Abstract. We prove the bigness of the Chow-Mumford line bundle associated to a $\bQ$-Gorenstein family of log Fano varieties of maximal variation with uniformly K-stable general geometric fibers. This result generalizes a recent theorem of Codogni and Patakfalvi to the logarithmic setting.

§ INTRODUCTION

Throughout this note, we work over an algebraically closed field $k$ of characteristic zero.

Let $f\colon (X,D)\to T$ be a flat morphism of relative dimension $n$ from a normal projective pair to a normal projective variety, such that $-(K_{X/T}+D)$ is $\bQ$-Cartier and $f$-ample. We define the Chow-Mumford line bundle (CM line bundle for short) to be the pushforward cycle

The main result of this article is the following theorem.

Let $f\colon (X,D)\to T$ be a flat morphism of relative dimension $n$ from a normal projective pair to a normal projective variety, such that $-(K_{X/T}+D)$ is $\bQ$-Cartier and $f$-ample. Assume that $D$ does not contain any fibers.

* Bigness: If each fiber $(X_t,D_t)$ is klt, the general geometric fibers $(X_t,D_t)$ are uniformly K-stable, and the variation of $f$ is maximal, then $\lambda_{f,D}$ is big.

* Ampleness: If all the geometric fibers $(X_t,D_t)$ are uniformly K-stable, then $\lambda_{f,D}$ is ample.

We compare this theorem with the main result of [8].

* Let $f\colon (X,D)\to T$ be as in <ref>. Then it is proved in <cit.> that:

* Pseudo-effectivity: If $T$ is smooth and $(X_t,D_t)$ is K-semi-stable for general geometric fibers $X_t$, then $\lambda_{f,D}$ is pseudo-effective.

* Nefness: If all fibers $X_t$ are normal and $(X_t,D_t)$ is K-semi-stable for all geometric fibers, then $\lambda_{f,D}$ is nef.

Hence, combining this result with our main theorem, we obtain a logarithmic version of <cit.>.

* In  <cit.>, respectively <cit.>, it is assumed that the very general geometric fiber is K-semi-stable, respectively uniformly K-stable. By [5] and [6], this is equivalent to the seemingly stronger assumption that the general geometric fiber is K-semi-stable, respectively uniformly K-stable.

* Compared to <cit.>, for the bigness statement we use the additional assumption that all fibers $(X_t,D_t)$ are klt. This hypothesis is natural for applications to moduli space of K-stable Fano varieties, where the families we consider have klt fibers (see <ref>).

The study of the CM line bundle is relevant to the construction of the moduli space of K-stable Fano varieties. It is conjectured that the functor $\sM_{n,v}^\text{Kss}$, sending $S\in \text{Sch}_k$ to

\begin{equation*}
\mathcal{M}^{\text{Kss}}_{n,v}(S)=\begin{Bmatrix}
\text{Proper flat morphisms } X\to S \text{ whose geometric fibers} \\ \text{are }n\text{-dimensional K-semistable } \bQ\text{-Fano varieties }\\ \text{of volume }v\text{, satisfying Kollar's condition }.
\end{Bmatrix}
\end{equation*}

is represented by an Artin stack of finite type over $k$ and admits a projective good moduli space $M_{n,v}^\text{Kps}$ (in the sense of [2]), whose closed points are in bijection with $n$-dimensional K-polystable $\bQ$-Fano varieties of volume $v$, and that the projectivity of $M_{n,v}^\text{Kps}$ is witnessed by a multiple of the CM line bundle. A large part of this conjecture is now verified:

The moduli functor $\sM_{n,v}^\text{Kss}$ is an Artin stack of finite type over $k$ and admits a separated good moduli space $M_{n,v}^\text{Kps}$ whose $k$-points parametrize K-polystable $\bQ$-Fano varieties of dimension $n$ and volume $v$.

The projectivity of $M_{n,v}^\text{Kps}$ by means of the CM line bundle is not proved yet. If we restrict to the subfunctor of smoothable $\bQ$-Fano varieties, it is known that the CM line bundle descends to a big and nef line bundle on the moduli space, and restricts to an ample one on open locus parametrizing smooth Fano manifolds [22].

Similar results are known about the subfunctor $\sM_{n,v}^\text{uKs}$ parametrizing uniformly K-stable $\bQ$-Fano varieties:

The functor $\sM_{n.v}^\text{uKs}$ is a separated Deligne-Mumford stack of finite type over $k$, and admits a coarse moduli space $M_{n,v}^\text{uKs}$ that is a separated algebraic space.

It is conjectured that $M_{n,v}^\text{uKs}$ is quasi-projective, and that a sufficiently divisible multiple of the CM line bundle descends to an ample line bundle on $M_{n,v}^\text{uKs}$.

The moduli functors $\sM_{n,v}^\text{Kss}$ and $\sM_{n,v}^\text{uKs}$ have natural extensions to the logarithmic setting, although it is not clear yet what the Kollar's condition should be. The logarithmic versions of the conjectures are expected to be true. In particular, as a consequence of <ref> we have:

If $M$ is a proper algebraic space with is the moduli space of some class of K-semi-stable log Fanos with the uniform K-stable locus $M^u\subseteq M$ being open, then the normalization of $M^u$ is q quasi-projective scheme over $k$.

We refer to <ref> for the precise definitions.

§.§ Overview of the proof

The proof of the bigness statement is based on the following idea. Let $(X,D)\to T$ be a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fibers. By [8], we know that $\lambda_{f,D}$ is a pseudo-effective divisor. Assume that the components of $D$ are $\bQ$-Cartier. Then for a small perturbation $D^\epsilon$ of $D$, the family $(X,D^\epsilon)\to T$ has the same properties. Hence the perturbed CM line bundle $\lambda_{f,D^\epsilon}$ remains pseudo-effective. By understanding the variation of $\lambda_{f,D}$ into $\lambda_{f,D^\epsilon}$, we will deduce that $\lambda_{f,D}$ belongs to the interior of the pseudo-effective cone. If the components of $D$ are not $\bQ$-Cartier, we use techniques from the MMP to run a similar analysis.

§.§.§ Curve base and $\bQ$-Cartier coefficients

The variation of $\lambda_{f,D}$ is easy to analyse when the base $T$ is a smooth curve, and all the reduced components $D^i$ of $D$ are $\bQ$-Cartier. Notice that

Let $D^\epsilon=D-\sum_i\epsilon_i D^i$ be a perturbed boundary. Then $\lambda_{f,D^\epsilon}$ is pseudo-effective for small values of $\epsilon$, which means $\deg\lambda_{f,D^\epsilon}\geq 0$. We calculate this degree as above:

\begin{eqnarray*}
-\deg \lambda_{f,D^\epsilon}&=&\left(-K_{X/T}-D+\sum_i\epsilon_iD^i\right)^{n+1}\\
&=&(-K_{X/T}-D)^{n+1}+(n+1)\sum_i\epsilon_i(-K_{X/T}-D)^n.D^i+O(\epsilon^2) \\
& = & -\deg\lambda_{f,D} +(n+1)\sum_i\epsilon_i(-K_{X/T}-D)^n.D^i+O(\epsilon^2)
\end{eqnarray*}

So for small values of $\epsilon$, the function $\epsilon\mapsto -\deg \lambda_{f,D^\epsilon}$ behaves like an affine polynomial with linear coefficients $(-K_{X/T}-D)^n.D^i$. Assume that these first-order derivatives $(-K_{X/T}-D)^n.D^i$ are all positive. Then $\deg \lambda_{f,D}$ cannot be too small in comparison to them, for otherwise $\deg\lambda_{f,D^\epsilon}<0$ for a small value of $\epsilon$.

We estimate these first order derivatives as follows. For positive integers $r_0,\dots,r_N$, let

$$D^{(r_\bullet)}=X^{(r_0)}\times_T \left(D^1\right)^{(r_1)}\times_T\dots\times_T \left(D^N\right)^{(r_N)}$$

and $L$ be the Cartier divisor on $D^{(r_\bullet)}$ given by the sum of the pullbacks of $-K_{X/T}-D$ restricted to the different factors. Then the self-intersection of $L$ depends only on $(-K_{X/T}-D)^{n+1}$, $(-K_{X/T}-D)^n.D^i$ and $r_i$ (<ref>). Moreover, if $r_\bullet$ is suitably chosen, we can infer some positivity of $L$ from the positivity of the sheaf

$$\det\left(f_*\OO_X(-K_{X/T}-D)\otimes \bigotimes_{i}f_*\OO_{D^i}(-K_{X/T}-D)\right).$$

The positivity of this determinant sheaf is a consequence of the maximal variation assumption via Kollar's ampleness lemma (<ref>). From the positivity of $L$, we deduce a positive lower bound for the first-order derivatives $(-K_{X/T}-D)^n.D^i$ (<ref>).

It is useful to replace $-K_{X/T}-D$ in the argument with a sufficient twist of $-K_{X/T}-D$ by the pullback of $\lambda_{f,D}$, since we obtain a nef divisor <cit.>. This replacement has technical significance, but does not affect the strategy.

§.§.§ General case

The CM line bundle behaves well with respect to base-change (<ref>); in particular, it holds that $\lambda_{f,D}.C=\deg\lambda_{f_C,D_C}$ for a smooth curve $C$ mapping to $T$. Hence if $T$ has higher dimension, we can base-change over a general curve $C$, apply the previous case and obtain $\lambda_{f,D}.C>0$. However, this does not suffice to prove that $\lambda_{f,D}$ is big, as the boundary of the cone of movable curves of $T$ need not be spanned by classes of movable irreducible curves. Nevertheless, this strategy still works if we keep a precise track of the positivity.

* First we need to estimate the derivatives $(-K_{X_C/C}-D_C)^n.D^i_C$. We can construct $D^{(r_\bullet)}$ and $L$ as before, and conclude to some positivity of $L$. However the base-change $D^{(r_\bullet)}\times_T C$ might not be flat over a general curve $C$, which creates difficulties. Thus we construct the product from a suitable birational model of $X$ (<ref>). Then we use the ampleness lemma to estimate the derivatives (<ref>), and we can garantee that these derivatives do not simultaneously go to zero when $[C]$ gets closer to the boundary of the movable cone (<ref>).

* Once we have a uniform control on the derivatives, we would like to perturb the boundary $D$. However the components $D^i$ might not be $\bQ$-Cartier. Using the techniques of the MMP, we produce a birational model $W$ of $X$ on which some components become $\bQ$-Cartier, and which remains a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with general geometric fibers (see <ref> for the precise statement). Then we are in position to perform the perturbation argument on $W$ (<ref>) and conclude.

As the repeated use of birational models suggests, our proof of <ref> is quite technical. We emphasize that the core idea is the one presented in <ref>, and that a large part of this paper is dedicated to the extension of this idea to the general case.

§.§ Organization of the paper

In <ref> we gather some notations, recall the characterization of K-stability in terms of the $\delta$-invariant and discuss base-change of divisors and the definition of the CM line bundle. The statement of the ampleness lemma is recalled in <ref>, and we gather some facts about the Knudsen-Mumford expansion in <ref>. In <ref> we show how to perturb a family of pairs so that a part of the boundary becomes $\bQ$-Cartier, while the other relevant properties of the family are preserved. <ref> is devoted to the proof of the main theorems, and the technical results are gathered in the appendix.

§.§ Acknowledgments

I would like to thank my advisor Zsolt Patakfalvi for his concern, his support and his help. I am also grateful to Giulio Codogni, Roberto Svaldi and Maciej Zdanowicz for many helpful conversations.

§ PRELIMINARIES

§.§ Notations and conventions

We follow the standard terminology of [16]. A pair $(X,D)$ is the data of a normal variety $X$ and an effective $\bQ$-Weil divisor $D$ such that $K_X+D$ is $\bQ$-Cartier. We refer to <cit.> for the definition of klt and lc pairs.

A pair $(X,D)$ is Fano if $X$ is projective and $-K_X-D$ is ample. A pair $(X,D)$ is weak log Fano if it is a klt projective pair such that $-K_X-D$ is big and nef. A pair $(X,D)$ is log Fano if it is a klt Fano pair. We say that $X$ is $\bQ$-Fano if $(X,0)$ is log Fano.

If $X$ is a Noetherian variety, an open subset $U\subset X$ is called big if $X-U$ has codimension at least $2$ in $X$.

A birational proper morphism $\pi\colon Y\to X$ between projective varieties is called small if the exceptional locus of $\pi$ has codimension at least $2$.

Let $X$ be a projective variety. We say that a smooth curve mapping to $X$ is a general movable curve if it is the normalization of a general curve in a family of curves covering $X$.

When fixing a general movable curve, we can always assume that it is not contained in a proper closed subset of $X$. By <cit.>, a $\bQ$-Cartier divisor $D$ on $X$ is big (resp. pseudo-effective) if and only if $D.C>0$ (resp. $D.C\geq 0$) for every general movable curve $C\to X$.

A $\bQ$-Gorenstein family of log Fano pairs $f\colon (X,D)\to T$ is the data of a flat morphism $f\colon X\to T$ between normal projective varieties, and of an effective Weil $\bQ$-divisors $D$, such that

* the fibers of $f$ are normal and connected,

* the support of $D$ does not contain any fiber,

* $(X_t,D_t)$ is klt for each $t\in T$ (the definition of the restricted divisor $D_t$ is given in <ref>), and

* $-K_{X/T}-D$ is an $f$-ample $\bQ$-Cartier divisor.

Let $f\colon (X,D)\to T$ be a $\bQ$-Gorenstein family of log Fano pairs. Then $f$ has maximal variation if there is a non-empty dense open subset $V\subset T$ such that for every point $t\in V$, the set $\{t'\in V\mid (X_t,D_t)\cong (X_{t'},D_{t'})\}$ is finite.

