$
are in
$\Sigma ^{\prime }_\sigma $
.
Let
$X_\Sigma $
be a toric variety. We call the divisor
$D_{\Sigma }:=X_\Sigma \setminus T$
the toric boundary of
$X_\Sigma $
, where T is the big torus.
Theorem 3.1. [Reference Hu, Liu and Yau10, Proposition 2.1.4]
Let
$\tilde {\psi }:X_{\Sigma '}\rightarrow X_{\Sigma }$
be a toric morphism induced by a map of fans
$\psi :\Sigma '\rightarrow \Sigma $
. Then,
-
• The image
$\tilde {\psi }(X_{\Sigma '})$
of
$\tilde {\psi }$
is a subvariety of
$X_\Sigma $
. It is realized as the toric variety corresponding to the fan
$\Sigma _\psi :=\Sigma \cap \psi (N^{\prime }_{\mathbb {R}})$
. -
• The fiber of
$\tilde {\psi }$
over a point
$y\in X_{\Sigma _{\psi }}$
depends only on the orbit
$O_\sigma ,\sigma \in \Sigma _\psi $
, that contains y. Denote this fiber by
$F_\sigma $
. Then it can be described as follows.Define
$\Sigma ^{\prime }_\sigma $
to be the set of cones
$\sigma '$
in
$\Sigma '$
, whose interior is mapped to the interior of
$\sigma $
. Let
$Ind(\sigma )$
be the index of
$\tilde {\psi }$
over
$O_\sigma $
. Then
$\psi ^{-1}(y)=F_\sigma $
is a disjoint union of
$Ind(\sigma )$
identical copies of connected reducible toric variety
$F^c_\sigma $
, whose irreducible components
$F^{\tau '}_\sigma $
are the toric variety associated to the relative star
$Star_\sigma (\tau ')$
of the primitive elements
$\tau '$
in
$\Sigma ^{\prime }_\sigma $
. -
• For
$\sigma \in \Sigma _\psi , \tilde {\psi }^{-1}(O_\sigma )=\tilde {O}_\sigma \times F^c_\sigma $
, where
$\tilde {O}_\sigma $
is a connected covering space of
$O_\sigma $
of order
$Ind(\sigma )$
.
Remark 3.2. Here, the term reducible toric variety means a reducible variety obtained by gluing a collection of toric varieties along some isomorphic toric orbits.
Theorem 3.3. [Reference Hu, Liu and Yau10, Remark 2.1.12]
If
$\psi $
is surjective, then
$Ind(\sigma )=1$
for all
$\sigma \in \Sigma $
.
For any toric variety
$X_\Sigma $
, it is well-known that there is a refinement
$\psi :\Sigma '\rightarrow \Sigma $
, that is, each cone of
$\Sigma $
is a union of cones in
$\Sigma '$
, such that
$\tilde {\psi }:X_{\Sigma '}\rightarrow X_\Sigma $
is a resolution of singularities.
Theorem 3.4. Let
$\tilde {\psi }:X_{\Sigma '}\rightarrow X_\Sigma $
be the resolution defined by a refinement
$\psi :\Sigma '\rightarrow \Sigma $
. Suppose V is a prime divisor of
$X_{\Sigma '}\setminus T_{N}$
. Then P is birationally equivalent to
$\mathbb {P}^r\times \tilde {\psi }(P)$
, where
$r=\mathrm {dim}V-\mathrm {dim}\tilde {\psi }(P)$
.
Proof. Because
$\tilde {\psi }$
is a toric morphism, every prime divisor of
$X_{\Sigma '}\setminus T_{N}$
corresponds to a 1-dimension cone of
$\Sigma '$
. Fix a cone
$\sigma \in \Sigma $
of dimension
$\geq 2$
, and suppose
$\sigma ^{\prime }_1,\sigma ^{\prime }_2,...\in \Sigma ^{\prime }_\sigma $
are the 1-dimensional cones that map to the interior of
$\sigma $
, which are clearly primitive. Let
$\sigma ^{\prime \prime }_1,\sigma ^{\prime \prime }_2,...\in \Sigma ^{\prime }_\sigma $
be other primitive cones. By Theorem 3.3 and Theorem 3.1,
$\tilde {\psi }^{-1}(O_\sigma )=O_\sigma \times F^c_\sigma $
, and the irreducible components of
$F^c_\sigma $
correspond to the cones
$\{\sigma ^{\prime }_1,\sigma ^{\prime }_2,...\}\cup \{\sigma ^{\prime \prime }_1,\sigma ^{\prime \prime }_2,...\}$
.
By comparing the dimension of exceptional locus, it is easy to see that the codimension 1 components of
$\tilde {\psi }^{-1}(O_\sigma )$
are equal to
$O_\sigma \times F^c_{\sigma '}$
, where the irreducible components of
$F^c_{\sigma '}$
are the toric variety associated to the relative stars
$\{Star_\sigma (\sigma ^{\prime }_1),Star_\sigma (\sigma ^{\prime }_2),...\}$
. Suppose P is the divisor defined by
$\sigma ^{\prime }_1$
. Then
$P\subset \tilde {\psi }^{-1}(O_\sigma )$
is a codimension 1 component and birational equivalent to
$O_\sigma \times F^{c}_{\sigma ^{\prime }_1}$
, where
$F^{c}_{\sigma ^{\prime }_1}$
is the toric variety associated to the relative star
$Star_\sigma (\sigma ^{\prime }_1)$
. Because every irreducible toric variety is birationally equivalent to
$\mathbb {P}^r$
for some
$r\in \mathbb {N}$
, the result follows.
3.2 Toroidal embeddings
Given a normal variety X and an open subset
$U_X\subset X$
, the embedding
$U_X\subset X$
is called toroidal if for every closed point
$x\in X$
, there exist a toric variety
$X_\sigma $
, a point
$s\in X_\sigma $
, and an isomorphism of complete local k-algebras
$$ \begin{align*}\hat{\mathcal{O}}_{X,x}\cong \hat{\mathcal{O}}_{X_\sigma,s},\end{align*} $$
such that the ideal of
$X\setminus U_X$
maps isomorphically to the ideal of
$X_\sigma \setminus T_\sigma $
, where
$T_\sigma $
is the big torus.
Given a normal variety X and a reduced divisor D on X, we call
$(X,D)$
a toroidal pair if
${U_X:=X\setminus D\subset D}$
is a toroidal embedding.
In this paper, we will assume that every irreducible component of
$X\setminus U_X$
is normal – that is,
$U_X\subset X$
a strict toroidal embedding.
Proposition 3.5 [Reference Kempf, Knudsen, Mumford and Saint-Donat14, Page 195].
