On the boundedness of $n$-folds with $\kappa(X)=n-1$

The author would like to thank Gabriele Di Cerbo and Roberto Svaldi for many fruitful discussions on this work. He would like to thank Stefan Patrikis for explaining the basics of Tate–Shafarevich groups. He would like to thank Christopher Hacon and Burt Totaro for helpful comments and feedback, and Javier Carvajal-Rojas, Mark Gross and János Kollár for answering his questions. Finally, he would like to thank the anonymous referee for the thorough feedback and the many suggestions to improve this work.

Throughout this paper, the base field will be an algebraically closed field of characteristic zero.

In this paper a contraction is a projective morphism of quasi-projective varieties $f\colon X\rightarrow Z$ with $f_{*}\mathcal{O}_{X}=\mathcal{O}_{Z}$. Notice that, if $X$ is normal, then so is $Z$. An elliptic fibration is a contraction whose general fiber is a smooth elliptic curve.

Let $\mathcal{R}$ be a subset of $[0,1]$. Then, we define the set of hyperstandard multiplicities associated to $\mathcal{R}$ as

When $\mathcal{R}=\{0,1\}$, we call it the set of standard multiplicities. Usually, with no mention, we assume $0,1\in\mathcal{R}$, so that $\Phi(\{0,1\})\subset\Phi(\mathcal{R})$. Furthermore, if $1-r\in\mathcal{R}$ for every $r\in R$, we have that $\mathcal{R}\subset\Phi(\mathcal{R})$. Now, assume that $\mathcal{R}\subset[0,1]$ is a finite set of rational numbers. Then, $\Phi(\mathcal{R})$ is a set of rational numbers satisfying the descending chain condition (DCC in short) whose only accumulation point is 1.

Let $X$ be a normal quasi-projective variety. We say that $D$ is a divisor on $X$ if it is a $\mathbb{Q}$-Weil divisor, i.e., $D$ is a finite sum of prime divisors on $X$ with coefficients in $\mathbb{Q}$. The support of a divisor $D=\sum_{i=1}^{n}d_{i}P_{i}$ is the union of the prime divisors appearing in the formal sum $\mathrm{Supp}(D)=\sum_{i=1}^{n}P_{i}$. Let $f\colon X\rightarrow Z$ be a projective morphism of quasi-projective varieties. Given a divisor $D=\sum d_{i}P_{i}$ on $X$, we define

We call $D^{v}$ and $D^{h}$ the vertical part and horizontal part of $D$, respectively. Let $D_{1}$ and $D_{2}$ be divisors on $X$. We write $D_{1}\sim_{Z}D_{2}$ (respectively $D_{1}\sim_{\mathbb{Q},Z}D_{2}$) if there is a Cartier (respectively $\mathbb{Q}$-Cartier) divisor $L$ on $Z$ such that $D_{1}-D_{2}\sim f^{*}L$ (respectively $D_{1}-D_{2}\sim_{\mathbb{Q}}f^{*}L$). Equivalently, we may also write $D_{1}\sim D_{2}$ over $Z$. The case of $\mathbb{Q}$-linear equivalence is denoted similarly.

A sub-pair $(X,B)$ is the datum of a normal quasi-projective variety and a divisor $B$ such that $K_{X}+B$ is $\mathbb{Q}$-Cartier. If $B\leq\operatorname{Supp}(B)$, we say that $B$ is a sub-boundary, and if in addition $B\geq 0$, we call it boundary. A sub-pair $(X,B)$ is called a pair if $B\geq 0$. A sub-pair $(X,B)$ is simple normal crossing (or log smooth) if $X$ is smooth, every irreducible component of $\operatorname{Supp}(B)$ is smooth, and étale locally $\operatorname{Supp}(B)\subset X$ is isomorphic to the intersection of $r\leq n$ coordinate hyperplanes in $\mathbb{A}^{n}$. A log resolution of a sub-pair $(X,B)$ is a birational contraction $\pi\colon X^{\prime}\rightarrow X$ such that $\mathrm{Ex}(\pi)$ is a divisor and $(X^{\prime},\pi^{-1}_{*}(\operatorname{Supp}(B))+\mathrm{Ex}(\pi))$ is log smooth. Here, $\mathrm{Ex}(\pi)\subset X^{\prime}$ is the exceptional set of $\pi$, i.e., the reduced subscheme of $X^{\prime}$ consisting of the points where $\pi$ is not an isomoprhism. If $(X,B)$ is a sub-pair and $f\colon X\rightarrow U$ is a morphism, we say that $(X,B)$ is log smooth over $U$ if $(X,B)$ is simple normal crossing, and every stratum of $(X,\operatorname{Supp}(B))$, including $X$ itself, is smooth over $U$.