Let $X$ be a normal variety and $D$ a Weil $\bQ$-divisor on $X$. For $c\in\bQ$, the $c$-coefficient part of $D$ is defined to be

where the sum runs through the set of prime Weil divisors $E$ of $X$. We have $D=\sum_{c\in\bQ} cD_c$. For simplicity, if $\{c\in\bQ \mid D_c\neq 0\}=\{c_1,\dots,c_m\}$, we let $D^i:=D_{c_i}$ so that $D=\sum_{i=1}^m c_iD^i$.
We will also denote by $D^i$ the corresponding reduced closed subscheme.

Let $D$ be a $\bQ$-Cartier divisor on a proper $k$-scheme $X$ of dimension $n$. We denote by $D^n=(D \cdots  D)$ its self-intersection. If $D$ is ample, we will also write $\text{vol}(D)=D^n$. If $\LL$ is a line bundle on $X$, we denote by $c_1(\LL)$ the associated Cartier divisor. We will write

$$[c_1(\LL)]^m.D^{n-m}=(\underbrace{c_1(\LL) \cdots  c_1(\LL)}_{m \text{ times}}\cdot \underbrace{D \cdots D}_{n-m \text{ times}}).$$

Let $f\colon X\to T$ be a morphism of proper $k$-schemes. We denote by $X^{(m)}$ the $m$-times fiber product of $X$ with itself over $T$. It comes with projection morphisms $p_i\colon X^{(m)}\to X$ for $i=1,\dots,m$ and the structural morphism $f^{(m)}\colon X^{(m)}\to T$. Given a line bundle $\LL$ on $X$, or a Cartier divisor $D$ on $X$, we write

$$\LL^{(m)}:=\bigotimes_{i=1}^m p_i^*\LL,\quad D^{(m)}:=\sum_{i=1}^m p_i^*D.$$

§.§ K-stability

In this section, we recall briefly one characterization of the $\delta$-invariant for log Fano pairs, and its relation with K-stability. We refer to [11] for the algebraic definition of K-stability in terms of test configurations.

Consider a $n$-dimensional weak log Fano pair $(X,D)$. Given a prime divisor $E$ over $X$, we let $A_{X,D}(E)$ be the log discrepancy of $E$ with respect to $(X,D)$, and define the quantity

where $\pi\colon Y\to X$ is a birational model on which $E$ appears.

Let $(X,D)$ be a log Fano pair. The $\delta$-invariant of $(X,D)$ is given by

where $E$ runs through the prime divisors over $X$.

The original definition of the delta invariant in [10] is formulated in terms of basis-type divisors of the anti-log-canonical linear system. However, the above one is more convenient for our purpose. The equivalence between the two definitions is proved in <cit.> and <cit.>.

The relation between the delta invariant and K-stability of log Fano pairs is given by the following theorem. In this article, we use this characterization as the definition of uniform K-stability.

Let $(X,D)$ be a Fano pair.

* $(X,D)$ is K-semistable if and only if $(X,D)$ is klt and $\delta(X,D)\geq 1$.

* $(X,D)$ is uniformly K-stable if and only if $(X,D)$ is klt and $\delta(X,D)>1$.

The next result will be useful in <ref>.

Let $(X,D)$ be a weak log Fano pair, and $\Gamma$ an effective $\bQ$-Cartier divisor supported on $\text{Supp}(D)$. Assume that

where $E$ runs through the divisors over $X$. Then for all rational $\epsilon >0$ small enough,

Replacing $\Gamma$ by a small multiple, we may assume that there is an effective $\bQ$-Cartier divisor $\Gamma'$ on $X$ such that
$\Gamma+\Gamma'\in |-K_X-D|_{\bQ}$. By assumption, there is an $a>0$ such that $A_{X,D}(E)\geq (1+a)S_{X,D}(E)$ for all divisors $E$. Choose $a>a'>0$ and define the function

for $\epsilon\in \bR$. Since $\lim_{\epsilon\to 0}f(\epsilon)=1+a'$, we can fix $\epsilon_0=\epsilon_0(a')>0$ such that for all $0<\epsilon<\epsilon_0$, we have $f(\epsilon)<1+a$. Since we assumed $\text{Supp}(\Gamma)\subset\text{Supp}(D)$, we can also arrange that $D-\epsilon_0\Gamma$ is effective.

Now let $E$ be a divisor over $X$, appearing on a birational model $\pi\colon Y\to X$. For any $\epsilon>0$ and any $x\in\bR_+$, observe that

\begin{eqnarray*}
\text{vol}(\pi^*(-K_X-D+\epsilon\Gamma)-xE) &\leq & \text{vol}(\pi^*(-K_X-D+\epsilon(\Gamma+\Gamma'))-xE)\\
\end{eqnarray*}

Integrating over $x$, we obtain

\begin{eqnarray*}
\int_0^{\infty}\text{vol}(\pi^*(-K_X-D)-xE)dx&\leq &\int_0^\infty\text{vol}((1+\epsilon)\pi^*(-K_X-D)-xE)dx\\
&=&(1+\epsilon)^{n+1}\int_0^\infty \text{vol}(\pi^*(-K_X-D)-xE)dx.
\end{eqnarray*}

This inequality implies

$$S_{X,D}(E)\leq (1+\epsilon)^{n+1}\frac{(-K_X-D)^n}{(-K_X-D+\epsilon\Gamma)^n} S_{X,D_\epsilon\Gamma}(E).$$

Now take $\epsilon<\epsilon_0$ and assume that $A_{X,D-\epsilon\Gamma}(E)\leq (1+a')S_{X,D-\epsilon\Gamma}(E)$. Then we have

\begin{eqnarray*}
A_{X,D}(E)\leq A_{X,D}(E)+\epsilon\cdot\text{ord}_E(\Gamma)=A_{X,D-\epsilon\Gamma}(E)&\leq& (1+a')S_{X,D-\epsilon\Gamma}(E)\\
& \leq & f(\epsilon)S_{X,D}(E)\\
\end{eqnarray*}

which is a contradiction. Thus $A_{X,D-\epsilon\Gamma}(E)>(1+a')S_{X,D-\epsilon\Gamma}$ for all $\epsilon<\epsilon_0$. Since $\epsilon_0$ does not depend on $E$, the proof is complete.

§.§ Base-change of divisors

In this subsection, we consider a flat morphism $f\colon X\to T$ between normal projective varieties, and an effective Weil $\bQ$-divisor $D$ on $X$, such that

* the fibers of $f$ are normal and connected,

* the support of $D$ does not contain any fiber.

Let $U\subset X$ be the smooth locus of $f$. By <cit.>, every Weil divisor on $X$ not containing any component of a fiber is Cartier over $U$. Moreover $U$ is relatively big over $T$, so it is a big open subset of $X$.

Let $u\colon S\to T$ be a morphism from a normal projective variety $S$. We define the divisorial pullback of $D$ along $u$ as follows. The open set $U_S$ is big in $X_S$. Since $D|_U$ is $\bQ$-Cartier, we let the divisorial pullback $D_S$ of $D$ along $u$ be the unique Weil $\bQ$-divisor extending the $\bQ$-Cartier divisor $(D|_U)_{U_S}$.

In particular, if $t\in T$ is a closed point, then $D_t$ is the unique Weil $\bQ$-divisor of $X_t$ extending the $\bQ$-Cartier divisor $D|_{U\cap X_t}$.

It follows from this definition that there is a $\bQ$-linear equivalence

\begin{equation}\label{eqn:base_change_of_pairs}
K_{X_S/S}+D_S\sim_\bQ v^*(K_{X/T}+D)
\end{equation}

where $v\colon X_S\to X$ is the induced morphism, see <cit.>.

In the situation of <ref>, let $S\to T$ be a morphism from a normal projective variety. If $D$ is Cartier, then the divisorial pullback of $D$ and the pullback of $D$ as Cartier divisor along $\sigma\colon X_S\to X$ agree.

If $U$ is the smooth locus of $X\to T$, then $\sigma^*D$ represents $D_{X_S}$ on $U_S$ by definition. A Cartier divisor on a normal variety is determined in codimension one, and $U_S$ is big. Thus $\sigma^*D$ represent the Weil divisor $D_{X_S}$.

In the situation of <ref>, there is a dense big open set $U\subseteq T$ over which all the possible unions of components of $D$ (with the reduced structure) are flat.

Let $E$ be a union of components of $D$ with the reduced structure. By generic flatness, the locus of $T$ over which $E$ is flat, is dense open. Pick a codimension one point $t\in T$, and any $x\in E$ such that $f(x)=t$. The morphism $\OO_{T,t}\to \OO_{E,x}$ is flat if and only if the uniformizer $\pi$ of $\OO_{T,t}$ is sent to a non-zero-divisor of $\OO_{E,x}$. Now if $\pi$ is a zero-divisor in $\OO_{E,x}$, then the components of $X_t$ passing through $x$ are contained in $E$. But by assumption $X_t$ is irreducible and $E$ does not contain any fiber, so this cannot happen. Thus $E\to T$ is flat at $x$. Since $x$ is arbitrary, we conclude that $t\in U$. Therefore $U$ is big.

In the situation of <ref>, there is a dense open set $V\subseteq U$ such that:

* for any $v\in V$ and $c\in\mathbb{Q}$, the divisorial restriction $(D_c)_v$ is equal to $(D_v)_c$.

* for any $v\in V$ and $c\in\mathbb{Q}$, the scheme-theoretic fiber $D_c\times k(v)$ is equal to the divisorial restriction $(D_c)_v$ with the reduced structure.

Base-changing if necessary, we may assume that $U=T$. Given a reduced Weil divisor $E$ not containing any fiber, we claim that the divisorial restriction $E_t$ is reduced for a general $t\in T$. In view of <ref>, we may assume that $E$ is Cartier. Since the claim is local on $X$, we may assume that $E$ is actually principal, say cut out by $s\in \OO(X)$. Then $\OO_X/(s)$ is reduced and flat over $T$; thus its fiber $\OO_{X}/(s)\otimes k(t)$ over a general $t\in T$ is reduced <cit.>. This means exactly that the divisor $E_t$ is reduced. If $E'$ is another reduced divisor not containing any fiber, such that $E$ and $E'$ have no common components, by applying the claim to $E+E'$ we see that the divisorial restrictions $E_t$ and $E'_t$ have no common component for a general $t\in T$. Let $E$ run through the coefficient parts $D_c$ of $D$ to obtain the first assertion.

If we consider $E$ as a reduced closed subscheme, the scheme-theoretic fiber $E\times k(t)$ has pure codimension one for all $t\in T$ <cit.> and is reduced for a general $t\in T$. Combining this and the first assertion, we obtain the second assertion.

In the situation of <ref>, let $C\to T$ be a smooth curve whose image intersect $V$. Let $Z:=X\times_T C$ and $D_Z=\sum_{c\in\bQ} c(D_Z)_c$ be the divisorial pullback of $D$. Then $(D_Z)_c$ is the divisorial pullback of $D_c$ for all $c\in\bQ$.

We have to check that any two distinct components parts $D_c$ and $D_{c'}$, have distinct divisorial pullbacks. Since these divisorial pullbacks are horizontal, this can be checked on a general fiber of $Z\to C$. Since $C$ meets $V$, the result follows from <ref>.

§.§ The CM line bundle

Let $f\colon X\to T$ be a flat morphism of relative dimension $n$ between normal projective varieties, let $D$ be an effective $\bQ$-divisor on $X$ such that $-(K_{X/T}+D)$ is $\bQ$-Cartier and $f$-ample. Assume also that the fibers of $f$ are normal and that $\text{Supp}(D)$ does not contain any fiber. Then the Chow-Mumford line bundle is defined by

By <cit.>, $\lambda_{f,D}$ is a $\bQ$-Cartier $\bQ$-Weil divisor. It is compatible with base-change in the following sense:

In the above situation, let $\tau\colon S\to T$ be a morphism from a normal variety $S$. Let $f_S\colon X_S\to S$ be the induced morphism and $D_S$ be the divisorial pullback in the sense of <ref>. Then
$\tau^*\lambda_{f,D}= \lambda_{f_S,D_S}$.

We refer to <cit.> for the proof and more background.

§ AMPLENESS LEMMA

Let $U$ be a normal $k$-variety that can be embedded as a big open subset of a projective variety. Let $W,Q_1,\dots, Q_s$ be vector bundles on $U$ of respective ranks $w,q_1,\dots,q_s$. Assume that there exist morphisms $\phi_i\colon W\to Q_i$ for $i=1,\dots,s$ which are generically surjective. Assume also that the classifying map

$$U(k)\to \left( \prod_{i=1}^s \emph{Gr}(w,q_i)(k)\right) \big/ \emph{GL}(k,w)$$

is finite-to-one on a dense subset of $U$. Then for any ample Cartier divisor $B$ on $U$, there exists a positive integer $m>0$ and a non-zero morphism

$$\emph{Sym}^{mq}\left( \bigoplus_{i=1}^{s^2w^2} W\right)\longrightarrow \OO_T(-B)\otimes \left( \bigotimes_{i=1}^s \det Q_i\right)^{\otimes m}$$

where $q=\sum_{i=1}^s q_i$.

This result is contained in the proof of <cit.>. More precisely, let $W'=\oplus_{i=1}^s W$, $Q'=\oplus_{i=1}^s Q_i$ and $\phi=\oplus_{i=1}^s \phi_i$. As explained in <cit.>, $\phi$ is generically surjective, and there is a dense open set of $U$ where the corresponding classifying map has finite fibers. Notice that during the proof of <cit.>, the assumption of weak positivity is not used until the last three lines; in particular, the equation $(5.5.5)$ holds without this assumption. Thus it applies to our setting, and we get a non-zero morphism

$$\text{Sym}^{mq}\left(\bigoplus_{i=1}^{\text{rank}(W')} W' \right)\longrightarrow \OO_U(-B)\otimes \left(\det Q' \right)^{\otimes m}$$

as desired.