Let
$U\subset X$
be a toroidal embedding of varieties and x a closed point of X. Then there exists an affine toric variety
$X_\sigma $
and an étale morphism
$\psi $
from an open neighborhood of
$x\in X$
to
$X_\sigma $
, such that locally at x (for the Zariski topology), we have
$U=\psi ^{-1}(T)$
, where T is the big torus.
A dominant morphism
$f:(Y,D_Y)\rightarrow (X,D)$
of toroidal pairs is called toroidal if for every closed point
$x\in X$
, there exist local models
$(X_\sigma ,s)$
at x,
$(X_\tau ,t)$
at
$f(x)$
and a toric morphism
$g:X_\sigma \rightarrow X_\tau $
so that the following diagram commutes:
where
$\hat {f}^\#$
and
$\hat {g}^\#$
are the algebra homomorphisms induced by f and g.
Corollary 3.6 [Reference Abramovich and Karu1, Corollary 1.6].
If
$f:(X,D)\rightarrow (Y,D_Y)$
and
$g:(Y,D_Y)\rightarrow (Z,D_Z)$
are toroidal morphisms, then
$g\circ f:(X,D)\rightarrow (Z,D_Z)$
is a toroidal morphism.
Definition 3.7 [Reference Abramovich and Karu1, Section 8.2].
Let
$f:(X,D)\rightarrow (Z,B)$
be a projective toroidal morphism between toroidal pairs with connected fibers. We say f is almost semistable if
-
• the morphism f is equidimensional,
-
• all the fibers of the morphism f are reduced,
-
• Z is smooth, and
-
• X has quotient singularities.
Theorem 3.8 (Almost Semistable Reduction).
Let
$X\rightarrow Z$
be a projective morphism between projective normal varieties and
$D\subset X$
be a divisor. Then there exists a proper, surjective, generically finite morphism of irreducible varieties
$b:Z'\rightarrow Z$
, a projective birational morphism of irreducible varieties
$a:X'\rightarrow (X\times _Z Z')^m$
, where
$(X\times _Z Z')^m$
is the main component of the fiber product
$X\times _Z Z'$
, and divisors
$B'\subset Z'$
,
$D'\subset X'$
, such that
-
•
$a^{-1}(D\times _Z Z')\cup f^{\prime -1}(B')\subset D'$
, and -
• the morphism
$f':(X',D')\rightarrow (Z',B')$
is almost semistable.
Proof. This is a direct result of [Reference Abramovich and Karu1, Theorem 2.1], [Reference Abramovich and Karu1, Proposition 4.4], [Reference Abramovich and Karu1, Proposition 5.1] and [Reference Abramovich and Karu1, Section 8.2].
Lemma 3.9 [Reference Abramovich and Karu1, Lemma 6.2].
Let
$f:(X,D)\rightarrow (Z,B)$
be an almost semistable morphism,
$g:C\rightarrow Z$
a morphism such that C is nonsingular and
$g^{-1}(B)$
is a normal crossing divisor. Define
$X_C=C\times _Z X$
and let
$g_C:X_C\rightarrow X,f_C:X_C\rightarrow C$
be the two projections.
Denote
$B_C=g^{-1}(B)$
and
$D_C=g_C^{-1}(D)$
. Then
$(C,B_C)$
and
$(X_C,D_C)$
are toroidal pairs, and
$f_C:(X_C,D_C)\rightarrow (C,B_C)$
is an equidimensional toroidal morphism with reduced fibers.
Lemma 3.10. Let X be a projective normal variety, and D a reduced divisor on X such that
$(X,D)$
is a toroidal pair. Suppose
$\Delta \leq D$
is a
$\mathbb {Q}$
-divisor such that
$(X,\Delta )$
is sub-lc.
If P is an lc place of
$(X,\Delta )$
, then P is birational equivalent to
$\mathbb {P}^r\times V$
, where V is the image of P in X and
$r=\mathrm {dim}X-\mathrm {dim}V-1$
.
Proof. Let P be an lc place of
$(X,\Delta )$
, and suppose x is a general point of the image of P on X. For the rest of the proof, we consider Zariski locally near x by replacing X with an open neighborhood of x.
Let
$X_\sigma $
be the affine toric variety defined in Proposition 3.5 and
$\sigma \subset \sigma _1$
a subdivision such that
$h_\sigma :X_{\sigma _1}\rightarrow X_\sigma $
is a resolution. Because
$\pi $
is étale,
$X_1:=X_{\sigma _1}\times _{X_\sigma }X$
is a log resolution of
$(X,D)$
. We have the following diagram:
Let
$D_1$
be the strict transform of D on
$X_1$
plus the h-exceptional divisor. Then
$(X_1,D_1)$
is log smooth, and
$h:(X_1, D_1)\rightarrow (X,D)$
is a toroidal morphism. By an easy computation of discrepancies on log smooth pairs, it is easy to see that P can be obtained by a sequence of blow-ups along strata of
$(X_1,D_1)$
. We will show that such morphism is étale locally equal to a toric morphism between toric varieties.
Suppose we have a sequence of blow-ups
$h_i:X_{i+1}\rightarrow X_i,1\leq i\leq k-1$
along a strata
$V_i$
of
$(X_i,D_i)$
, where
$D_{i+1}$
is the strict transform of
$D_i$
plus the
$h_i$
-exceptional divisor, so that P is a divisor on
$X_k$
. Next, we show that there is a Cartesian diagram
where
-
• the horizontal arrows are étale morphisms,
-
•
$\sigma _j$
is a refinement of
$\sigma _1$
, -
•
$X_{\sigma _j}\rightarrow X_{\sigma _1}$
is the corresponding toric morphism, and -
• near any closed point of
$g_j^{-1}x_1$
, we have
$U_{j}:=X_j\setminus D_j=\pi _{j}^{-1}T_{j}$
, where
$T_{j}$
is the big torus of
$X_{\sigma _{j}}$
,
for all
$1\leq j\leq k$
.
Suppose it is true for
$j=i$
. Let
$X_{\sigma _{i+1}}\rightarrow X_{\sigma _i}$
be the toric morphism determined by blowing up
$X_{i}$
along the image of
$V_i$
on
$X_{\sigma _i}$
. Because blowing up is uniquely determined by local equations and both
$X_{i+1}\rightarrow X_i$
and
$X_{\sigma _{i+1}}\rightarrow X_{\sigma _i}$
are obtained by blowing up the same subvariety étale locally, then there is a natural étale morphism
$\pi _{i+1}:X_{i+1}\rightarrow X_{\sigma _{i+1}}$
such that near any closed point of
$g_j^{-1}x$
, we have
$U_{i+1}=\pi _{i+1}^{-1}T_{i+1}$
, where
$T_{i+1}$
is the big torus of
$X_{\sigma _{i+1}}$
. Because the composition of
$X_{\sigma _{i+1}}\rightarrow X_{\sigma _i}$
and
$X_{\sigma _i}\rightarrow X_{\sigma _1}$
is a toric morphism, the claim is true for
$j=i+1$
.