Let $(X_{1},B_{1})$ and $(X_{2},B_{2})$ be two pairs. We say that they are crepant to each other if there exist a normal variety $Y$ and birational morphisms $p\colon Y\rightarrow X_{1}$ and $q\colon Y\rightarrow X_{2}$ so that $p^{*}(K_{X_{1}}+B_{1})=q^{*}(K_{X_{2}}+B_{2})$.

Let $(X,B)$ be a sub-pair, and let $\pi\colon X^{\prime}\rightarrow X$ be a birational contraction from a normal variety $X^{\prime}$. Then, we can define a sub-pair $(X^{\prime},B^{\prime})$ on $X^{\prime}$ via the identity

where we assume that $\pi_{*}(K_{X^{\prime}})=K_{X}$. We call $(X^{\prime},B^{\prime})$ the log pull-back or trace of $(X,B)$ on $X^{\prime}$. The log discrepancy of a prime divisor $E$ on $X^{\prime}$ with respect to $(X,B)$ is defined as $a_{E}(X,B)\coloneqq 1-\operatorname{mult}_{E}(B^{\prime})$. Let $\epsilon$ be a non-negative number. We say that a sub-pair $(X,B)$ is $\epsilon$-sub-log canonical (resp. $\epsilon$-sub-klt) if $a_{E}(X,B)\geq\epsilon$ (resp. $a_{E}(X,B)>\epsilon$) for every $\pi$ and every $E$ as above. If $\epsilon=0$, we drop it from the notation. When $(X,B)$ is a pair, we say that $(X,B)$ is $\epsilon$-log canonical or $\epsilon$-klt, respectively. Notice that, if $(X,B)$ is log canonical (resp. klt), we have $\operatorname{coeff}(B)\subset[0,1]$ (resp. $\operatorname{coeff}(B)\subset[0,1)$).

Given a sub-pair $(X,B)$ and an effective $\mathbb{Q}$-Cartier divisor $D$, we define the log canonical threshold of $D$ with respect to $(X,B)$ as

Let $X$ be a normal variety, and consider the set of all proper birational morphisms $\pi\colon X_{\pi}\rightarrow X$, where $X_{\pi}$ is normal. This is a partially ordered set, where $\pi^{\prime}\geq\pi$ if $\pi^{\prime}$ factors through $\pi$. We define the space of Weil b-divisors as the inverse limit

where $\mathrm{Div}(X_{\pi})$ denotes the space of Weil divisors on $X_{\pi}$. Then, we define the space of $\mathbb{Q}$-Weil b-divisors $\mathbf{Div}_{\mathbb{Q}}(X)\coloneqq\mathbf{Div}(X)\otimes\mathbb{Q}$. In the following, by b-divisor, we will mean a $\mathbb{Q}$-Weil b-divisor. Equivalently, a b-divisor $\mathbf{D}$ can be described as a (possibly infinite) sum of geometric valuations $V_{i}$ of $k(X)$ with coefficients in $\mathbb{Q}$,

such that for every normal variety $X^{\prime}$ birational to $X$, only a finite number of the $V_{i}$ can be realized by divisors on $X^{\prime}$. The trace $\mathbf{D}_{X^{\prime}}$ of $\mathbf{D}$ on $X^{\prime}$ is defined as

where $c_{X^{\prime}}(V_{i})$ denotes the center of the valuation on $X^{\prime}$.