§ ABOUT THE KNUDSEN-MUMFORD EXPANSION

We recall a special case of <cit.>. Let $f\colon X\to T$ be a projective morphism between Noetherian schemes of relative dimension $n$. We do not require that $f$ is flat. Let $L$ be an $f$-very ample Cartier divisor. Then for every $q\gg 0$, there is an isomorphism

$$\det f_*\OO_X(qL)\cong \det Rf_*\OO_X(qL)\cong \bigotimes_{i=0}^{n+1}\sM_i^{\otimes \binom{q}{i}}$$

where $\sM_i\in\text{Pic}(T)$. We call this expression the Knudsen-Mumford expansion of $\OO_X(qL)$, and refer to the $\sM_{i}$ as the coefficients of the expansion. This isomorphism is moreover functorial: if $S\to T$ is a morphism from a Noetherian scheme, then it holds that

$$\det (f_S)_*\OO_{X_S}(qL_S)\cong \bigotimes_{i=0}^{n+1}(\sM_i)_S^{\otimes \binom{q}{i}}, \quad q\gg 0.$$

Let $f\colon X\to T$ be a flat morphism of relative dimension $n$ between normal projective varieties, let $D$ be an effective $\bQ$-divisor on $X$ such that $-(K_{X/T}+D)$ is $\bQ$-Cartier and $f$-ample. Assume also that the fibers of $f$ are normal and that $\text{Supp}(D)$ does not contain any fiber. Let $s$ be such that $s(-K_{X/T}-D)$ is very ample over $T$. Then one can show that

where $\sM_{n+1}$ is the leading coefficient of the Knudsen-Mumford expansion of $s(-K_{X/T}-D)$. See <cit.>.

The following proposition characterizes the numerical class of $\sM_{n+1}$ in several situations.

Let $f\colon X\to T$ be an equidimensional projective morphism of relative dimension $n$ between Noetherian proper schemes (we do not require $f$ to be flat). Let $A$ be an $f$-very ample Cartier divisor on $X$, and $\sM_{n+1}$ be the leading coefficient of the Knudsen-Mumford expansion of $\OO_X(A)$.

* Assume that $A$ is nef. For any smooth curve $C\to T$, it holds that

* Assume that $A$ is nef and $X$ is generically reduced. Let $X'\to X$ be the normalization morphism and $A'$ be the pullback of $A$. Then $\sM_{n+1}.C= f'_*((A')^{n+1}).C$ for a general movable curve $C\to T$.

* Assume that $T$ is normal and $f$ is flat with normal fibers. Then for any smooth curve $C\to T$, we have $\sM_{n+1}.C= f_*(E^{n+1}).C=A_C^{n+1}$.

Fix a smooth curve $C\to T$. In any case, since both $qA$ and $qA_C$ are relatively very ample for $q\gg 0$, both sheaves $f_*\sO_X(qA)$ and $(f_C)_*\OO_{X_C}(qA_C)$ are locally free with vanishing $R^i$, $i>0$. It follows from the functoriality of the Knudsen-Mumford expansion that

\begin{equation}\label{eqn:KM_over_curve}
\det\left[f_*\sO_X(qA)\right].C=\deg \det\left[(f_C)_*\sO_{X_C}(qA_C)\right], \quad q\gg 0.
\end{equation}

With this set-up:

* Assume that $A$ is nef. The left-hand side of equation <ref> is given by

$$\sum_{i=0}^{n+1}\binom{q}{i}\sM_i.C=\frac{q^{n+1}}{(n+1)!}\sM_{n+1}.C +O(q^n),$$

where $\sM_i$ are the Knudsen-Mumford coefficients of $\sO_X(A)$. Now consider the right-hand side of the same equation. By Riemann-Roch, for $q$ large enough we have

$$h^0(X,\OO_{X_C}(qA_C))=\deg \det\left[(f_C)_*\OO_{X_C}(qA_C)\right]+\chi(C,\OO_C)\cdot \text{rk }(f_C)_*\OO_{X_C}(qA_C).$$

Since $A_C$ is nef, we have

$$h^0(X, \sO_{X_C}(qA_C))=\frac{q^{n+1}}{(n+1)!}A_C^{n+1}+O(q^n)$$

by <cit.>. Since the fibers of $f$ are $n$-dimensional, $(q\mapsto \text{rk }(f_C)_*\OO_{X_C}(qA_C))$ is a polynomial in $q$ of degree at most $n$. Hence

$$\deg \det\left[(f_C)_*\OO_{X_C}(qA_C)\right]=\frac{q^{n+1}}{(n+1)!}A_C^{n+1}+O(q^n).$$

It follows by comparing the leading coefficients in <ref> that $A_C^{n+1}=\sM_{n+1}.C$.

* Assume that $A$ is nef and $X$ generically reduced. The normalization morphism $X'\to X$ is finite, so $A'$ is nef and relatively ample over $T$. Say that $sA'$ is relatively very ample for some $s>0$, and let $\sM'_{n+1}$ be the leading coefficient of the Knudsen-Mumford polynomial of $\OO_{X'}(sA')$. Since $X$ is generically reduced, the normalization $X'\to X$ is an isomorphism away from a closed subset $Z\subsetneq X$. If $C\to T$ is a smooth curve which intersects $f(X-Z)$, the pullback morphism $(X')_C\to X_C$ is birational. Using the first assertion, we obtain that

By <cit.> it holds that $\sM'_{n+1}=f'_*(sA')^{n+1}$, so the second assertion follows.

* Assume that $T$ is normal and $f$ is flat with normal fibers. Then both $X$ and $X_C$ are normal. It follows from <cit.> that for $q\gg 0$

$$\det f_*\OO_X(qA)=\frac{q^{n+1}}{(n+1)!}f_*(A^{n+1})+O(q^n)$$

$$\det (f_C)_*\OO_{X_C}(qA_C)=\frac{q^{n+1}}{(n+1)!}(f_C)_*(A^{n+1}_C)+O(q^n)$$

in the Chow rings of $T$ and $C$ respectively.	It follows that $\sM_{n+1}=f_*(A^{n+1})$, and by intersecting with $C$ that

$$\sM_{n+1}.C=f_*(A^{n+1}).C=\deg (f_C)_*(A_C^{n+1})=E_C^{n+1}$$

as claimed.

§ PERTURBATION OF THE FAMILY

Consider a $\bQ$-Gorenstein family $(X,D)\to T$ of log Fano pairs of maximal variation, with uniformly K-stable general fibers. We show in <ref> that we can find a model of $(X,D)$ with the same properties over $T$, and on which some components of the boundary $D$ are $\bQ$-Cartier.

We need a few preliminary lemmas. For the first one, we use the terminology of [3].

Let $f\colon X\to T$ be a projective morphism between quasi-projective normal varieties. Let $D$ be an effective $\bQ$-Cartier $\bQ$-divisor on $X$ such that $(X,D)$ is (klt) weak log Fano over $T$. Assume that $D_1,\dots,D_m$ are effective $\bQ$-Cartier $\bQ$-divisors on $X$ with supports contained in the support of $D$. Then there exists a polyhedron $P\subset\bR^m$ with the following properties:

* $P$ contains the origin, and its interior $\emph{int}(P)$ is contained in $(\bR_{>0})^m$; and

* for every rational vector $(\epsilon_1,\dots,\epsilon_m)\in \emph{int}(P)$, the log canonical models of $(-K_X-D+\sum_i \epsilon_iD_i)$ over $T$ have isomorphic underlying variety.

Fix a general very ample divisor $A$ on $X$, which has no component in common with $D$. Since $X$ is of Fano type over $T$, there is a $\bQ$-boundary $\Delta$ such that $(X,\Delta)$ is klt, $\Delta$ is big over $T$ and $a(-K_X-D-A)\sim_{\bQ,T} K_X+\Delta$ for some small rational $a>0$. Replacing $A$ by a general member of its linear system, we may assume that $(X,\Delta+aA)$ is also klt <cit.>. Notice that $K_X+\Delta+aA+\sum_i\epsilon_i D_i\sim_{\bQ,T}a(-K_X-D)+\sum_i \epsilon_iD_i$. So a log canonical model of $(X,\Delta+aA+\sum_i\epsilon_i D_i)$ over $T$ is also a $(-K_X-D+\sum_i \epsilon_iD_i)$-log canonical model of $X$ over $T$. Therefore it is equivalent to prove that for $\epsilon_i$ as in the statement, the pairs $(X,\Delta+aA+\sum_i\epsilon_i D_i)$ have a log canonical model over $T$ whose underlying variety is isomorphic.

Let us write $A':=aA$ and define the affine cone $V:=\Delta+\sum_i\bR_+ D_i$ in $\text{Weil}(X)_\bR$. Since $(X,\Delta+A')$ is klt, there is an open Euclidean neighborhood $\sU$ of $\Delta\in V$ such that for all $\Gamma\in \sU$, the pair $(X,\Gamma+A')$ is klt. Also, since $K_X+\Delta+A'\sim_{\bQ,T}a(-K_X-D)$ is big over $T$, we may shrink $\sU$ so that $K_X+\Gamma+A'$ is big over $T$ for all $\Gamma\in \sU$. With the notations of <cit.>, this implies that $\sU\subset \sE_{A',f}(V)$. Combining Corollary 1.1.5 and Theorem E of [3], we obtain that there are finitely birational contractions $\psi_i\colon X\dashrightarrow Z_i$ over $T$, $i=1,\dots,n$, and a decomposition

$$\sE_{A',f}(V)=\bigcup_{i=1}^n \sW_i$$

where each $\sW_i=\sW_{\psi_i,A',f}(V)$ is a rational polytope, such that for each $\Gamma\in \sW_i$, the underlying variety of a weak log canonical model of $(X,\Gamma+A')$ over $T$ is isomorphic to $Z_i$.

By <cit.>, for every $\Gamma\in \sU$ the pair $(X,\Gamma+A')$ has a log canonical model over $T$. Since relative log canonical models are in particular relative weak log canonical models, we obtain that the underlying variety of a log canonical model of $(X,\Gamma+A')$ is isomorphic to one of the $Z_i$. The decomposition of $\sE_{A',f}(V)$ induces a finite decomposition of $\sU$. Thus we may find a polyhedron $P\subset \bR^m$ containing the origin, with non-empty interior $\text{int}(P)\subset(\bR_{>0})^m$ such that

$$\left(\bigcup_{(\epsilon_1,\dots,\epsilon_m)\in \text{int}(P)} \Delta +\sum_i\epsilon_i D_i\right)\subset \sU\cap \sW_i$$

for some $i$. This finishes the proof.

Let $(X,D)\to T$ be a $\bQ$-Gorenstein family $(X,D)\to T$ of log Fano pairs of maximal variation. Write $D=\sum_i c_iD^i$ as in <ref>. Then there is a rational number $r>0$ such that for all $i$ such that $D^i$ is $\bQ$-Cartier, and all rational $\epsilon\in (-r;r)$, the family $(X,D+\epsilon D^i)\to T$ has maximal variation.

Take $r=\min_{i\neq j}\{\frac{1}{2}|c_i-c_j|\}$.

Let $f\colon (X,D)\to T$ be a flat equidimensional morphism from a normal projective pair to a smooth projective variety. Assume that every fiber $(X_t,D_t)$ is klt. Then:

* $(X,D)$ is klt, and

* for any closed point $t\in T$, if $H_1,\dots,H_d$ $(d=\dim T)$ are general Cartier divisors such that $\bigcap_i H_i=\{t\}$, then $(X,D+\sum_if^*H_i)$ is dlt.

Let $t\in T$ be a closed point, and let $H_1,\dots,H_d$ $(d=\dim T)$ be general Cartier divisors on $T$ such that $\bigcap_iH_i=\{t\}$.

To begin with, we prove that $(X,D)$ is klt. Indeed, we can choose the $H_1,\dots,H_{d-1}$ in a general linear system, so the iterated hyperplane sections $X^m:=\bigcap_{i=1}^mf^*H_i$ are normal varieties [26] for $m\leq d-1$. By inversion of adjunction, since $(X_t,D_t)$ is assumed to be klt, we obtain that $(X^{d-1},D|_{X^{d-1}}+f^*H_d)$ is plt along $X_t$ <cit.>. Hence $(X^{d-1},D|_{X^{d-1}})$ is klt along $X_t$. We repeat this argument to obtain that $(X,D)$ is klt along $X_t$. The choice of $t$ was arbitrary, so we conclude that $(X,D)$ is klt.

Using inversion of adjunction for lc pairs <cit.>, a similar argument shows that $(X,D+\sum_{i\leq m}f^*H_i)$ is lc for all $m\leq d$.

We can be more precise about the singularities of $(X,D+\sum_{i=1}^df^*H_i)$: we claim that the pair is dlt if the $H_i$ are suitably chosen. Indeed, we can choose the $H_m$ inductively with the property that

$$\text{for each } I\subseteq \{1,\dots,m\}, \text{ the intersection }X^I:=\bigcap_{i\in I}f^*H_i \; \text{ is irreducible and normal.}$$

Each of these conditions is satisfied for a general $H_m$ passing through $t$, except for the condition on $H_d$ that $\bigcap_{i=1}^d f^*H_i=X_t$ is irreducible and normal. But this is satisfied for any choice of $H_d$, since $X_t$ is assumed to be irreducible and normal.

Notice that $(X,D+\sum_{i=1}^df^*H_i)$ is snc at every generic point of the $X^I$. Moreover, by a similar argument to above, every pair $(X^I,D|_{X^I})$ is klt.