Now we have the following Cartesian diagram
By assumption, P is a divisor on
$X_k$
and
$\pi _k$
is étale near a general point of P. Then P is equal to the pull-back of a divisor
$P_{\sigma _k}$
on
$X_{\sigma _k}$
. Because
$\sigma _k\rightarrow \sigma $
is a refinement, by Theorem 3.4,
$f_\sigma |_{P_{\sigma _k}}$
is birationally equivalent to a
$\mathbb {P}^r$
-bundle. Because the diagram is Cartesian,
$f|_P$
is also birationally equivalent to a
$\mathbb {P}^r$
-bundle. Then P is birationally equivalent to
$V\times \mathbb {P}^r$
.
4 Moduli of polarized log Calabi–Yau pairs
In this section, we recall some definitions and results on the moduli of stable pairs and polarized log Calabi–Yau pairs, see [Reference Kollár13], [Reference Kollár and Xu16], [Reference Birkar5] and [Reference Birkar6]. We fix a natural number d and positive rational numbers
$c,v$
.
Definition 4.1. A log Calabi–Yau pair is an slc pair
$(X,\Delta )$
such that
$K_X+\Delta \sim _{\mathbb {Q}} 0$
.
A polarized log Calabi–Yau pair consists of a log Calabi–Yau pair
$(X,\Delta )$
and an effective ample integral divisor
$N\geq 0$
such that
$(X,\Delta +uN)$
is slc for any sufficiently small positive real number
$u\ll 1$
.
A
$(d,c,v)$
-polarized log Calabi–Yau pair is a polarized log Calabi–Yau pair
$(X,\Delta ),N$
such that
$\mathrm {dim}X=d,\Delta =cD$
for some integral divisor D, and
$\mathrm {vol}(N)=v$
.
Let
$f:X\rightarrow S$
be a flat morphism of schemes with
$S_2$
fibers of pure dimension. A closed subscheme
$D\subset X$
is a relative Mumford divisor over S if there is an open subset
$U\subset X$
such that
-
• codimension of
$X_s\setminus U_s$
is
$\geq 2$
for every
$s\in S$
, -
•
$D|_U$
is a Cartier divisor, -
•
$\mathrm {Supp}(D|_U)$
does not contain any irreducible component of any fiber
$U_s$
, -
• D is the closure of
$D|_U$
, and -
•
$X\rightarrow S$
is smooth at the generic points of
$X_s\cap D$
for every
$s\in S$
.
Definition 4.2. Let S be a reduced scheme. A
$(d,c,v)$
-polarized log Calabi–Yau family over S consists of a projective morphism
$f:X\rightarrow S$
of schemes, a
$\mathbb {Q}$
-divisor
$\Delta $
and an integral divisor N on X such that
-
•
$(X,\Delta +uN)\rightarrow S$
is a stable family for some rational number
$u>0$
with fibers of pure dimension d, -
•
$\Delta =cD$
, where
$D\geq 0$
is a relative Mumford divisor, -
•
$N\geq 0$
is a relative Mumford divisor, -
•
$K_{X/S}+\Delta \sim _{\mathbb {Q},S}0$
, and -
• for any fiber
$X_s$
of f,
$\mathrm {vol}(N|_{X_s})=v$
.
Lemma 4.3. There exist a positive rational number t and a natural number r both depending only on
$d,c,v$
such that
$rc,rt\in \mathbb {N}$
satisfying the following. Assume
$(X,\Delta ),N$
is a
$(d,c,v)$
-polarized log Calabi–Yau pair. Then
-
•
$(X,\Delta +tN)$
is an slc pair, -
•
$\Delta +tN$
uniquely determines
$\Delta ,N$
and -
•
$r(K_X+\Delta +tN)$
is very ample with for
$$ \begin{align*}h^j(mr(K_X+\Delta+tN))=0\end{align*} $$
$m,j>0$
.
Proof. This is Lemma 7.7 in the first arXiv version of [Reference Birkar6].
The following definition comes from Chapter 7 in the first arXiv version of [Reference Birkar6].
Definition 4.4. Let
$t,r$
be as in Lemma 4.3. To simplify notation, let
$\Theta =(d,c,v,t,r)$
. A strongly embedded
$\Theta $
-polarized log Calabi–Yau family over a reduced scheme S is a
$(d,c,v)$
-polarized log Calabi–Yau family
$f:(X,\Delta ),N\rightarrow S$
together with a closed embedding
$g:X\rightarrow \mathbb {P}^n_S$
such that
-
•
$n=h^0(r(K_{X_s}+\Delta _s+tN_s))$
for a closed point
$s\in S$
, -
•
$(X,\Delta +tN)\rightarrow S$
is a stable family, -
•
$f=\pi g$
, where
$\pi $
denotes the projection
$\mathbb {P}^n_S\rightarrow S$
, -
• letting
$\mathcal {L}:=g^*\mathcal {O}_{\mathbb {P}^n_S}(1)$
, we have
$R^qf_*\mathcal {L}\cong R^q\pi _*\mathcal {O}_{\mathbb {P}^n_S}(1)$
for all q, and -
• for every
$s\in S$
, we have
$$ \begin{align*}\mathcal{L}_s\cong \mathcal{O}_{X_s}(r(K_{X_s}+\Delta_s+tN_s)).\end{align*} $$
We denote the family by
$f:(X\subset \mathbb {P}^n_S ,\Delta ),N\rightarrow S$
.
Define the functor
$\mathcal {E}^s\mathcal {PCY}_{\Theta }$
on the category of reduced schemes by setting
$$ \begin{align*}\mathcal{E}^s\mathcal{PCY}_{\Theta}(S) =\{\text{strongly embedded }\Theta\text{-polarized log Calabi--Yau families over }S\}.\end{align*} $$
Proposition 4.5. The functor
$\mathcal {E}^s\mathcal {PCY}_{\Theta }$
is represented by a reduced separated scheme
$\mathcal {S}:=E^sPCY_{\Theta }$
together with a universal family
$(\mathcal {X}\subset \mathbb {P}^n_{\mathcal {S}},\mathcal {D}),\mathcal {N}\rightarrow \mathcal {S}$
.
Proof. This is Proposition 7.8 in the first arXiv version of [Reference Birkar6].
5 Boundedness of log places
The main result in this section is the following.