Given a b-divisor $\mathbf{D}$ over $X$, we say that $\mathbf{D}$ is a b-$\mathbb{Q}$-Cartier b-divisor if there exists a birational model $X^{\prime}$ of $X$ such that $\mathbf{D}_{X^{\prime}}$ is $\mathbb{Q}$-Cartier on $X^{\prime}$, and for any model $r\colon X^{\prime\prime}\rightarrow X^{\prime}$, we have $\mathbf{D}_{X^{\prime\prime}}=r^{\ast}\mathbf{D}_{X^{\prime}}$. When this is the case, we will say that $\mathbf{D}$ descends to $X^{\prime}$ and write $\mathbf{D}=\overline{\mathbf{D}_{X^{\prime}}}$. We say that $\mathbf{D}$ is b-effective, if $\mathbf{D}_{X^{\prime}}$ is effective for any model $X^{\prime}$. We say that $\mathbf{D}$ is b-nef, if it is b-$\mathbb{Q}$-Cartier and, moreover, there exists a model $X^{\prime}$ of $X$ such that $\mathbf{D}=\overline{\mathbf{D}_{X^{\prime}}}$ and $\mathbf{D}_{X^{\prime}}$ is nef on $X^{\prime}$. The notion of b-nef b-divisor can be extended analogously to the relative case.

A generalized sub-pair $(X,B,\mathbf{M})/Z$ over $Z$ is the datum of:

• •

  a normal variety admitting a projective morphism $X\rightarrow Z$;

• •

  a divisor $B$ on $X$;

• •

  a b-$\mathbb{Q}$-Cartier b-divisor $\mathbf{M}$ over $X$ which descends to a nef$/Z$ $\mathbb{Q}$-Cartier divisor $\mathbf{M}_{X^{\prime}}$ on some birational model $X^{\prime}\rightarrow X$.

Moreover, we require that $K_{X}+B+\mathbf{M}_{X}$ is $\mathbb{Q}$-Cartier. If $B$ is effective, we say that $(X,B,\mathbf{M})/Z$ is a generalized pair. The divisor $B$ is called the boundary part of $(X,B,\mathbf{M})/Z$, and $\mathbf{M}$ is called the moduli part. In the definition, we can replace $X^{\prime}$ with a higher birational model $X^{\prime\prime}$ and $\mathbf{M}_{X^{\prime}}$ with $\mathbf{M}_{X^{\prime\prime}}$ without changing the generalized pair. Whenever $\mathbf{M}_{X^{\prime\prime}}$ descends on $X^{\prime\prime}$, then the datum of the rational map $X^{\prime\prime}\dashrightarrow X$, $B$, and $\mathbf{M}_{X^{\prime\prime}}$ encodes all the information of the generalized pair.

Let $(X,B,\mathbf{M})/Z$ be a generalized sub-pair and $\rho\colon Y\rightarrow X$ a projective birational morphism. Then, we may write

Given a prime divisor $E$ on $Y$, we define the generalized log discrepancy of $E$ with respect to $(X,B,\mathbf{M})/Z$ to be $a_{E}(X,B,\mathbf{M})\coloneqq 1-\operatorname{mult}_{E}(B^{\prime})$. If $a_{E}(X,B,\mathbf{M})\geq 0$ for all divisors $E$ over $X$, we say that $(X,B,\mathbf{M})/Z$ is generalized sub-log canonical. Similarly, if $a_{E}(X,B+M)>0$ for all divisors $E$ over $X$ and $\lfloor B\rfloor\leq 0$, we say that $(X,B,\mathbf{M})/Z$ is generalized sub-klt. When $B\geq 0$, we say that $(X,B,\mathbf{M})/Z$ is generalized log canonical or generalized klt, respectively.

We recall the statement of the canonical bundle formula. We refer to [FG14] for the notation involved and a more detailed discussion about the topic. Let $(X,B)$ be a sub-pair. A contraction $f\colon X\rightarrow T$ is an lc-trivial fibration if

• (i)

  $(X,B)$ is a sub-pair that is sub-log canonical over the generic point of $T$;

• (ii)

  $\mathrm{rank}f_{*}\mathcal{O}_{X}(\lceil\mathbf{A}^{*}(X,B)\rceil)=1$, where $\mathbf{A}^{*}(X,B)$ is the b-divisor defined in Example 2.1; and

• (iii)

  there exists a $\mathbb{Q}$-Cartier divisor $L_{T}$ on $T$ such that $K_{X}+B\sim_{\mathbb{Q}}f^{*}L_{T}$.