Now let $E$ be an exceptional divisor over $X$ whose center $c_X(E)$ does not belong to the snc locus of $(X,D+\sum_if^*H_i)$. Let $J:=\{i \mid c_X(E)\in \text{Supp}(f^*H_i)\}$. If $J=\emptyset$, then $a(E;X,D+\sum_if^*H_i)=a(E;X,D)>-1$. If $J$ is non-empty, then $c_X(E)$ defines a point of codimension $\geq 1$ in $X^J$, not contained in any $\text{Supp}(f^*H_i|_{X^I})$ for $i\notin J$. If $a(E;X,D+\sum_{i=1}^df^*H_i)\leq -1$, then by a repeated application of <cit.> we obtain a divisor $E^J$ over $X^J$ such that $-1\geq a(E^J;X^J,D|_{X^J}+\sum_{i\notin J}f^*H_i)=a(E^J;X^J,D|_{X^J})$. This is a contradiction, since $(X^J,D|_{X^J})$ is klt. This shows that $(X,D+\sum_if^*H_i)$ is dlt.

Let $f\colon (X,D)\to T$ be a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fiber. Assume that $T$ is smooth. Then there exists a positive number $r>0$ with the following property. For every coefficient part $\Gamma:=D^i$ of $D$ (as in <ref>), there exists a small proper birational morphism $\nu\colon W\to X$ such that:

* the strict transform $\Gamma_W$ of $\Gamma$ is $\bQ$-Cartier, and

* for any rational $0<\epsilon <r$, the family $(W,D_W-\epsilon \Gamma_W)\to T$ is a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fibers. (Here $D_W$ denotes the strict transform of $D$.)

Since there are finitely many coefficient part of $D$, we only need to prove the result for a fixed $\Gamma$.

We construct $\nu\colon W\to X$, using the technique of dlt blow-up (see <cit.>).

* The pair $(X,D)$ is klt by <ref>, so by <cit.> there is a small proper birational morphism $\mu\colon Y\to X$ where $Y$ is a $\mathbb{Q}$-factorial projective variety. Denote by $D_Y$ the strict transform of $D$, and $\Gamma_Y$ the strict transform of $\Gamma$. We have

$$\mu^*(K_X+D)\sim_\bQ K_Y+D_Y.$$

* For $\epsilon>0$, run a $(-K_Y-D_Y+\epsilon\Gamma_Y)$-MMP over $X$ to obtain a relative log canonical model. By <ref>, this model $W$ is the same for all $0<\epsilon\ll 1$. Denote by $p\colon Y\dashrightarrow W$ the induced morphism and $D_W:=p_*D_Y,\Gamma_W:=p_*\Gamma_Y$.

Our construction is pictured by the following diagram:

(Y,D_Y,\Gamma_Y)\arrow[d, "\mu"] \arrow[r, dashrightarrow, "p"] & (W,D_W,\Gamma_W)\arrow[dl, "\nu"]\arrow[ddl, "g", bend left] \\
(X, D,\Gamma) \arrow[d, "f"] &&\\
\end{tikzcd}$$

We must show that for small $\epsilon >0$, the morphism $(W,D_W-\epsilon\Gamma_W)\to T$ is flat between normal projective varieties, of maximal variation, with (klt) log Fano fibers and uniformly K-stable general geometric fibers, and that $\Gamma_W$ is $\bQ$-Cartier. First we establish the global properties of $W$ and $\Gamma_W$.

* The morphism $\nu\colon W\to X$ is small. Indeed, $\mu$ is small and $p$ extracts no divisors. Notice that $D_W$ is equal to the strict transform of $D$.

* Since $W$ is the end product of an MMP, it is normal. Moreover, since $\nu$ is small and $(X,D)$ is klt by <ref>, $(W,D_W)$ is klt and hence $W$ is Cohen-Macaulay.

* The $\bQ$-divisor $\Gamma_W$ is $\bQ$-Cartier. Indeed, $p_*(-K_Y-D_Y+\epsilon \Gamma_Y)=-K_W-D_W+\epsilon\Gamma_W$ is $\bQ$-Cartier by construction. It holds that $\nu^*(-K_X-D)=-K_W-D_W$, and so $-K_W-D_W$ is also $\bQ$-Cartier. Therefore $\Gamma_W$ is $\bQ$-Cartier.

* Let $\epsilon$ be a rational number. We have

$$\epsilon \Gamma_Y\equiv_X -K_Y-(D_Y-\epsilon\Gamma_Y)$$

and thus

$$\epsilon \Gamma_W\equiv_X \underbrace{-K_W-(D_W-\epsilon\Gamma_W)}_{\text{ample over } X}$$

Hence $\Gamma_W$ is a $\mathbb{Q}$-Cartier $\mathbb{Q}$-divisor which is ample over $X$. Furthermore,

\begin{eqnarray*}
-K_W-(D_W-\epsilon\Gamma_W) & = & p_*(-K_Y-(D_Y-\epsilon\Gamma_Y)) \\
& = & p_*(p^*\nu^*(-K_X-D)+\epsilon\Gamma_Y) \\
& = & \nu^*(-K_X-D)+\epsilon\Gamma_W.
\end{eqnarray*}

Now by <cit.>,

$$-K_W+g^*K_T=-K_{W/T}, \quad -K_X+f^*K_T=-K_{X/T}$$

and therefore $-K_{W/T}-D_W+\epsilon\Gamma_W$ is ample over $T$ for $0<\epsilon\ll 1$.

Now we study the morphism $W\to T$.

* By Stein factorization, the fibers of $\nu$ are connected. Since the fibers of $f$ are irreducible, we deduce that $g$ has connected fibers.

To establish some properties of the fibers $W_t$, we work in a neighborhood of an arbitrary closed point $t\in T$. Let $H_1,\dots,H_d$ be general Cartier divisors such that $\bigcap_i H_i=\{t\}$.

* Since $\nu$ is small, it is an isomorphism over the snc-locus of $(X,D+\sum_if^*H_i)$ <cit.>. Since $(W,D_W+\sum_{i\leq m}g^*H_i)$ is crepant to $(X,D+\sum_{i\leq m}f^*H_i)$, we obtain from <ref> that $(W,D_W+\sum_ig^*H_i)$ is dlt. Notice that $(W,D_W+\sum_ig^*H_i)$ is also lc <cit.>.

* From <cit.> we deduce that every irreducible component of $W_t=\bigcap_{i=1}^dg^*H_i$ is normal of codimension $d$. Assume that $W_t$ has at two different components; by connectedness of the fibers of $W\to T$, the two components must intersect, and the intersection is a union of lc centers of $(W,D_W+\sum_{i=1}^dg^*H_i)$ <cit.>. This contradicts <cit.>. So $W_t$ is irreducible and normal of codimension $d$.

* Since $W$ is Cohen-Macaulay, $T$ smooth and the fibers $W_t$ equidimensional, the morphism $g\colon W\to T$ is flat <cit.>.

* Assume that some fiber $W_t$ is contained in the support of $D_W$. Since $W_t$ dominates $X_t$ and $D_W$ dominates $D$, we obtain that $X_t$ is contained in the support of $D$, which is impossible. Thus $\text{Supp}(D_W)$ contains no fibers of $W\to T$.

Finally we study the pairs $(W_t,(D_W-\epsilon\Gamma_W)_t)$.

* Fix a closed point $t\in T$. For a general choice of $H_1,\dots,H_d$ cutting $t$ on $T$, we know that $(W,D_W+\sum_ig^*H_i)$ is dlt. By <cit.>, we obtain that the pair $(W_t,(D_W)_t)$ is dlt. Since $\lfloor (D_W)_t\rfloor=0$, we obtain that $(W_t,(D_W)_t)$ is klt <cit.>. Hence $(W_t,(D_W-\epsilon\Gamma_W)_t)$ is klt for every $0<\epsilon=\epsilon(t)\ll 1$.

* The property of being klt is open on the base and $T$ is quasi-compact. Thus if $0<\epsilon\ll 1$, for every $t\in T$ the fiber $(W_t,(D_W-\epsilon\Gamma_W)_t)$ is klt.

* Since $\mu$ is small, for a general $t\in T$ the morphism $W_t\to X_t$ is small and $(D_W)_t$ is the strict transform of $D_t$. Fix one such $t$ for which $(X_t,D_t)$ is uniformly K-stable. Then for every prime divisor $E$ over $W_t$, we have

$$A_{W_t,(D_W)_t}(E)=A_{X_t,D_t}(E),\quad A_{W_t,(D_W)_t}(E)=S_{X_t,D_t}(E),$$

so by <ref> we have

Moreover $(W_t,(D_W)_t)$ is a weak log Fano pair. Thus we may apply <ref> to obtain that

for all $\epsilon>0$ small enough. Since $(W_t,(D_W)_t-\epsilon\Gamma_t)$ is a log Fano pair, we conclude by <ref> that $(W_t,(D_W)_t-\epsilon\Gamma_t)$ is uniformly K-stable for all $\epsilon>0$ small enough. By the openess of the uniform K-stable locus <cit.>, we conclude that: for all $0<\epsilon \ll 1$, the general fiber of the family $(W,D_W-\epsilon\Gamma_W)\to T$ is uniformly K-stable.

* For a general $t\in T$, the pair $(X_t,D_t)$ is a log canonical model of $(W_t,(D_W)_t)$. Thus $(W_t,(D_W)_t)\cong (W_u,(D_W)_u)$ implies that $(X_t,D_t)\cong (X_u,D_u)$. Moreover by <ref>, for $\epsilon$ small enough, $(W_t,(D_W-\epsilon\Gamma_W)_t)\cong (W_u,(D_W-\epsilon\Gamma_W)_u)$ if and only if $(W_t,(D_W)_t)\cong (W_u,(D_W)_u)$. Therefore $(W,D_W-\epsilon\Gamma_W)\to T$ has maximal variation for $0<\epsilon<\ll 1$.

This shows that $(W,D_W-\epsilon \Gamma_W)\to T$ has the required properties for all rational numbers $0<\epsilon\ll 1$.

§ PROOF OF THE MAIN RESULTS

We divide the proof of the main results in several steps. In <ref>, we set-up the notational framework of the proof. The argument using the ampleness lemma is given in <ref>. The estimates of the derivatives (see <ref>) is obtained in <ref>, and the perturbation argument is given in <ref>.

§.§ General notations

Let $T$ be a smooth variety and $f\colon (X,D=\sum_{i=1}^N c_iD^i)\to T$ be a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fibers. We introduce the following additional notations, and shall use them for the rest of this section.

* Let $n:=\dim X-\dim T$ and $v:=((-K_{X/T}-D)_t)^n$. We write $\delta:=\delta(X_{\bar{\eta}}, D_{\bar{\eta}})$ where $\eta$ is the generic point of $T$. Then $\delta$ is the very general value of $\delta(X_t,D_t)$ <cit.>.

* We let $\lambda_{f,D}:=-f_*(c_1(-K_{X/T}-D)^{n+1})$ be the CM line bundle provided by the family $(X,D)\to T$.

* The restrictions of the morphism $f$ to the support of the $D^i$ (with the reduced structure) are denoted $f_i\colon D^i\to T$. We also write $D^0=X$ and $f_0=f$. Notice that each $f_i$ is surjective.

* Fix a rational number $\alpha>\max \{1,	\frac{\delta}{(\delta-1)v(n+1)}\}$.

In the situation of <ref>, let $\iota\colon C\to T$ be a smooth curve. Form the Cartesian square

Z \arrow[r, "\sigma"]\arrow[d, "h"] & X\arrow[d, "f"] \\
C\arrow[r, "\iota"] & T
\end{tikzcd}$$

Note that $Z$ is normal because the fibers of $h$ are normal. Let $D_Z$ be the divisorial pullback of $D$, with coefficient parts $D^i_Z$ (see <ref>; it is more convenient for the proof to write $D_Z$ instead of $D_C$). According to <ref> and to <ref>, we have

\begin{equation}\label{eqn:base_change_for_CM_line_bundle}
K_{Z/C}+D_Z\sim_{\bQ}\sigma^*(K_{X/T}+D), \quad \iota^*\lambda_{f,D}=\lambda_{h,D_Z}.
\end{equation}

§.§ Application of the ampleness lemma

In the situation of <ref>, let $M:=-K_{X/T}-D+2\alpha f^*\lambda_{f,D}$. Then for $r$ divisible enough, there exists a positive integer $d$ such that the line bundle

$$ \bigotimes_{i\geq 0}\; \det (f_i)_*\OO_{D^i}(drM)$$

is big on $T$.

The argument is very similar to <cit.> and to <cit.>. We let $V\subseteq U$ be the open subsets of $T$ given by <ref> and <ref>. By the maximal assumption, shrinking $V$ if necessary, we may assume that the isomorphism classes of fibers of $(X,D)\times_T V\to V$ are finite.

If $r$ is a fixed integer divisible by the Cartier index of $M$, for an arbitrary $d\in\bZ$ we write

$$\MM_d:=\OO_X(dr(-K_{X/T}-D+2\alpha f^*\lambda_{f,D})),\quad \MM_d^{D_i}:=\MM_d|_{D^i}.$$

We choose an integer $r\geq 2$ such that for every $d>0$:

* $-dr(K_{X/T}+D)$ and $dr\alpha\lambda_{f,D}$ are Cartier;

* $\MM_d=\OO_X(dr(-K_{X/T}-D+2\alpha f^*\lambda_{f,D}))$ is $f$-very ample;

* $R^jf_*\MM_d=0$ for all $j\geq 1$;

* for each $i\geq 1$: $(R^j(f_i)_*\MM_r^{D_i})|_V=0$ for all $j\geq 1$;

* for each $i\geq 1$: $(f_*\MM_1)|_V\to ((f_i)_*\MM_1^{D^i})|_V$ is surjective.

These conditions imply that $f_*\MM_1$ and $((f_i)_*\MM_1^{D_i})|_V$ are locally free and compatible with base-change. In particular,

* if $s:=\text{rk }f_*\MM_r$, then $s=h^0(\MM_r|_{X_t})$ for all $t\in T$.