Theorem 5.1. Fix a natural number d and positive rational numbers
$c,v$
. Then there exist a natural number l and a bounded family of projective varieties
$\mathcal {W}\rightarrow \mathcal {T}$
both depending only on
$d,c,v$
, such that:
Suppose X is a normal quasi-projective variety,
$(X,\Delta )$
is an lc pair, and
$f:X\rightarrow C$
is a projective morphism of relative dimension d over a smooth curve C (possibly non-proper), such that
-
•
$K_X+\Delta \sim _{\mathbb {Q},C} 0$
, and -
• there is a divisor N on X such that a general fiber
$(X_g,\Delta _g),N_g$
is a
$(d,c,v)$
-polarized log Calabi–Yau pair.
Let
$0\in C$
be a closed point and P an lc place of
$(X,\Delta +\mathrm {lct}(X,\Delta ;f^*0)f^*0)$
. Then there is a closed point
$t\in \mathcal {T}$
and a rational map
$\mathcal {W}_t\dashrightarrow P$
which is a finite cover over the generic point of P with degree less or equal to
$\mathrm {min}\{l,\mathrm {mult}_{P}f^*0\}$
.
Lemma 5.2. Let
$(\mathcal {X},\mathcal {D}')\rightarrow \mathcal {S}$
be a locally stable morphism over a smooth variety
$\mathcal {S}$
, and
$\mathcal {D}$
be a
$\mathbb {Q}$
-divisor such that
$\mathcal {D}\leq \mathcal {D}'$
and
$K_{\mathcal {X}}+\mathcal {D}$
is
$\mathbb {Q}$
-Cartier. Then the set
$$ \begin{align*}\{V\ |\ V\text{ is a log center of }(\mathcal{X}_s,\mathcal{D}_s)\text{ for some closed point }s\in\mathcal{S}\}\end{align*} $$
is bounded.
Proof. After passing to a stratification of
$\mathcal {S}$
, we may assume that
$(\mathcal {X},\mathcal {D})\rightarrow \mathcal {S}$
has a fiberwise log resolution
$\xi : \mathcal {Y}\rightarrow \mathcal {X}$
and
$\mathcal {S}$
is smooth. Define
$\mathcal {D}_{\mathcal {Y}}$
by
$K_{\mathcal {Y}}+\mathcal {D}_{\mathcal {Y}}\sim _{\mathbb {Q}}\xi ^*(K_{\mathcal {X}}+\mathcal {D})$
. Then we have
$$ \begin{align*}K_{\mathcal{Y}_s}+\mathcal{D}_{\mathcal{Y}_s}\sim_{\mathbb{Q}}\xi^*(K_{\mathcal{X}_s}+\mathcal{D}_s)\end{align*} $$
for any closed point
$s\in \mathcal {S}$
. It is easy to see that every log center of
$(\mathcal {X}_s,\mathcal {D}_s)$
is dominated by a log center of
$(\mathcal {Y}_s,\mathcal {D}_{\mathcal {Y}_s})$
.
By the construction,
$(\mathcal {Y},\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}}))$
is log smooth over
$\mathcal {S}$
, and we denote its strata by
$\mathcal {V}_i,i\in I$
. Then
$\mathcal {V}_i\rightarrow \mathcal {S}$
is smooth for all
$i \in I.$
Because
$(\mathcal {Y}_s,\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}_s}))$
is log smooth for all
$s\in \mathcal {S}$
, then any log center of
$(\mathcal {Y}_s,\mathcal {D}_{\mathcal {Y}_s})$
is
$V_{i}|_{\mathcal {Y}_s}$
for some
$i\in I$
. Then any log center of
$(\mathcal {X}_s,\mathcal {D}_s)$
is isomorphic to
$\xi (V_i)|_{\mathcal {X}_s}$
for some
$i\in I$
, and the set of families
$\xi (V_i)\rightarrow \mathcal {S},i\in I$
parametrizes all log center of
$(\mathcal {X}_s,\mathcal {D}_s)$
. The result follows.
Lemma 5.3. Let
$f:X\rightarrow T$
be a flat morphism from a normal variety to a smooth curve T. Let
$\pi :S\rightarrow T$
be a ramified cover and
$Y\rightarrow X\times _T S$
the normalization of the main component, and denote the projection
$Y\rightarrow S$
by
$f_Y$
.
Fix a closed point
$t\in T$
, and let
$s\in \pi ^{-1}t$
be a closed point. Suppose P is an irreducible component of
$f^*t$
and Q is an irreducible component of the preimage of P in Y such that
$f_Y(Q)=s$
. Denote the ramified index of
$\pi $
along s by
$r_S$
, the multiplicity of
$f^*t$
along P by
$m_P$
. Then the degree of the finite morphism
$\pi _Q:Q\rightarrow P$
is less or equal to
$\mathrm {min}\{r_S,m_P\}$
.
Proof. By assumption, we have the following diagram:
Denote the ramified index of
$\pi _Y$
along the generic point of
$ Q$
by
$r_Q$
and
$\mathrm {mult}_{Q}f_Y^*s$
by
$m_Q$
.
Next, we calculate the multiplicity of
$\pi _Y^*f^*t$
along Q. By the definition of the ramified index, we have
$$ \begin{align*}\mathrm{mult}_Q\pi_Y^*f^*t=m_P\mathrm{mult}_Q\pi_Y^*P=m_Pr_Q.\end{align*} $$
However, since
$\pi _Y f=f_Y \pi $
, we have
$$ \begin{align*}\mathrm{mult}_Qf_Y^*\pi^*t=r_S\mathrm{mult}_Qf_Y^*s=r_Sm_Q.\end{align*} $$
Then we have
$m_Pr_Q=r_Sm_Q$
.
Choose a general point
$x\in P$
. The degree of
$\pi _Q$
is equal to the number of points in
$\pi _Q^{-1}(x)$
. By comparing the preimages of x in Y (with multiplicity), we have
$$ \begin{align*}\mathrm{deg}(\pi_Q)r_Q\leq r_S.\end{align*} $$
After multiplying both sides by
$m_Q$
, we have
$$ \begin{align*}\mathrm{deg}(\pi_Q)r_Qm_Q\leq r_Sm_Q=r_Qm_P.\end{align*} $$
Then we have
$\mathrm {deg}(\pi _Q)m_Q\leq m_P$
. Since
$r_Q,m_Q$
are positive integers,
$\mathrm {deg}(\pi _Q)$
is less or equal to
$\mathrm {min}\{r_S,m_P\}$
.
Proof of Theorem 5.1.
Suppose
$(X,\Delta )\rightarrow C$
is a fibration, and N is a divisor on X such that
-
•
$K_X+\Delta \sim _{\mathbb {Q},C}0$
, and -
• a general fiber
$(X_g,\Delta _g),N_g$
is a
$(d,c,v)$
-polarized log Calabi–Yau pair.