Condition (ii) above is automatically satisfied if $B$ is effective over the generic point of $T$. Given a sub-pair $(X,B)$ and an lc-trivail fibration $f\colon X\rightarrow T$, there exist b-divisors $\mathbf{B}$ and $\mathbf{M}$ over $T$ such that the following linear equivalence relation, known as the canonical bundle formula, holds

For a prime divisor $P\subset T$, its coefficient in $\mathbf{B}_{T}$ is given by the formula $1-\mathrm{lct}_{\eta_{P}}(X,B;f^{*}P)$, where the symbol $\mathrm{lct}_{\eta_{P}}$ denotes the log canonical threshold over the generic point $\eta_{P}$ of $P$. Then, we set $\mathbf{M}_{T}\coloneqq L_{T}-(K_{T}+\mathbf{B}_{T})$. If $f^{\prime}\colon X^{\prime}\rightarrow T^{\prime}$ is a higher model of $f$ with morphisms $\phi\colon X^{\prime}\rightarrow X$ and $\psi\colon T^{\prime}\rightarrow T$, one repeats this algorithm with $(X^{\prime},B^{\prime})$ and $L_{T^{\prime}}$, where $K_{X^{\prime}}+B^{\prime}=\phi^{*}(K_{X}+B)$ and $L_{T^{\prime}}=\psi^{*}L_{T}$. We refer to [PS09]*§ 7 for more details.

The b-divisor $\mathbf{B}$ is often called the boundary part in the canonical bundle formula; it is a canonically defined b-divisor. Furthermore, if $B$ is effective, then so is $\mathbf{B}_{T}$. The b-divisor $\mathbf{M}$, in turn, is often called the moduli part in the canonical bundle formula, and it is in general defined only up to $\mathbb{Q}$-linear equivalence. The linear equivalence in (2.1) holds at the level of b-divisors: namely,

where $\mathbf{K}$ denotes the canonical b-divisor of $T$. Let $I$ be a positive integer such that $I(K_{X}+B)\sim 0$ along the generic fiber of $f$. Then, by [PS09]*Construction 7.5, we may choose $\mathbf{M}$ in its $\mathbb{Q}$-linear equivalence class so that

The moduli b-divisor $\mathbf{M}$ is expected to detect the variation of the fibers of the morphism $f$. In this direction, we have the following statement.

Here we recall the notion of boundedness for a set of pairs, and we introduce a suitable notion of boundedness for fibrations.

Here, we recall a few facts about the geometric Tate–Shafarevich group. We will limit ourselves ot state only those facts regarding the Tate–Shafarevich group that will be needed in the article. We refer to [DG94, Gro94] for a detailed development of the theory we need. An introduction to the topic over a one-dimensional base can be found in [Sha65]. Finally, all the needed facts about étale cohomology and group cohomology can be found in [Mil80, Mil06].

Let $Y$ be a variety defined over a field of characteristic zero, and let $k(Y)$ be its field of fractions. Let $\pi\colon X\rightarrow Y$ be an elliptic fibration, that is, a contraction whose general fiber is an elliptic curve. Then, the generic fiber $X_{{\eta_{Y}}}$ is a genus 1 curve over $k(Y)$ with possibly no $k(Y)$-rational points. Let $J(X)_{{\eta_{Y}}}$ denote its Jacobian. Then, $X_{{\eta_{Y}}}$ is a principal homogeneous space over $J(X)_{{\eta_{Y}}}$ defined over $k(Y)$. Furthermore, we can consider a projective model $\rho\colon J(X)\rightarrow Y$ realizing $J(X)_{{\eta_{Y}}}$ over $Y$.