We may also assume that:

* the multiplication maps

$$\text{Sym}^df_*\MM_1\to f_*\MM_{d} \quad \text{and} \quad \text{Sym}^d(f_*\MM_1)|_V\to ((f_i)_*\MM_{d}^{D_i})|_V$$

are surjective.

Now that $r$ is chosen, we can find $d>0$ such that:

* For all $t\in T$, the kernel
$$K_t:=\ker \left[\text{Sym}^dH^0( \MM_1|_{X_t})\longrightarrow H^0(\MM_{d}|_{X_t})\right]$$
generates $\mathcal{I}_t(d)$, where $\mathcal{I}_t$ is the ideal sheaf of $X_t$ for the embedding
$$\varphi_{\MM_1|_{X_t}}\colon X_t\hookrightarrow \bP^{s-1}.$$

Here $\varphi_{\MM_1|_{X_t}}$ is only defined up to the action of GL$_s(k)$ on the target.

Hence, writing $w:=\text{rk } \text{Sym}^df_*\MM_1$ and $q_0:=\text{rk } f_*\MM_{d}$, we see that the orbit of $K_t$ in $\text{Gr}(w,q_0)/\text{GL}_s(k)$ determines the projective embedding of $X_t$ up to linear automorphisms of $\bP^{s-1}$.

* Similarly, for all $v\in V$ and $i\geq 1$, the kernel
$$K_t^{D^i}:=\ker \left[\text{Sym}^dH^0(\MM_1|_{X_v})\longrightarrow H^0(\MM_{d}^{D_i}|_{(D_i)_v})\right]$$
generates $\mathcal{I}_{v,i}(d)$, where $\mathcal{I}_{v,i}$ is the ideal sheaf of $(D_i)_v$ for the embedding

$$\varphi_{\MM_1^{D_i}|_{(D_i)_v}}\colon (D_i)_v\hookrightarrow X_v\overset{\varphi_{\MM_1|_{X_v}}}{\longrightarrow} \bP^{s-1},$$

Here $\varphi_{\MM_1^{D_i}|_{(D_i)_v}}$ is only defined up to the action of GL$_s(k)$.

Hence, writing $q_i:=\text{rk } ((f_i)_*\MM_{d}^{D_i})|_V$, we see that the orbit of $K_t^{D_i}$ in $\text{Gr}(w,q_i)/\text{GL}_s(k)$ determines the projective embedding of $(D_i)_t$ up to linear automorphisms of $\bP^{s-1}$.

Having choosen $r$ and $d$ with these properties, we let

$$W:=\text{Sym}^d(f_*\MM_1)|_U, \quad Q_0:=(f_*\MM_{d})|_U, \quad Q_i :=\left((f_i)_*\MM_{d}^{D_i}\right)\Big|_U \; (i\geq 1).$$

The sheaves $W$ and $Q_0$ are locally free by construction. For $i\geq 1$, the sheaf $(f_i)_*\MM_{d}^{D_i}$ is torsion-free, hence locally free at codimension one points. Since it is locally free over $V$, we may restrict $U$, but keeping it a big open set, so that $Q_i$ becomes locally free for all $i\geq 0$.

By construction, on $U$ there are morphisms $W\to Q_i$ for $i\geq 0$ which are surjective over $V$. We claim that the corresponding classifying map

$$\xi\colon U(k)\longrightarrow \left( \prod_{i\geq 0} \text{Gr}(w,q_i)(k)\right)\big/ \text{GL}(s,k)$$

is finite-to-one over $V(k)$ (here $\text{GL}_s(k)$ acts diagonally). Fix $t\in V(k)$. For $t'\in V(k)$, the equality $\xi(t)=\xi(t')$ holds if and only if: $X_t$ and $X_{t'}$ have isomorphic embeddings into $\bP^{s-1}$, and that under this isomorphism $(D_i)_t$ is sent to $(D_i)_{t'}$, for every $i\geq 1$. As explained at the beginning of the proof, there are only finitely many such $t'$. Thus $\xi$ is finite-to-one on $V(k)$. Hence by <ref>, given an ample line bundle $B$ on $T$, there is a non-zero morphism

\begin{equation*}
\text{Sym}^{qm}\left( \bigoplus_{i=1}^w \text{Sym}^d(f_*\MM_1)\right)\Big|_U \to \OO_U(-B_U)\otimes \left( \bigotimes_{i\geq 0}\; \left(\det (f_i)_*\MM_{d}^{D^i}\right)^{\otimes m}\right)\Big|_U
\end{equation*}

for some integer $m>0$. Since $U$ is a big dense open subset of $T$, and since both sides are restrictions of reflexive sheaves, this map extends to a non-zero morphism

\begin{equation}\label{eqn: Ampleness lemma II}
\text{Sym}^{qm}\left( \bigoplus_{i=1}^w \text{Sym}^d(f_*\MM_1)\right) \to \OO_T(-B)\otimes \left( \bigotimes_{i\geq 0}\; \left(\det (f_i)_*\MM_{d}^{D^i}\right)^{\otimes m}\right)
\end{equation}

Notice that as the right-hand side is a line bundle, this map is generically surjective.

Now let $\iota\colon C\to T$ be a general curve. We use the notations of <ref>. As $f_*\MM_1$ is compatible with base-change, we obtain

$$\iota^*f_*\MM_1\cong h_*\OO_Z(\sigma^*rM)\cong h_*\OO_Z(r(-K_{Z/C}-D_Z+2\alpha h^*\lambda_{h,D_Z})). $$

As $C$ is general, the general geometric fiber of $(Z,D_Z)\to C$ is uniformly K-stable. Thus by <cit.>, the divisor $-K_{Z/C}-D_Z+2\alpha h^*\lambda_{h,D_Z}$ is $h$-ample and nef. Moreover, since we can write

$$r(-K_{Z/C}-D_Z+2\alpha h^*\lambda_{h,D_Z})=K_{Z/C}+D_Z+\underbrace{(r+1)(-K_{Z/C}-D_Z+\alpha h^*\lambda_{h,D_Z})+(r-1)\alpha h^*\lambda_{h,D_Z},}_{\text{ nef and } h\text{-ample}}$$

we may apply <cit.> to obtain that $\iota^*f_*\MM_1$ is nef. Hence the pullback of $\text{Sym}^{qm}\left( \bigoplus_{i=1}^w \text{Sym}^d(f_*\MM_1)\right)$ to $C$ is also nef. By generality of $C$, the restriction of <ref> to $C$ is generically surjective, so we obtain that $\OO_T(-B)\Big|_C\otimes \left( \bigotimes_{i\geq 0}\; \left(\det (f_i)_*\MM_{d}^{D^i}\right)^{\otimes m}\right)\Big|_C$ is nef. This shows that the line bundle $\OO_T(-B)\otimes \bigotimes_{i\geq 0}\; \left(\det (f_i)_*\OO_{D^i}(drM)\right)^{\otimes m}$ is pseudo-effective. The conclusion follows as $B$ is ample.

§.§ Estimation of the derivatives

In the situation of <ref>:

* We keep the notation $M:=-K_{X/T}-D+2\alpha f^*\lambda_{f,D}$ used in <ref>;

* According to <ref>, we can fix a positive integer $q$ such that the divisor $qM$ is Cartier and such that the line bundle $\bigotimes_{i\geq 0}\; \det (f_i)_*\OO_{D^i}(qM)$ is big.

* We write $\NN=\OO_X(qM)$ and $\NN^{D^i}=\sN|_{D^i}$.

* If $C\to T$ is a smooth curve and $Z=X\times_T C$ as in <ref>, then we let $\NN_Z:=\sigma^*\NN$. Recall from <ref> that
$\sN_Z\cong\OO_Z(q(-K_{Z/C}-D_Z+2\alpha h^*\lambda_{f,D_Z}))$, and this line bundle is nef if $C$ meets the locus of uniformly K-stable fibers <cit.>.

We aim to give a lower bound to the intersection numbers $[c_1(\sN_Z)]^{\dim D^i_Z}.D^i_Z$. As explained in <ref>, the idea is to construct a product $D^{(r_\bullet)}$ over $T$ and then base-change over a general curve. In view of <ref>, we want the pullback of $D^{(r_\bullet)}$ to be flat over the curve. Hence it would be convenient that the restricted morphisms $D^i\to T$ are flat already. To achieve this, we pass to a birational model of $X$. Unfortunately this makes the notation quite cumbersome.

In the situation of <ref>. Let $r_i$ $(i=0,\dots ,N)$ be the generic ranks of $(f_i)_*\NN^{D^i}$.

* We let

$$D^{(r_\bullet)}:= \left( D^0\right)^{(r_0)}\times_T \dots \times_T \left(D^N\right)^{(r_N)}.$$

The projection morphism from $D^{(r_\bullet)}$ to the $i^\text{th}$ $D^j$-factor is denoted by $p^{ij}\colon D^{(r_\bullet)}\to D^j$. We denote

$$D^{(r_\bullet)}_\text{red}:=\left(D^{(r_\bullet)}\right)_\text{red},\quad D^{(r_\bullet)}_\text{norm}:=\text{normalization of } D^{(r_\bullet)}_\text{red}.$$

We define the line bundles

$$\sN^{(r_\bullet)}_\text{red}:=\text{pullback of }\sN^{(r_\bullet)}\text{ to }D^{(r_\bullet)}_\text{red},
\quad \sN^{(r_\bullet)}_\text{norm}:=\text{pullback of }\sN^{(r_\bullet)}\text{ to }D^{(r_\bullet)}_\text{norm}.$$

* Next fix a small $\bQ$-factorial proper model $\mu\colon Y\to X$. There exists one since $X$ has a klt structure; see the first step of <ref>. Denote by $D_Y=\sum_{i=1}^N c_iD_Y^i$ the strict transform of $D$. Let $\mu^i\colon D_Y^i\to D^i$ be the induced birational morphisms, with $\mu^0=\mu$. We write

$$D_Y^{(r_\bullet)}:= \left(D_Y^0\right)^{(r_0)}\times_T \dots \times_T \left(D_Y^N\right)^{(r_N)}.$$

The projection morphism from $D_Y^{(r_\bullet)}$ to the $i^\text{th}$ $D^j_Y$-factor is denoted by $p^{ij}_Y\colon D_Y^{(r_\bullet)} \to D_Y^j$.

We define the line bundles

$$ \NN_Y^{(r_\bullet)}=\bigotimes_{i,j}\left(p^{ij}_Y\right)^*\NN_Y^{D_Y^j}.$$

* If $\iota \colon C\to T$ is a smooth curve, we denote by $Y_C$, respectively $D^i_{Y_C},D_{Y_C}^{(r_\bullet)}, \NN_{Y_C}^{(r_\bullet)}, \mu^i_C, p^{ij}_{Y_C},$ the scheme-theoretic pullbacks of $Y$, respectively $D^i_Y, D_Y^{(r_\bullet)}, \NN_Y^{(r_\bullet)},\mu^i, p^{ij}_Y$ along $\iota$. Notice that

$$D_{Y_C}^{(r_\bullet)}=D_Y^{(r_\bullet)}\times_T C=\left(D_{Y_C}^0\right)^{(r_0)}\times_C \dots \times_C \left(D_{Y_C}^N\right)^{(r_N)}$$

and that the projection morphisms $D_{Y_C}^{(r_\bullet)}\to D^j_{Y_C}$ are exactly the $p^{ij}_{Y_C}$. Notice also that if $\sN^{D^j_{Y_C}}$ is the pullback of $\sN^{D^j_Y}$ along $D^j_{Y_C}\to D^j_Y$, then

$$\sN_{Y_C}^{(r_\bullet)}\cong \bigotimes_{i,j}\left(p^{ij}_{Y_C}\right)^*\sN^{D^j_{Y_C}}.$$

The construction of parts $(a)$ and $(b)$ is summarized by the following diagram, where the arrow $D_Y^{(r_\bullet)}\to D_\text{red}^{(r_\bullet)}$ exists by <ref><ref>.

\begin{equation*}
\begin{tikzcd}
\left(D^{j}_Y, \sN^{D^j_Y}\right)\arrow[dd, "\mu^j" left] & \left(D_Y^{(r_\bullet)},\sN_Y^{(r_\bullet)}\right)\arrow[dr]\arrow[l, "p^{ij}_Y" above]\arrow[dd, "\mu^{(r_\bullet)}" left] & & \left(D^{(r_\bullet)}_\text{norm},\sN_\text{norm}^{(r_\bullet)}\right)\arrow[dl]\\
& &  \left(D^{(r_\bullet)}_\text{red},\sN_\text{red}^{(r_\bullet)}\right)\arrow[dl] &\\
\left(D^j,\sN^{D^j}\right) & \left(D^{(r_\bullet)},\sN^{(r_\bullet)}\right)\arrow[l, "p^{ij}" above] & &
\end{tikzcd}
\end{equation*}

Next we establish some properties of these product varieties and these product line bundles.

In the situation of <ref>:

*  $D^{(r_\bullet)}$ is equidimensional over $T$, and every component dominates $T$; moreover there is a big open set of $T$ over which $D^{(r_\bullet)}$ is flat and reduced;

*  $D_Y^{(r_\bullet)}$ is reduced, flat and equidimensional over $T$, and every components dominates $T$;

*  $\mu^{(r_\bullet)}_Y\colon D^{(r_\bullet)}_Y\to D^{(r_\bullet)}$ factors through $D^{(r_\bullet)}_\text{red}$;

*  $\mu^{(r_\bullet)}$ is an isomorphism over each generic point of $D^{(r_\bullet)}$, and every component of $D^{(r_\bullet)}_Y$ dominates a component of $D^{(r_\bullet)}$;

*  $D_{Y_C}^{(r_\bullet)}$ is flat equidimensional over $C$, and it is reduced if $C$ is general.