By Lemma 4.3, there exist a positive rational number t and a natural number r such that
$(X_g,\Delta _g+tN_g)$
is an slc pair and
$r(K_{X_g}+\Delta _g+tN_g)$
is very ample without higher cohomology. By [Reference Hartshorne9, §3, Theorem 12.11],
$r(K_{X_U}+\Delta _U+tN_U)$
is very ample over an open subset
$U\subset C$
, and it defines a closed embedding
$g:X_U\hookrightarrow \mathbb {P}^n_U$
, where
$n=h^0(r(K_{X_g}+\Delta _g+tN_g))$
. Also because
$(X_U,\Delta _U+tN_U)\rightarrow U$
is a stable family, then
$f_U:(X_U\subset \mathbb {P}^n_U,\Delta _U),N_U\rightarrow U$
is a strongly embedded polarized log Calabi–Yau family over U. Since
$\mathcal {E}^s\mathcal {PCY}_{\Theta }$
has a fine moduli space
$\mathcal {S}$
with the universal family
$(\mathcal {X}\subset \mathbb {P}^n_{\mathcal {S}},\mathcal {D}),\mathcal {N}\rightarrow \mathcal {S}$
according to Proposition 4.5, we have
$(X_U,\Delta _U)\cong (\mathcal {X},\mathcal {D})\times _{\mathcal {S}} U$
, where
$U\rightarrow \mathcal {S}$
is the moduli map defined by
$f_U$
. We denote this moduli map by
$\phi _U$
. Note
$(\mathcal {X},\mathcal {D})\rightarrow \mathcal {S}$
is a
$(d,c,v)$
-polarized log Calabi–Yau family over
$\mathcal {S}$
. In particular,
$K_{\mathcal {X}/\mathcal {S}}+\mathcal {D}\sim _{0,\mathcal {S}}0$
is
$\mathbb {Q}$
-Cartier.
After replacing
$\mathcal {S}$
by a dense open subset, we may assume that
$\mathcal {S}$
is smooth and there is a fiberwise log resolution
$\xi :(\mathcal {Y},\mathcal {D}_{\mathcal {Y}})\rightarrow (\mathcal {X},\mathcal {D})$
over
$\mathcal {S}$
, where
$\mathcal {D}_{\mathcal {Y}}$
is the
$\mathbb {Q}$
-divisor such that
$K_{\mathcal {Y}/\mathcal {S}}+\mathcal {D}_{\mathcal {Y}}\sim _{\mathbb {Q}}\xi ^*(K_{\mathcal {X}/\mathcal {S}}+\mathcal {D})$
. Then
$\mathcal {Y}$
is smooth. Let
$\mathcal {S}'$
be a smooth compactification of
$\mathcal {S}$
such that
$\mathcal {S}'\setminus \mathcal {S}$
is a divisor and
$(\mathcal {S}',\mathcal {S}'\setminus \mathcal {S})$
is log smooth,
$\mathcal {Y}'$
a smooth compactification of
$\mathcal {Y}$
such that
$\mathcal {Y}\rightarrow \mathcal {S}$
extends to a projective morphism
$\mathcal {Y}'\rightarrow \mathcal {S}'$
and
$\mathcal {Y}'\setminus \mathcal {Y}$
is pure of codimension 1. Let
$\mathcal {R}'$
be the reduced divisor whose support is equal to the sum of
$\mathcal {Y}'\setminus \mathcal {Y}$
and the closure of
$\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}})$
in
$\mathcal {Y}'$
. Then
$(\mathcal {Y}',\mathcal {R}')\times _{\mathcal {S}'} \mathcal {S}\cong (\mathcal {Y},\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}}))$
. Because
$(\mathcal {Y},\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}}))$
is log smooth over
$\mathcal {S}$
, then
$(\mathcal {Y}',\mathcal {R}')\rightarrow \mathcal {S}'$
is almost semistable over
$\mathcal {S}$
.
By Theorem 3.8, there is a generically finite cover
$\tau :\bar {\mathcal {S}}\rightarrow \mathcal {S}'$
and a birational morphism
$\psi :\bar {\mathcal {Y}}\rightarrow \mathcal {Y}'\times _{\mathcal {S}'}\bar {\mathcal {S}}$
such that
$$ \begin{align*}\chi:(\bar{\mathcal{Y}},\bar{\mathcal{R}})\rightarrow (\bar{\mathcal{S}},\bar{\mathcal{B}})\end{align*} $$
is an almost semistable morphism, where
$\bar {\mathcal {B}}\supset \bar {\mathcal {S}}\setminus \tau ^{-1}\mathcal {S}$
and
$\bar {\mathcal {R}}\supset \chi ^{-1}\bar {\mathcal {B}}\cup \psi ^{-1}(\mathcal {D}'\times _{\mathcal {S}'}\bar {\mathcal {S}})$
are reduced divisors. Perhaps after replacing
$\mathcal {S}$
by a dense open subset, we may assume there is a
$\mathbb {Q}$
-divisor
$\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}}$
on
$\bar {\mathcal {Y}}$
such that
-
•
$\tau $
is a finite cover over
$\mathcal {S}$
, -
• every component of
$\mathrm {Supp}(\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})$
is horizontal over
$\bar {\mathcal {S}}$
, -
•
$(K_{\bar {\mathcal {Y}}}+\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})|_{\bar {\mathcal {Y}}_{\bar {\mathcal {S}}^o}}\sim _{\mathbb {Q},\bar {\mathcal {S}}^o} 0$
, where
$\bar {\mathcal {S}}^o:=\tau ^{-1}\mathcal {S},\bar {\mathcal {Y}}_{\bar {\mathcal {S}}^o}:=\bar {\mathcal {Y}}\times _{\bar {\mathcal {S}}}\bar {\mathcal {S}}^o$
, and -
• for every point
$\bar {p}\in \bar {\mathcal {S}}^0$
(not necessarily closed), the fiber of
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})\rightarrow \bar {\mathcal {S}}$
over
$\bar {p}$
is crepant birationally equivalent to the base change of the fiber of
$(\mathcal {X},\mathcal {D})\rightarrow \mathcal {S}$
over
$\tau (\bar {p})$
.