For some finite Galois extension $K\colon k(Y)$, the variety $X_{{\eta_{Y}}}$ acquires a $K$-rational point. In particular, $X_{{\eta_{Y}}}$ and $J(X)_{{\eta_{Y}}}$ are non-canonically isomorphic over $K$. From a geometric point of view, it means that we can find a finite Galois morphism $Y^{\prime}\rightarrow Y$ such that $X\times_{Y}Y^{\prime}$ admits a rational section.

Now, let $E$ be an elliptic curve defined over $k(Y)$, and let $L\colon k(Y)$ be a finite Galois extension. We want to consider all the elliptic fibrations $\pi\colon X\rightarrow Y$ so that $X_{{\eta_{Y}}}$ is a principal homogeneous space over $E$ admitting an $L$-rational point. That is, we are interested in elliptic fibrations $\pi\colon X\rightarrow Y$ that acquire a rational section after the base change $\mathrm{Spec}(L)\rightarrow Y$ and whose associated Jacobian fibration has $E$ as generic fiber. Said otherwise, $\pi$ is an elliptic fibration whose generic fiber becomes isomorphic to $E_{L}\coloneqq E\otimes_{k(Y)}L$ as $L$-schemes. This set is parametrized by the cohomology group $H^{1}(\mathrm{Gal}(L\colon k(Y)),E(L))$. More precisely, this group parametrizes the isomorphism classes of the generic fibers $X_{{\eta_{Y}}}$ so that $\mathrm{Pic}^{0}(X_{{\eta_{Y}}})\simeq E$ and $X_{{\eta_{Y}}}$ has an $L$-rational point. Equivalently, the group $H^{1}(\mathrm{Gal}(L\colon k(Y)),E(L))$ parametrizes the birational classes of the elliptic fibrations with base $Y$ and associated Jacobian fibration corresponding to $E$ that acquire a rational section after the base change $\mathrm{Spec}(L)\rightarrow Y$. Notice that, by standard properties of group cohomology, the order of the elements of $H^{1}(\mathrm{Gal}(L\colon k(Y)),E(L))$ is finite and divides $[L\colon k(Y)]$.

Now, if we want to consider all the principal homogeneous spaces over $E$, we need to consider all possible finite Galois extensions $L\colon k(Y)$. This information is encoded in the Weil–Châtelet group $WC(E)\coloneqq H^{1}(\mathrm{Gal}(\overline{k(Y)}\colon k(Y)),E(\overline{k(Y)}))$, which is the direct limit of all possible groups $H^{1}(\mathrm{Gal}(L\colon k(Y)),E(L))$ as above. In particular, we have the following geometric consequence.

The Weil–Châtelet group $WC(E)$ is a very large group that parametrizes all the birational classes of all elliptic fibrations $\pi\colon X\rightarrow Y$ whose Jacobian fibration is a compactification of $E$ over $Y$. That is, $WC(E)$ parametrizes all birational classes of elliptic fibrations $\pi\colon X\rightarrow Y$ whose geometric generic fiber is isomorphic to $E_{\overline{k(Y)}}$ as $\overline{k(Y)}$-schemes. This group admits a natural subgroup, called the Tate–Shafarevich group, which is denoted by ${}_{Y}(E)$. For a formal definition of ${}_{Y}(E)$, we refer to [DG94]. Here, we limit ourselves to the following characterization as a set:

where $X_{C}\rightarrow Y$ is some proper model of the curve $C$ defined over $k(Y)$ [Gro94]*§ 3. Thus, ${}_{Y}(E)$ imposes pretty restrictive conditions on the type of singular fibers that can occur. In particular, multiple fibers cannot occur over codimension 1 points of the base.

Here we collect the technical statements that will be needed in the course of the main proofs.

We recall that, given a sub-pair $(W,\Delta)$ and a morphism $\pi\colon W\rightarrow U$, we say that $(W,\Delta)$ is log smooth over $U$ if $W$ is smooth, $\operatorname{Supp}(\Delta)$ is a divisor with simple normal crossings, and every stratum of $(W,\operatorname{Supp}(\Delta))$ (including $W$) is smooth over $U$.