Assertion <ref> is proved in <cit.>, and assertion <ref> will follow immediately from assertion <ref>.

Next observe that the pair $(X,D)$ is klt by <ref>, so $(Y,D_Y)$ is klt and hence $Y$ is Cohen-Macaulay. The divisors $D^i_Y$ are $\bQ$-Cartier because $Y$ is $\bQ$-factorial, so each $D^i_Y$ is also Cohen-Macaulay <cit.>. Hence all morphisms $D^i_Y\to T$ are flat <cit.>. This implies that the morphism $D_Y^{(r_\bullet)}\to T$ is flat. The fibers of $D_Y^{(r_\bullet)}\to T$ have the same dimension, so the morphism is equidimensional. If one component of $D^{(r_\bullet)}_Y$ does not dominate $T$, then it belongs to the non-flat locus, which is empty. The generic fiber if reduced, so <ref> implies that $D_Y^{(r_\bullet)}$ is reduced. This proves the assertion <ref>, and assertion <ref> is proved similarly.

To conclude, we must prove assertion <ref>. Let $V\subseteq X$ be the open subset over which $\mu\colon Y\to X$ is an isomorphism. Since $X$ is normal, $V$ is big. Thus $V_i:=V\cap D^i$ is dense in $D^i$ for all $i$. So $\mu^{(r_\bullet)}$ is an isomorphism over the open set $\sV:=V^{(r_0)}\times_T V_1^{(r_1)}\times_T\dots \times_T V_N^{(r_N)}\subset D^{(r_\bullet)}$. Let $K(T)$ be the function field of $T$. Then $(V_i)_{K(T)}$ is dense in $(D^i)_{K(T)}$ for all $i$, so $\sV_{K(T)}$ is a dense open set in the product $\left(D^{(r_\bullet)}\right)_{K(T)}$. Since every component of $D^{(r_\bullet)}$ dominates $T$, every generic point of $D^{(r_\bullet)}$ lifts to $D^{(r_\bullet)}_{K(T)}$ and must belong to $\sV_{K(T)}$. This shows that $\sV$ is dense in $D^{(r_\bullet)}$, and so $\mu^{(r_\bullet)}$ is an isomorphism over each generic point of $D^{(r_\bullet)}$. A component of $D_Y^{(r_\bullet)}$ that is contracted by $\mu^{(r_\bullet)}$ must belong to a product of the form $\left(D_Y^0\right)^{(r_0)}\times_T \dots \times_T \left(D_Y^N\right)^{(r_N)}$ with one factor $D_Y^i$ replaced by $D_Y^i-f_i^{-1}V_i$. Such a product has dimension strictly smaller than $D_Y^{(r_\bullet)}$, so by equidimensionality no component is contained in it. So assertion <ref> follows.

In the situation of <ref>,

* $\sN_\emph{norm}^{(r_\bullet)}$ is relatively ample over $T$ and pseudo-effective;

* for a general movable curve $C\to T$, the line bundle $\sN_{Y_C}^{(r_\bullet)}$ is nef.

Let $\sU\subset T$ be a non-empty open subset with the property that for all $t\in T$, the pair $(X_t,D_t)$ is uniformly K-stable. Let $C\to T$ be a smooth curve whose image intersects $\sU$. Denote by $\sN_Z$ the pullback of $\sN$ on $Z=X\times_T C$, and $\sD^i_C=D^i\times_C T$. Then by <cit.>, $\sN_Z$ and $\sN^{\sD^i_C}=\sN_Z|_{\sD^i_C}$ are nef. Thus the product line bundle

on $D^{(r_\bullet)}\times_T C$, is nef. By definition $\sN^{(r_\bullet)}_{Y_C}$ is a pullback of $\sN_Z^{(r_\bullet)}$, so it is also nef. This proves the second assertion.

By construction, the line bundle $\sN^{(r_\bullet)}$ is relatively ample. Let $\tilde{C}\to D^{(r_\bullet)}$ be a smooth curve that meets $g^{-1}\sU$. If $\tilde{C}$ is contracted by $g$, then $\sN^{(r_\bullet)}.\tilde{C}>0$ by relative ampleness. Otherwise, let $C\to T$ be the normalization of $g(\tilde{C})$. Then $\tilde{C}\to D^{(r_\bullet)}$ factors through $D^{(r_\bullet)}\times_T C$, on which the pullback of $\sN^{(r_\bullet)}$ is nef. Thus $\tilde{C}.\sN^{(r_\bullet)}\geq 0$. This shows that $\sN^{(r_\bullet)}$ is pseudo-effective. Since $\sN^{(r_\bullet)}_\text{norm}$ is the pullback of $\sN^{(r_\bullet)}$ through the finite morphism $D^{(r_\bullet)}_\text{norm}\to D^{(r_\bullet)}$, the first assertion follows.

We are now ready to estimate the intersection numbers $[c_1(\sN_Z)]^{\dim D^i_Z}.D^i_Z$ (where $Z=X\times_T C$). The first part of the proof is similar to the proof of <cit.>.

In the situation of <ref>, there are an ample Cartier divisor $A$ on some component of $D_Y^{(r_\bullet)}$ and a numerical constant $e=e(X,D,q)$ with the following property: for every general movable curve $C\to T$, it holds that

$$e(N+1)\cdot\emph{vol}(A_C)\leq \sum_{j=0}^N[c_1\left(\NN_Z\right)]^{\dim D^j_Z}.D^j_Z.$$

In particular, for every general movable curve $C\to T$,

\begin{equation}\label{eqn:Estimation_for_all_volumes}
\exists \; j=j(C)\geq 0 \quad : \quad \left[c_1\left(\NN_Z\right)\right]^{\dim D^j_Z}.D^j_Z\geq e\cdot \emph{vol}(A_C).
\end{equation}

We denote by $g\colon D^{(r_\bullet)}\to T$, $g_\text{red}\colon D^{(r_\bullet)}_\text{red}\to T$  and $g_\text{norm}\colon D^{(r_\bullet)}_\text{norm}\to T$ the structural morphisms. By <ref>, the morphism $D^{(r_\bullet)}\to T$ is equidimensional and every component dominates $T$. Thus $g_\text{red}\colon D_\text{red}^{(r_\bullet)}\to T$ and $g_\text{norm}\colon D_\text{norm}^{(r_\bullet)}\to T$ are also equidimensional morphisms, and every component of $D^{(r_\bullet)}_\text{red},D^{(r_\bullet)}_\text{norm}$ dominates $T$.

By <ref> and by the proof of <ref>, there is a big open subset $U\subseteq T$ over which $D^{(r_\bullet)}$ is flat and reduced, and the sheaves $(f_i)_*\left(\sN^{D^i}\right)|_U$ are locally free. Let $U^0:=g^{-1}U$. Since $U^0$ is reduced, it embeds as an open subset of $D_\text{red}^{(r_\bullet)}$.  Therefore the open set $U^0$ is big in $D_\text{red}^{(r_\bullet)}$ and meets every component, and so the preimage of $U^0$ in $D_\text{norm}^{(r_\bullet)}$ is big and meets every components.

& (D_\text{norm}^{(r_\bullet)},\NN_\text{norm}^{(r_\bullet)})\arrow[d]  \\
& (D_\text{red}^{(r_\bullet)},\NN_\text{red}^{(r_\bullet)}) \arrow[d] \\
U^0 \arrow[d]\arrow[r, hook] \arrow[ur, hook] & (D^{(r_\bullet)} ,\NN^{(r_\bullet)})\arrow[d, "g"] &\\
U\arrow[r, hook] & T
\end{tikzcd}$$

Let us write $\LL:=\bigotimes_{i\geq 0} \; \det (f_i)_*\NN^{D^i}$.  On $U$, there is an embedding

$$ \LL|_U \hookrightarrow \bigotimes_{i\geq 0}\bigotimes_{j=1}^{r_i}\;(f_i)_*\left(\NN^{D^i}|_U\right)\cong (g|_{U^0})_*\left(\NN^{(r_\bullet)}|_{U^0}\right)$$

where the first arrow is given by the natural embedding of $\det$ into the appropriate tensor power, and the isomorphism is given by <cit.>. By adjunction, we obtain a morphism

\begin{equation}\label{eqn:Pullback}
(g_\text{red}|_{U^0})^*\sL|_U=(g|_{U^0})^*\sL|_U\longrightarrow \NN^{(r_\bullet)}|_{U^0}=\sN_\text{red}^{(r_\bullet)}|_{U^0}.
\end{equation}

Since $U^0$ dominates $T$, the map <ref> is non-zero. We may pull back this map to the normalization $D_\text{norm}^{(r_\bullet)}$. Since the preimage $U^0$ in $D_\text{norm}^{(r_\bullet)}$ is big, by reflexivity this pullback morphism extends to a non-zero morphism

$$(g_\text{norm})^*\LL\longrightarrow \NN_\text{norm}^{(r_\bullet)}$$

which induces a non-zero map

\begin{equation}\label{eqn:Pullback_on_normalization}
\NN_\text{norm}^{(r_\bullet)}\otimes (g_\text{norm})^*\LL\longrightarrow\left(\NN_\text{norm}^{(r_\bullet)}\right)^{\otimes 2}.
\end{equation}

The line bundle $\NN_\text{norm}^{(r_\bullet)}$ is relatively ample over $T$ and pseudo-effective by <ref>. Moreover $g^*\LL$ is the pullback of a big divisor by <ref>. Thus the left-hand side of <ref> is big on every component by <ref>. Hence $\NN_\text{norm}^{(r_\bullet)}$ is big on at least one component. This implies that $\sN^{(r_\bullet)}_\text{red}$ is big on at least one component.

By <ref>, $\mu^{(r_\bullet)}\colon D^{(r_\bullet)}_Y\to D^{(r_\bullet)}$ is an isomorphism in codimension zero and factors through $D_\text{red}^{(r_\bullet)}$. Hence we obtain that $\NN_Y^{(r_\bullet)}$ is big on one component $P$ of $D_Y^{(r_\bullet)}$. So we may write

$$\left(\NN_Y^{(r_\bullet)}|_P\right)^{\otimes m}\cong\OO_P(mA+mE)$$

where $A$ is ample and $E$ is effective on $P$, for some $m>0$.

Now fix a general movable curve $C\to T$ with the following properties. Firstly, no component of its preimage in $P$ is contained in the support of $E$. Secondly, the line bundles $\sN_Z$ and $\sN_{Y_C}^{(r_\bullet)}$ are nef (see <ref>). Thirdly, the induced morphisms $\mu_C^j\colon D^j_{Y_C}\to \sD^j_Z:=D^j\times_T C$ are birational for all $j$; this is acheviable since $\mu^j\colon D^j_Y\to D^j$ are all birational. Fourthly, for all $j$ the product $\sD^j_Z$ agrees with the $j$-coefficient part $D^j_Z$ of $D_Z$ in codimension one; this is achievable by combining <ref> and <ref>.

With such a curve $C\to T$ fixed, we write

$$\OO_{P_C}(mA_C+mE_C)\cong \left(\NN_{Y_C}^{(r_\bullet)}\Big|_{P_C}\right)^{\otimes m}\cong \left(\bigotimes_{i,j}\left(p^{ij}_{Y_C}\right)^*\NN^{D^j_{Y_C}}\right)\Big|_{P_C}^{\otimes m}.$$

Since $A$ is ample, its pullback $A_C$ is also ample. By our choice of $C$, the divisor $E_C$ is effective. Thus

$$\text{vol}(A_C)\leq \text{vol}(A_C+E_C)=\left[  c_1\left( \sN^{(r_\bullet)}_{Y_C}\Big|_{P_C}\right)\right]^{\dim P_C}.$$

It holds by equidimensionality that $\dim P_C=\dim D_{Y_C}^{(r_\bullet)}$. Using <cit.>, we see that

$$\left[c_1\left( \NN_{Y_C}^{(r_\bullet)}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}}=\left[  c_1\left( \sN^{(r_\bullet)}_{Y_C}\Big|_{P_C}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}}+\sum_{P'}\left[  c_1\left( \sN^{(r_\bullet)}_{Y_C}\Big|_{P'}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}}$$

where $P'$ runs through the component of $D_{Y_C}^{(r_\bullet)}$ not contained in $P_C$. Since $\sN_{Y_C}^{(r_\bullet)}$ is nef, the sum over $P'$ is non-negative, and therefore we obtain

\begin{equation}\label{eqn:Estimation_volume_I}
\text{vol}(A_C)\leq\left[c_1\left( \NN_{Y_C}^{(r_\bullet)}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}}
\end{equation}

On the other hand, by <ref> we have:

\begin{equation}\label{eqn:Estimation_volume_II}
\left[c_1\left( \NN_{Y_C}^{(r_\bullet)}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}}=\sum_{i=0}^Nd_i \left[c_1(\NN^{D^j_{Y_C}})\right]^{\dim D^j_{Y_C}}\prod_{j\neq i}\left[c_1\left(\NN^{D^j_{Y_C}}_t\right)\right]^{\dim D^j_{Y_C}-1}
\end{equation}

for some numerical constants $d_i=d_i(X,D,q)>0$. The right-hand side of <ref> can be simplified: observe that

$$\left[c_1\left(\NN^{D^j_{Y_C}}\right)\right]^{\dim D^j_{Y_C}}=\left[c_1\left(\NN^{\sD^j_Z}\right)\right]^{\dim \sD^j_Z}=\left[c_1\left(\NN^{D^j_Z}\right)\right]^{\dim D^j_Z}=\left[c_1\left(\NN_Z\right)\right]^{\dim D^j_Z}.D^j_Z.$$

Indeed, the first equality holds because $\mu^i_C$ is birationa, while the second equality holds because  $\sD^j_Z$ and $D^j_Z$ are equal in codimension 1 and $\NN$ has full support (see <cit.>).
We also have

for $t\in C$ closed. Since $\NN^{D^j_Y}\cong (\mu^j)^*\NN^{D^j}$ is the pullback of a relatively ample line bundle, the volume of $(\NN^{D^j_Y})_t$ is strictly positive. Moreover $\NN_Z$ is nef so the quantities $\left[c_1\left(\NN_Z\right)\right]^{\dim D^j_Z}.D^j_Z$ are non-negative. Therefore, setting $a=\max_i\left\{\text{vol}\left(\NN^{D^i_Y}_t\right)^{r_i-1}\right\}$ and $b=\max_i\{d_i\}$, it follows from <ref> and <ref> that

$$\text{vol}(A_C)\leq \left[c_1\left( \NN_{Y_C}^{(r_\bullet)}\right)\right]^{\dim D_{Y_C}^{(r_\bullet)}} \leq ab\sum_{i=0}^N \left[c_1\left(\NN_Z\right)\right]^{\dim D^i_Z}.D^i_Z.$$

Notice that $ab$ depends only on $(X,D)$ and $q$, so put $e=(ab(N+1))^{-1}$ to conclude.