Because
$K_{\mathcal {Y}/\mathcal {S}}+\mathcal {D}_{\mathcal {Y}}\sim _{\mathbb {Q}}\xi ^*(K_{\mathcal {X}/\mathcal {S}}+\mathcal {D})$
, a general fiber of
$(\mathcal {X},\mathcal {D})\rightarrow \mathcal {S}$
is slc,
$(\mathcal {Y},\mathrm {Supp}(\mathcal {D}_{\mathcal {Y}}))$
is log smooth over
$\mathcal {S}$
, and
$\bar {\mathcal {R}}\supset \chi ^{-1}\bar {\mathcal {B}}\cup \psi ^{-1}(\mathcal {D}'\times _{\mathcal {S}'}\bar {\mathcal {S}})$
. Then we have
$\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}}\leq \bar {\mathcal {R}}$
. Since
$\chi :(\bar {\mathcal {Y}},\bar {\mathcal {R}})\rightarrow (\bar {\mathcal {S}},\bar {\mathcal {B}})$
is almost semistable,
$\bar {\mathcal {Y}}$
is
$\mathbb {Q}$
-factorial according to [Reference Kollár and Mori15, Proposition 5.15]. Also because every component of
$\mathrm {Supp}(\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})$
is horizontal over
$\bar {\mathcal {S}}$
, then
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}},\geq 0})\rightarrow \bar {\mathcal {S}}$
is locally stable.
Note we replace
$\mathcal {S}$
by a dense open subset. Then after applying the same argument on the complementary set, we get a stratification of
$\mathcal {S}$
. We also replace U by an open subset so that
$\phi _U:U\rightarrow \mathcal {S}$
is still a morphism.
Let
$\bar {C}$
be the closure of
$\bar {U}:=U\times _{\mathcal {S}}\bar {\mathcal {S}}^o$
. Then there is a finite cover
$\pi :\bar {C}\rightarrow C$
. We choose
$\bar {0}$
to be a closed point of
$\pi ^{-1}(0)$
. Because
$\bar {\mathcal {S}}$
is proper, the moduli map
$\phi _U:U\rightarrow \mathcal {S}$
defines a morphism
$\bar {\phi }:\bar {C}\rightarrow \bar {\mathcal {S}}$
. Define
$\bar {Y}:=\bar {\mathcal {Y}}\times _{\bar {\mathcal {S}}}\bar {C}$
and
$\bar {D}'=\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}} \times _{\bar {\mathcal {S}}} \bar {C}$
. It is easy to see that
$\bar {f}':(\bar {Y},\bar {D}')\rightarrow \bar {C}$
is the base change of
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})\rightarrow \bar {\mathcal {S}}$
via
$\bar {\phi }:\bar {C}\rightarrow \bar {\mathcal {S}}$
. Because
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}},\geq 0})\rightarrow \bar {\mathcal {S}}$
is locally stable, then
$\bar {f}':(\bar {Y},\bar {D}^{\prime }_{\geq 0})\rightarrow \bar {C}$
is also locally stable. Let
$\bar {R}$
be the base change of
$\bar {\mathcal {R}}$
on
$\bar {Y}$
; by Lemma 3.9,
$(\bar {Y},\bar {R})\rightarrow (\bar {C},\bar {0})$
is a also toroidal morphism with reduced fibers. Because
$\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}}\leq \bar {\mathcal {R}}$
, we have
$\bar {D}'\leq \bar {R}$
.
Define
$(\bar {Y}_{\bar {U}},\bar {D}^{\prime }_{\bar {U}}):=(\bar {Y},\bar {D}')\times _{\bar {C}}\bar {U}$
. Then
$(\bar {Y}_{\bar {U}},\bar {D}^{\prime }_{\bar {U}})$
is equal to the pull-back of
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})\times _{\bar {\mathcal {S}}}\bar {\mathcal {S}}^o$
via
$\bar {\phi }|_{\bar {U}}:\bar {U}\rightarrow \bar {\mathcal {S}}^o$
. Because
$(K_{\bar {\mathcal {Y}}}+\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})|_{\bar {\mathcal {Y}}_{\bar {\mathcal {S}}^o}}\sim _{\mathbb {Q},\bar {\mathcal {S}}^o} 0$
, then there is a
$\mathbb {Q}$
-divisor
$\bar {D}$
on
$\bar {Y}$
such that
-
•
$\bar {D}|_{\bar {Y}_{\bar {U}}}=\bar {D}^{\prime }_{\bar {U}}$
, -
•
$\bar {D}\leq \bar {D}'$
, -
•
$\mathrm {Supp}(\bar {D})$
does not contain the whole fiber
$\bar {Y}_{\bar {0}}$
and -
•
$K_{\bar {Y}}+\bar {D}\sim _{\mathbb {Q},\bar {C}}0.$
It is easy to see that
$\bar {D}$
is the largest
$\mathbb {Q}$
-divisor on
$\bar {Y}$
such that
$\bar {D}\leq \bar {D}'$
and
$K_{\bar {Y}}+\bar {D}\sim _{\mathbb {Q},\bar {C}}0$
.
Because
$\bar {f}':(\bar {Y},\bar {D}^{\prime }_{\geq 0})\rightarrow \bar {C}$
is locally stable and
$\bar {D}\leq \bar {D}'$
, then
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
is sub-lc and
$K_{\bar {Y}}+\bar {D}+\bar {Y}_{\bar {0}}\sim _{\mathbb {Q},\bar {C}}K_{\bar {Y}}+\bar {D} +(\bar {f}')^* \bar {0}\sim _{\mathbb {Q},\bar {C}}0.$
Let
$\bar {X}$
be the normalization of the main component of
$X\times _C \bar {C}$
,
$\pi _X$
denote the projection
$\bar {X}\rightarrow X$
and
$\bar {f}$
denote the projection
$\bar {X}\rightarrow \bar {C}$
. We replace
$\Delta $
by
$\Delta +\mathrm {lct}(X,\Delta ;f^*0)f^*0$
. Then we may assume
$\mathrm {lct}(X,\Delta ;f^*0)=0$
. By the Hurwitz’s formula, there is a
$\mathbb {Q}$
-divisor
$\bar {\Delta }$
such that
$$ \begin{align*}K_{\bar{X}}+\bar{\Delta}\sim_{\mathbb{Q}}\pi_X^*(K_{X}+\Delta).\end{align*} $$
Note we only add a
$\mathbb {Q}$
-divisor which is vertical over C. Then the generic fiber of
$(X,\Delta )\rightarrow C$
is unchanged.
Suppose P is an lc place of
$(X,\Delta )$
. Let
$X'\rightarrow X$
be a dlt modification of
$(X,\Delta )$
such that P is a divisor on
$X'$
,
$\bar {X}'$
the normalization of the main component of
$X'\times _C\bar {C}$
, and
$\bar {P}$
an irreducible component of the preimage of P on
$\bar {X}'$
. By [Reference Kollár12, 2.41],
$\bar {P}$
is an lc place of
$(\bar {X},\bar {\Delta })$
.
By Lemma 5.3, we have
$$ \begin{align*}\mathrm{deg}(\bar{P}\rightarrow P)\leq \mathrm{min}\{\mathrm{deg}(\pi),\mathrm{mult}_Pf^*0\}.\end{align*} $$
By the definition of
$\bar {C}$
,
$\mathrm {deg}(\pi )$
is equal to the degree of the finite cover
$\bar {\mathcal {S}}\rightarrow \mathcal {S}$
. Let l be the degree of the finite morphism
$\bar {\mathcal {S}}\rightarrow \mathcal {S}$
. Then
$\mathrm {min}\{\mathrm {deg}(\pi ),\mathrm {mult}_Pf^*0\}$
is less or equal to
$\mathrm {min}\{l,\mathrm {mult}_Pf^*0\}$
. Thus, we only need to prove that
$\bar {P}$
is birationally bounded.
Note
$(\bar {Y}_{\bar {U}},\bar {D}^{\prime }_{\bar {U}})$
is equal to the pull-back of
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})\times _{\bar {\mathcal {S}}}\bar {\mathcal {S}}^o$
via
$\bar {\phi }|_{\bar {U}}:\bar {U}\rightarrow \bar {\mathcal {S}}$
. Because
$(X_U,\Delta _U)\cong (\mathcal {X},\mathcal {D})\times _{\mathcal {S}} U$
and
$\bar {X}$
is the normalization of the main component of
$X\times _ C \bar {C}$
, then
$(\bar {X},\bar {\Delta })\times _{\bar {C}} \bar {U}'\rightarrow \bar {U'}$
is isomorphic to the pull-back of
$(\mathcal {X},\mathcal {D})\times _{\mathcal {S}}\bar {\mathcal {S}}^o$
via
$\bar {\phi }|_{\bar {U'}}:\bar {U}'\rightarrow \bar {\mathcal {S}}$
for an open subset
$\bar {U}'\subset \bar {C}$
. Also because the fiber of
$(\bar {\mathcal {Y}},\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}})\rightarrow \bar {\mathcal {S}}$
over
$\bar {p}$
is crepant birationally equivalent to the base change of the fiber of
$(\mathcal {X},\mathcal {D})\rightarrow \mathcal {S}$
over
$\tau (\bar {p})$
for every point
$\bar {p}\in \bar {\mathcal {S}}^0$
, then the generic fiber of
$\bar {f}:(\bar {X},\bar {\Delta })\rightarrow \bar {C}$
is crepant birationally equivalent to the generic fiber of
$\bar {f}':(\bar {Y},\bar {D}+\bar {Y}_{\bar {c}})\rightarrow \bar {C}$
. Since
$K_{\bar {X}}+\bar {\Delta }\sim _{\mathbb {Q},\bar {C}}0,K_{\bar {Y}}+\bar {D}+\bar {Y}_{\bar {0}}\sim _{\mathbb {Q},\bar {C}}0$
, then there is a
$\mathbb {Q}$
-divisor F on
$\bar {C}$
such that
$(\bar {X},\bar {\Delta }+\bar {f}^*F)$
is crepant birationally equivalent to
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
. Since both
$(\bar {X},\bar {\Delta })$
and
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
have an lc place dominating
$\bar {0}$
, then
$\bar {0}\not \in \mathrm {Supp}(F)$
. After replacing
$\bar {C}$
by an open neighborhood of
$\bar {0}$
, we may assume that
$(\bar {X},\bar {\Delta })$
is crepant birationally equivalent to
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
. In particular, a divisor
$\bar {P}$
is an lc place of
$(\bar {X},\bar {\Delta })$
if and only if it is an lc place of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
.
Recall that
$(\bar {Y},\bar {R})\rightarrow (\bar {C},\bar {0})$
is a toroidal morphism and
$\bar {D}\leq \bar {R}$
. Since
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
is sub-lc, by Lemma 3.10,
$\bar {P}$
is birationally equivalent to
$V\times \mathbb {P}^r$
, where V is the image of
$\bar {P}$
on
$\bar {Y}$
. Because
$\bar {P}$
is an lc place of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
, then V is an lc center of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
.
To prove
$\bar {P}$
is birationally bounded, we only need to show that all lc centers of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
are bounded. Let W be the normalization of an irreducible component of
$\bar {Y}_{\bar {0}}$
such that V is contained in the image of W on
$\bar {Y}_{\bar {0}}$
.
If V has codimension 1 in
$\bar {Y}$
, then
$\bar {P}$
is just W. Since
$\bar {Y}_{\bar {0}}$
is in a bounded family
$\bar {\mathcal {Y}}\rightarrow \bar {\mathcal {S}}$
and W is the normalization of an irreducible component of
$\bar {Y}_{\bar {0}}$
, P is birationally bounded.
If V has codimension
$\geq 2$
in
$\bar {Y}$
, by applying the adjunction on
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
, we have
$$ \begin{align*}(K_{\bar{Y}}+\bar{D}+\bar{Y}_{\bar{0}})|_{W}=K_{W}+\bar{D}_{W}.\end{align*} $$
By the inverse of adjunction (see [Reference Kollár12, Theorem 4.9]), an lc center of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
intersecting W corresponds to an lc center of
$(W,\bar {D}_{W})$
, hence also an lc center of
$(\bar {Y}_{\bar {0}},\bar {D}^{\prime }_{\bar {Y},\bar {0}})$
. Let
$s\in \bar {\mathcal {S}}$
be the image of
$\bar {C}$
in
$\bar {\mathcal {S}}$
. By the definition of
$\bar {Y}$
, we have the isomorphism
$$ \begin{align*}(\bar{Y}_{\bar{0}},\bar{D}^{\prime}_{\bar{Y},\bar{0}})\cong (\bar{\mathcal{Y}}_s,\bar{\mathcal{D}}^{\prime}_{\bar{\mathcal{Y}}_s}).\end{align*} $$
By Lemma 5.2, all lc centers of
$(\bar {\mathcal {Y}}_s,\bar {\mathcal {D}}^{\prime }_{\bar {\mathcal {Y}}_s})$
are in a bounded family. Then all lc centers of
$(\bar {Y},\bar {D}+\bar {Y}_{\bar {0}})$
are in a bounded family.
6 Proof of main theorems
Lemma 6.1. Fix a natural number d and positive rational numbers
$\epsilon ,c\in (0,1)$
. Suppose
$(X,\Delta )$
is an
$\epsilon $
-lc pair of dimension d,
$-(K_X+\Delta )$
is big and nef and
$\mathrm {coeff}(\Delta )\geq c$
. Then
$(X,\Delta )$
is log bounded.