The next lemma shows that $\text{vol}(A_C)$ cannot converge to zero when $[C]$ gets close to the boundary of the movable cone.

Let $A$ be the ample $\bQ$-Cartier divisor on the component $P$ of $D_Y^{(r_\bullet)}$ given by <ref>. Then there exists a big $\bQ$-Cartier divisor $\Psi$ such that for a general movable curve $C\to T$, we have $\emph{vol}(A_C)=\Psi.C$.

By <ref> the scheme $P$ is generically reduced and the morphism $P\to T$ is equidimensional, say of relative dimension $N$. So we may apply <ref> to $(P,A)\to T$. Namely, let $A'$ be the pullback of $A$ to the normalization $P'$ of $P$, and let $f'\colon P'\to T$ be the induced morphism. Then for a general smooth curve $C\to T$, we have $A_C^{N+1}=f_*'(A')^{N+1}.C$. By <ref>, $f_*'(A')^{N+1}$ is big. We take $\Psi=f_*'(A')^{N+1}$.

§.§ Variation of the boundary

In this subsection, we follow <ref> and let $\Psi$ be the big $\bQ$-Cartier divisor on $T$ obtained in <ref>.

Given a general smooth curve $C\to T$, the inequality <ref> gives a lower bound for some $(c_1[\sN_Z])^{\dim D^j_Z}.D^j_Z$. As explained in <ref>, we derive a lower bound on $ \lambda_{f,D}.C$. The case $j=0$ is the easiest.

Let $C\to T$ be a smooth curve. Assume that $\left[c_1\left(\NN_Z\right)\right]^{n+1}\geq e\cdot (\Psi.C)$ for some $e>0$. Then
$\lambda_{f,D}.C\geq e_0\cdot (\Psi.C)$
for some numerical constant $e_0=e_0(X,D,q,e)>0$.

Recall the fact that $\sN_Z\cong\OO_Z(q(-K_{Z/C}-D_Z+2\alpha h^*\lambda_{f,D_Z}))$ (see <ref>). We have

\begin{eqnarray*}
\frac{1}{q^{n+1}}\left[c_1\left(\NN_Z\right)\right]^{n+1} & = & (-K_{Z/C}-D_Z)^{n+1} + (n+1)2\alpha(h^*\lambda_{h,D_Z} \cdot (-K_{Z/C}-D_Z)^n) \\
& = & \deg_C h_*((-K_{Z/C}-D_Z)^{n+1})+(n+1)2\alpha \cdot \deg\lambda_{h,D_Z} \cdot ((-K_{Z/C}-D_Z)_t)^n) \\
& = & \deg_C \lambda_{h,D_Z} \cdot \left[-1+2\alpha v(n+1)\right] =( \lambda_{f,D}.C)\cdot  \left[\underbrace{-1+2\alpha v(n+1)}_{>0 \text{ by choice of }\alpha}\right]
\end{eqnarray*}

We let $e_0=e(q^{n+1}[2\alpha v(n+1)-1])^{-1}$ to obtain the desired inequality.

If $j>0$, we wish to relate $(c_1[\sN_Z])^{\dim D^j_Z}.D^j_Z$ to the a first-order derivative of $\lambda_{f,D}.C$ as the component $D^j_Z$ is perturbed. Since $D^j_Z$ might not be $\bQ$-Cartier, we introduce a birational model where it is $\bQ$-Cartier.

In the situation of <ref>. By <ref>, we may fix $r_{X,D}\in (0;1)$ with the property that for every coefficient part $D^i$ of $D$, there is a small birational morphism $W_i\to X$ such that for all rational numbers $\epsilon\in (0;r_{X,D})$, the family $(W_i,D_{W_i}-\epsilon D^i_{W_i})\to T$ is a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fibers.

Fix an index $j>0$ and any smooth curve $\iota\colon C\to T$. Write $\nu\colon W:=W_j\to X$ and $\Gamma:=D^j$. We let

$$V:=W\times_X Z\cong W\times_T C.$$

With the notations of <ref>, we obtain the diagrams

(V,D_V,\Gamma_V)\arrow[dd, bend right=80, "g" left]\arrow[r, "\tau"]\arrow[d,"\mu"] & (W,D_W,\Gamma_W)\arrow[d, "\nu" left]\arrow[dd, bend left=80, "f'" right] \\
(Z,D_Z,D^j_Z)\arrow[r, "\sigma"] \arrow[d, "h"] & (X,D,\Gamma) \arrow[d, "f" left] \\
C \arrow[r, "\iota"] & T
\end{tikzcd}$$

where $D_Z$ is the divisorial base-change of $D$, $D_Z^j$ the $c_j$-coefficient part of $D_Z$, and where the other $\bQ$-divisors are defined as follows:

* let $D_W$ and $\Gamma_W$ be the $\nu$-strict transforms of respectively $D$ and $\Gamma$;

* let $D_V$ be defined on $V$ by the equality $K_{V/C}+D_V=\tau^*(K_{W/T}+D_W)$;

* let $\Gamma_V$ be defined on $V$ by $\Gamma_V=\tau^*\Gamma_W$. In particular, it is $\bQ$-Cartier.

Define the polynomial $F(C,j)\in \bR[t]$ by

$$F(C,j)(t):=(-K_{V/C}-D_{V}+t\Gamma_{V})^{\dim V}=(-K_{V/C}-D_{V}+t\Gamma_{V})^{n+1}.$$

We will also use the ad hoc notation

$$\mm_2\left(\sum_{l=0}^ra_jt^j\right):=\max_{l\geq 2}\{|a_l|\}. $$

We need to relate the intersection products one can do on $V$, to the intersection products one can do on $Z$ and on $T$. This is the purpose of the next three lemmas.

In the situation of <ref>,

* $\mu^*(K_{Z/C}+D_Z)\sim_\bQ K_{V/C}+D_V$.

Moreover, if $C\to T$ is a general movable curve, then

* $\mu\colon V\to Z$ is small birational;

* $D_V,\Gamma_V$ are the strict transforms of $D_Z$ and $D^j_Z$ respectively;

* $\Gamma_V$ is the divisorial pullback of $\Gamma_W$ and the $c_j$-coefficient part of $D_V$.

* for every rational $0<\epsilon<r_{X,D}$, the family $(V,D_V-\epsilon\Gamma_V)\to C$ is a $\bQ$-Gorenstein family of log Fanos of maximal variation with uniformly K-stable general geometric fibers.

Since $\sigma^*(K_{X/T}+D)\sim_\bQ K_{Z/C}+D_Z$ and $K_{W/T}+D_W\sim_\bQ \nu^*(K_{X/T}+D)$, we obtain that $K_{V/C}+D_V\sim_\bQ \mu^*(K_{Z/C}+D_Z)$. This proves part $(a)$.

By <ref>, the morphism $W_t\to X_t$ is small birational for a general $t\in T$. So if $C$ meets the open locus of such $t\in T$, the birational morphism $\mu\colon V\to Z$ is small as well. In this case $D_V,\Gamma_V$ are the strict transforms of $D_Z$ and $D^i_Z$. By <ref>, if $C$ is general movable then $D^j_Z$ is the divisorial pullback of $\Gamma$, and the $c_j$-coefficient part of $D_Z$. So $\Gamma_V$ is the $c_j$-coefficient part of $D_V$ and the divisorial pullback of $\Gamma_W$. This proves parts $(b),(c)$ and $(d)$.

For all rational $0<\epsilon<r_{X,D}$, the family $(W,D_W-\epsilon\Gamma_W)\to T$ is a $\bQ$-Gorenstein family of log Fanos of maximal variation with uniformly K-stable general geometric fibers. Since the fibers are normal and $C$ is normal, $V$ is also normal. If $C$ meets the open locus where the fibers are uniformly K-stable and of maximal variation, then part $(e)$ holds.

In the situation of <ref>, if $ C\to T$ is general movable, we have

$$(-K_{V/C}-D_V)^{n+1}=(-K_{Z/C}-D_Z)^{n+1} \quad \text{and}\quad (-K_{V/C}-D_V)^n.\Gamma_V= (-K_{Z/C}-D_Z)^n.D^j_Z.$$

By part $(a)$ of <ref>, it holds that $K_{V/C}+D_V\sim_\bQ \mu^*(K_{Z/C}+D_Z)$. The first equality follows. By part $(c)$ of <ref>, if $C$ is general movable then $\mu$ restricts to a birational morphism $\mu|_{\Gamma_V}\colon \Gamma_V\to D^j_Z$, and the second equality follows.

In the situation of <ref>, for each $j>0$ there is a non-empty finite family of $\mathbb{Q}$-Cartier divisors $\{\Upsilon_{jl}\}_{l=2}^{n+1}$ on $T$ such that $\mm_2(F(C,j))=\max_l\{|\Upsilon_{jl}.C|\}$ for any smooth curve $C\to T$.

Fix an index $j>0$ and let $(f'\colon W\to T, D_W,\Gamma_W)$ be as in <ref>. The fibers of $f'$ are normal, and for small positive values of $\epsilon$, the $\bQ$-Cartier divisor $-K_{W/T}-D_W+\epsilon\Gamma_W$ is relatively ample over $T$. Thus by <ref>, for any smooth curve $C\to T$ we have:

$$f'_*(-K_{W/T}-D_W+\epsilon\Gamma_W)^{n+1}.C=(-K_{V/C}-D_{V}+\epsilon\Gamma_{V})^{n+1}=F(C,j)(\epsilon). $$

Write $f'_*(-K_{W/T}-D_W+\epsilon\Gamma_W)^{n+1}=\sum_{l=0}^{n+1}\epsilon^l\Upsilon_{jl}$ in the Chow ring of $T$. We obtain

$$\mm_2(F(C,j))=\max_{l\geq 2}\{ |\Upsilon_{jl}.C|\}$$

as claimed. Notice that the family $\{\Upsilon_{jl}\}_{l=2}^{n+1}$ is non-empty since $n\geq 1$.

We are now able to treat the case $j>0$.

Let $C\to T$ be a smooth curve such that all the assertions of <ref> are satisfied. Assume that $\left[c_1\left(\NN_Z\right)\right]^n.D^j_Z\geq e\cdot(\Psi.C)$ for some $j>0$ and $e>0$. Then
$\lambda_{f,D}.C\geq e_1\cdot (\Psi.C)$
where $e_1=e_1(X,D,q,e)>0$ is a numerical constant.

By assumption on $C\to T$, the results of <ref> and <ref> hold. Thus we have

\begin{equation}\label{eqn:Value_at_0}
\end{equation}

\begin{equation}\label{eqn:Derivate_estimate_I}
F'(C,j)(0)= (n+1)(-K_{Z/C}-D_Z)^n.D^j_Z.
\end{equation}

A direct calculation gives

\begin{equation}\label{eqn:Explicit _form_of_volume}
\left[c_1\left(\NN_Z\right)\right]^n.D^j_Z=q^n(-K_{Z/C}-D_Z)^n.D^j_Z+2n\alpha \deg_C\lambda_{h,D_Z}\left[c_1\left(\NN^{D^j}_t\right)\right]^{n-1}
\end{equation}

Combining <ref>, <ref> and the hypothesis on $\left[c_1\left(\NN_Z\right)\right]^n.D^j_Z$, we obtain that

\begin{equation}\label{eqn:Derivate_estimate_II}
F'(C,j)(0)\geq \frac{(n+1)e}{q^n}(\Psi.C) - \frac{2n(n+1)\alpha}{q^n}\left(\left[c_1\left(\NN_t\right)\right]^{n-1}.D^j_t\right)\deg_C \lambda_{h,D_Z}.
\end{equation}

On the other hand, for any rational $0<\epsilon <r_{X,D}$, the family $(V,D_V-\epsilon\Gamma_V)\to C$ is a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with uniformly K-stable general geometric fibers. Thus

\begin{equation}\label{eqn:Negativeness}
-\deg_C\lambda_{h,D_V-\epsilon\Gamma_V}=F(C,i)(\epsilon)\leq 0 \quad \forall \; \epsilon\in (0,r_{X,D})
\end{equation}

by <cit.>. We combine <ref> and <ref> to get a negativity condition on $F(C,i)(0)=\deg \lambda_{h,D_Z}$.