Proof. By the main theorem of [Reference Birkar4], X is bounded. Then there exist a natural number n, two constants
$V_1,V_2$
depending only on d and
$\epsilon $
, and a very ample divisor H on X defining an embedding
$X\subset \mathbb {P}^n$
such that
$H^d\leq V_1$
and
$H^{d-1}\cdot K_{X}\geq -V_2$
. Because
$\mathrm {coeff}(\Delta )\geq c$
, we have
$$ \begin{align*} \begin{aligned} cH^{d-1}\cdot\mathrm{Supp} (\Delta) & \leq H^{d-1}\cdot \Delta\\ & = H^{d-1}\cdot(K_{X}+\Delta)-H^{d-1}\cdot K_{X} \\ & \leq -H^{d-1}\cdot K_{X}\\ & \leq V_2. \end{aligned} \end{align*} $$
By the boundedness of the Chow variety, both X and
$\mathrm {Supp}(\Delta )$
are parametrized by a subscheme of the Hilbert scheme. Then
$(X,\Delta )$
is log bounded.
Proof of Theorem 1.1.
Because f has reduced fibers, then
$X_0=f^*0$
, and we have
$K_X+\Delta +X_0\sim _{\mathbb {Q},C}0$
. By adjunction, we have
$K_{X_0}+\Delta _0\sim _{\mathbb {Q}} (K_X+\Delta +X_0)|_{X_0}$
. Because
$(X_0,\Delta _0)$
is slc, then its normalization
$(X_0^\nu ,\Delta _0^\nu )$
is lc. Also because X is a normal variety, by inverse of adjunction,
$(X,\Delta +f^*0)$
is lc over an open neighborhood of
$0\in C$
. It is easy to see that every irreducible component of
$X_0$
is an lc place of
$(X,\Delta +f^*0)$
. After replacing C by an open neighborhood of
$0$
, we may assume
$(X,\Delta +f^*0)$
is lc.
Because
$\mathrm {mult}_Pf^*0=1$
for every irreducible component
$P\subset X_0$
, by Theorem 5.1, there exists a bounded family
$\mathcal {W}\rightarrow \mathcal {T}$
and a finite dominant rational map
$\mathcal {W}_t\dashrightarrow P$
whose degree is a factor of
$\mathrm {min}\{l,\mathrm {mult}_Pf^*0\}=1$
. Then
$\mathcal {W}_t\dashrightarrow P$
is a birational map, which means P is birationally bounded.
Proof of Corollary 1.2.
By the proof of Theorem 1.1,
$(X,\Delta +X_0)$
is lc over an open neighborhood of
$0\in C$
, and every irreducible component of
$X_0$
is an lc place of
$(X,\Delta +X_0)$
. After replacing C by an open neighborhood of
$0$
, we may assume
$(X,\Delta +X_0)$
is lc. Also because
$X_0=f^*0$
is a Cartier divisor, then
$(X,\Delta )$
is lc, and its lc centers are not contained in
$X_0$
. After replacing C by an open neighborhood of
$0$
, we may assume every lc center of
$(X,\Delta )$
dominates C.
Because a general fiber
$(X_g,\Delta _g)$
of f is
$\epsilon $
-lc, by inverse of adjunction,
$(X,\Delta +X_g)$
is plt in a neighborhood of
$X_g$
. Also because every lc center of
$(X,\Delta )$
dominates C, then
$(X,\Delta )$
is klt. Because
$-(K_X+\Delta )$
is ample over C, let
$B\in |-(K_X+\Delta )|_{\mathbb {Q}/C}$
be a general member. Then
$(X,\Delta +B)$
is klt and
$K_X+\Delta +B\sim _{\mathbb {Q},C}0$
. Thus, X is Fano type over C.
Because
$-(K_X+\Delta +X_0)\sim _{\mathbb {Q},C} -(K_X+\Delta )$
is ample over C, X is Fano type over C and
$\mathrm {coeff}(\Delta +X_0)\subset (c\mathbb {N}\cap [0,1])\cup \{1\}$
is in a finite set, by [Reference Birkar3, Theorem 1.8]. After replacing C by an open neighborhood of
$0$
, there exist a natural number l depending only on
$d,c$
and a
$\mathbb {Q}$
-divisor
$\Lambda $
on X such that
-
•
$\Lambda \geq \Delta +X_0$
, -
•
$l(K_X+\Lambda ) \sim _{C} 0$
and -
•
$(X,\Lambda )$
is lc.
Because
$\Lambda \geq \Delta +X_0$
, then every irreducible component of
$X_0$
is an lc place of
$(X,\Lambda )$
.
Since
$l(K_X+\Lambda ) \sim _{C} 0$
, then
$l(K_{X_g}+\Lambda _g) \sim 0$
, where
$(X_g,\Lambda _g)$
is a general fiber of
$(X,\Lambda )\rightarrow C$
. Because
$(X,\Delta +X_0)$
is lc, then
$(X_g,\Lambda _g)$
is a Calabi–Yau pair and
$\mathrm {coeff}\Lambda _g \subset \frac {1}{l}\mathbb {N}$
.
Because a general fiber
$X_g$
is
$\epsilon $
-lc,
$-(K_{X_g}+\Delta _g)$
is ample, and
$\mathrm {coeff}(\Delta _g)\geq c$
, by Lemma 6.1,
$(X_g,\Delta _g)$
is log bounded. Then there exist a natural number m depending only on
$d,\epsilon $
and an open subset
$U\subset C$
such that
$-m(K_{X_u}+\Delta _{u})$
is very ample without higher cohomology for every
$u\in U$
. Thus,
$-m(K_{X_U}+\Delta _U)$
is relatively very ample over U. Choose a general member
$N\in |-m(K_{X_U}+\Delta _U)|$
. Because
$N_g$
is ample and
$(X_g,\Delta _g)$
is log bounded, then
$\mathrm {vol}(N_g)=N_g^d$
is in a finite set. To prove the result, we may assume
$\mathrm {vol}(N_g)=v$
is fixed. Because
$N\in |-m(K_{X_U}+\Delta _U)|$
is a general member, then there is a sufficiently small positive rational number t such that
$(X_g,\Lambda _g+tN_g)$
is lc. Thus,
$(X_g,\Lambda _g),N_g$
is a
$(d,\frac {1}{l},v)$
-polarized log Calabi–Yau pair. Then apply Theorem 1.1.
Acknowledgements
The author would like to thank his advisor Christopher D. Hacon and his postdoctoral advisor Caucher Birkar for their encouragement and constant support. He would also like to thank Jingjun Han, Chen Jiang, Xiaowei Jiang and Jihao Liu for their helpful comments.
Competing interests
The authors have no competing interest to declare.
Financial Support
This work was supported by grants from Tsinghua University, Yau Mathematical Sciences Center.
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