Let $\beta_0=\max_{i>0}\left\{\left[c_1\left(\MM^{D^i}_t\right)\right]^{n-1}, t\in C(k)\right\}$, and for convenience let us write

$$w=\Psi.C, \quad a=\frac{(n+1)e}{q^n}, \quad b=\frac{2n(n+1)\alpha}{q^n}\beta_0.$$

Therefore <ref> implies that

\begin{equation}\label{eqn:Estimate_of_F'}
F'(C,j)(0)\geq aw - b\deg_C \lambda_{h,D_Z}
\end{equation}

Assume that

\begin{equation}\label{eqn:Degree_CM_I}
\deg_C\lambda_{h,D_Z}\leq \frac{a}{2b}w.
\end{equation}

Then the estimate <ref> implies

$$F'(C,j)(0)\geq \frac{a}{2}w>0.$$

Thus we know by <ref> that $F(C,j)(t)$ must be negative in a neighborhood of $t=0$, and we have a positive lower bound on the linear part. This gives a condition on $F(C,j)(0)$ that depends on the quadratic part of $F(C,j)$: we apply <ref> with

$$G=\frac{a}{2}w, \quad H=\mm_2(F(C,j)), \quad l=r_{X,D}, \quad d=n+1\geq 2,$$

and we obtain that $F(C,j)$ takes a strictly positive value on $[0;r_{X,D}/2)$ if

\begin{equation}\label{eqn:Estimate_derivation_I}
F(C,j)(0)> \max\left\{-\frac{ar_{X,D}}{2}w, -\frac{a^2}{4n}\frac{w^2}{\mm_2(F(C,j))} \right\},
\end{equation}

where we set $\frac{w^2}{\mm_2(F(C,j))}=+\infty$ if $\mm_2(F(C,j))=0$. Obviously <ref> and <ref> cannot simultaneously hold. To conclude, we need to modify the right-hand side of <ref> so that the only quantity that depends on $C$ is $w=(\Psi.C)$.

Let $\{\Upsilon_{ij}\}_{i,j}$ be the $\bQ$-Cartier divisors on $T$ given by <ref>. Consider the function

$$\overline{\text{Mov}}(T)_\bR-\{\bold{0}\}\to \bR, \quad \gamma\mapsto \frac{\max_{i,j} |\Upsilon_{ij}.\gamma|}{\Psi.\gamma}.$$

This function is well-defined since $\Psi$ is big and hence defines a strictly positive functional on $\overline{\text{Mov}}(T)_\bR-\{\bold{0}\}$. It is also continuous and invariant under $\bR^*_+$-scaling of its argument. So it admits a maximum which is strictly positive, since the numerator is not zero for all movable curves. Thus there exists $\beta_1>0$ such that

$$\frac{\mm_2(F(C,j))}{w}=\frac{\max_{i} |\Upsilon_{ij}.C|}{\Psi.C}<\beta_1$$

for all general movable curve $C$ and $j>0$. So <ref> implies that if

\begin{equation}\label{eqn:Estimate_derivation_II}
F(C,j)(0)> \max\left\{-\frac{ar_{X,D}}{2b}, -\frac{a^2\beta_1}{4b^2n} \right\}\cdot w
\end{equation}

then $F(C,j)$ takes a strictly positive value on $[0;r_{X,D}/2)$. This contradicts <ref>, so <ref> and <ref> cannot hold simultaneously. In other words,

$$\deg_C\lambda_{h,D_Z}=-F(C,j)(0)\geq \min \left\{\frac{a}{2b},\frac{ar_{X,D}}{2b}, \frac{a^2\beta_1}{4b^2n} \right\}\cdot w.$$

The quantity $e_1=\min \left\{\frac{a}{2b},\frac{ar_{X,D}}{2b}, \frac{a^2\beta_1}{4b^2n} \right\}$ depends only on $(X,D),q$ and $e$. Therefore the proof is complete.

§.§ Proof of <ref>

Let $\tau\colon T'\to T$ be a resolution of singularities. Then the induced family $f_{T'}\colon (X_{T'},D_{T'})\to T'$ is again a $\bQ$-Gorenstein family of log Fano pairs of maximal variation with general geometric uniformly K-stable fibers. The morphism $\tau$ is birational and $\tau^*\lambda_{f,D}=\lambda_{f_{T'},D_{T'}}$ by <ref>. So $\lambda_{f,D}$ is big if and only if $\lambda_{f_{T'},D_{T'}}$ is big. Thus we may assume that $T$ is smooth to begin with.
Let $C\to T$ be a general movable curve. By <ref> and <ref>, the hypothesis of either <ref> or <ref> is fullfilled, with a constant $e$ that depends only on $X,D$ and $q$. Thus there is a constant $c=c(X,D,q)>0$ such that $\lambda_{f,D}.C\geq c\cdot (\Psi.C)$. As $\Psi$ is big, the result follows.

By the Nakai-Moishezon theorem it is enough to prove that for all normal varieties $V$ mapping finitely to $T$, we have $(\lambda_{f,D}|_V)^{\dim V}>0$. Let $V'\to V$ be a resolution of singularities. By <ref> and since $(\lambda_{f,D}|_V)^{\dim V}=(\lambda_{f,D}|_{V'})^{\dim V'}$, we may replace $f\colon (X,D)\to T$ by $f_{V'}\colon (X_{V'},D_{V'})\to {V'}$. By assumption all the closed fibers of $f_{V'}$ are uniformly K-stable, hence klt. So all the fibers of $f_{V'}$ are klt. Therefore we are in position to apply point (c) of <ref>.

§.§ Proof of <ref>

We give an application of <ref> to the moduli theory of Fano varieties. More precisely, we generalize <cit.> to the logarithmic setting.

Let $\fC$ be a class of K-semi-stable log Fano varieties over $k$. A pseudo-functor $\sM_\fC$ is called a moduli functor for $\fC$ if for every normal scheme $T$ of finite type over $k$,

\begin{equation*}
\sM_\fC(T)=
\begin{Bmatrix}
\begin{array}{c|c}
\begin{tikzcd}
(X, \Delta)\arrow[d, "f" left] \\ T \end{tikzcd}
\begin{tikzcd}[row sep=0.02cm,
/tikz/column 1/.append style={anchor=base west},]
(a)\; f \text{ is a flat projective morphism,}\\
(b)\; \text{the fibers of }f\text{ are connected, pure-dimensional and normal,}  \\
(c)\; \Delta \text{ is a }\bQ\text{-Weil divisor whose support does not contain any fiber,} \\
(d)\; K_{X/T}+\Delta \text{ is }\bQ\text{-Cartier,} \\
(e)\; \text{for each geometric point }t\in X, (X_t,\Delta_t)\in \fC.
\end{tikzcd}
\end{array}
\end{Bmatrix}
\end{equation*}

with arrows given by Cartesian diagrams and divisorial base-change.

The hypothesis on the families contained in $\sM_\fC$ guarantee that there is a sensible way to define the restrictions and the pullbacks of the $\bQ$-Weil divisor $\Delta$ (see <ref>). We do not require anything for the value of $\sM_\fC$ over non-normal test schemes.

In the situation of <ref>, we say that a proper algebraic space $M$ is a moduli space for $\fC$, with open uniform K-stable locus $M^u\subseteq M$, if

* $M^u\subseteq M$ is an open sub-algebraic space,

* there is a morphism $\mu\colon \sM_\fC\to M$ which is universal among morphisms from $\sM_\fC$ to algebraic spaces,

* $\mu(k)$ sends exactly the uniformly K-stable varieties in $\fC(k)$ to $M^u(k)$,

* $\mu(k)$ is surjective and bijective when restricted to the uniformly K-stable varieties,

* there is a generically finite, proper cover $\pi\colon Z\to M$ by a scheme, given by a family $((X,\Delta)\to Z)\in \sM_\fC(Z)$, such that

* $\pi$ is finite over $M^u$, and

* some positive multiple of the CM line bundle $\lambda_{f,\Delta}$ on $Z$ is numerically equivalent to a line bundle that descends to $M$.

The proof is identical to the one of <cit.>. The only change is that we use <ref>.(c) instead of <cit.>.

§ APPENDIX

We gather some technical results that are used in the text.

Let $f\colon X\to T$ be an equidimensional proper morphism of relative dimension $n$ between projective $k$-schemes. Assume that $T$ is smooth. Let $A$ be an ample $\bQ$-Cartier divisor on $X$. Then the cycle $f_*A^{n+1}$ is $\bQ$-Cartier and big.

Replacing $A$ by a multiple, we may assume that $A$ is very ample and $f_*A^{n+1}$ Cartier. Then the linear system defined by $f_*A^{n+1}$ on $T$ is base-point free and separates points. The result now follows from <cit.>.

Let $f\colon X\to T$ be proper morphism between normal projective $k$-schemes. Let $A$ be a pseudo-effective relatively ample $\bQ$-Cartier divisor on $X$, and $B$ a big $\bQ$-Cartier divisor on $T$. Then $A+f^*B$ is big on every component of $X$.

We may assume that $X$ is integral. Write $B\sim_\bQ C+E$ where $C$ is ample and $E$ effective. Fix an ample divisor $H$ on $X$. Choose $\epsilon'\in \bQ_+^*$ small enough such that $\epsilon'A+f^*C$ is ample on $X$. Then choose $\epsilon\in\bQ_+^*$ small enough such that $A+\epsilon H$ is effective, and $\epsilon'A+f^*C-(1-\epsilon')\epsilon H$ is still ample. We write

$$A+f^*B \sim_\bQ f^*E+(1-\epsilon')(A+\epsilon H) + (\epsilon'A+f^*C-(1-\epsilon')\epsilon H)$$

so $A+f^*B$ is the sum of an effective and an ample $\bQ$-divisors. By <cit.>, it is big.

Let $f\colon X\to T$ be a flat morphism between Noetherian schemes. Assume that $T$ is $S_1$ and integral, and that the generic fiber of $f$ is reduced. Then $X$ is reduced.

Let $x$ be an associated point of $X$. By assumption, the local map $\OO_{T,f(x)}\to \OO_{X,x}$ is flat. If $f(x)$ is not the generic point $\eta$ of $T$, then $\OO_{T,f(x)}$ has dimension at least one, and so its maximal ideal contains a non-zero divisor. By flatness, the image of this element is also a non-zero divisor in the maximal ideal of $\OO_{X,x}$. This contradicts the fact that $x$ is an associated point, so $f(x)=\eta$. Now $X_\eta$ is reduced, so $x$ cannot be an embedded associated point. Therefore $X$ is reduced.

Let $X_i$ $(i=1,\dots, r)$ be proper $k$-schemes of dimensions $n_i$. Set $\mathcal{X}:=X_1\times_k\dots \times_k X_r$, with projections $p_i$ onto its factors. There is a positive numerical constant $c=c(n_1,\dots,n_r)$ with the following property: if $L_i$ are Cartier divisors on $X_i$ and $L:=\sum_{i=1}^rp_i^*L_i$, then

$$L^{\dim \mathcal{X}}=c\prod_{i=1}^r L_i^{n_i}.$$

By induction on $r$, it suffices to consider the case $r=2$. In this case we have

as claimed. In the general case, the precise form of the constant $c$ is
$\prod_{i=1}^r \binom{\sum_{k\geq i}n_k}{n_i}$.

Let $X_i\to T$ $(i=1,\dots,r)$ be flat morphisms from proper $k$-schemes of dimension $1+n_i$ to a common smooth $k$-curve. Set $\mathcal{X}:=X_1\times_T \dots \times_T X_r$ with projections $p_i$ onto its factors. Then there are positive numerical constants $d_i=d(n_1,\dots, n_r)$  with the following property: if $L_i$ are Cartier divisors on $X_i$ and $L:=\sum_{i=1}^r p_i^*L_i$, then

$$L^{\dim\mathcal{X}}=\sum_{i=1}^r d_i L_i^{n_i+1}\prod_{j\neq i}(L_j)_t^{n_j}.$$

Notice that $\dim \mathcal{X}=1+\sum_j n_j$. Thus $L^{\dim\mathcal{X}}$ is the sum of terms of the form $(p_1^*L_1)^{i_1}\dots (p_r^*L_r)^{i_r}$ where $\sum_j i_j=1+\sum_j n_j$. Such a term a zero as soon as $i_j>1+n_j$ for some $j$. On the other hand, by the pigeon-hole principle, at least one $i_j$ is greater or equal to $1+n_j$. Thus:

$$L^{1+\sum_j n_j} = \sum_{i=1}^r\binom{1+\sum_j n_j}{1+n_i}(p_i^*L_i)^{1+n_i} \left( \sum_{j\neq i}p_j^*L_j\right)^{\sum_{j\neq i}n_j}.$$

Notice that the fiber of $p_i \colon \mathcal{X}\to X_i$ above $x_i$ is given by the fiber product $(X_2)_{t_i}\times_k \dots\times_k (X_r)_{t_i}$ where $t_i$ is the image of $x_i$ through $X_i\to T$. Applying <ref> and using flatness, we get

$$L^{\dim \mathcal{X}}=\sum_{i=1}^r \binom{1+\sum_j n_j}{1+n_i}L_i^{1+n_i}c(n_1,\dots,\widehat{n_i},\dots,n_r)\prod_{j\neq i}(L_j)_t^{n_j}$$

where $t\in T$ is any closed point. Put $d_i(n_1,\dots,n_r):=\binom{1+\sum_j n_j}{1+n_i}c(n_1,\dots,\widehat{n_i},\dots,n_r)$ to conclude.

Let $G>0, H\geq 0$ and $l\in (0;1)$ be positive real numbers, and $d\geq 2$ be an integer.
Then for every choice of real numbers $a_0,\dots,a_d$ satisfying

$$\max\left\{-\frac{Gl}{4}, -\frac{G^2}{4H(d-1)}\right\}< a_0\leq 0, \quad a_1\geq N, \quad\text{and}\quad |a_i|\leq H \;\forall i\geq 2,$$

the polynomial $p(t)=\sum_{i=0}^d a_it^i$ takes a strictly positive value in the interval $(0;l/2)$. (If $H=0$ we set $\frac{G^2}{4H(d-1)}=+\infty$).

Let $a_0,\dots ,a_d$ be real numbers satisfying the prescribed conditions.
We have, for $0<t<1$:

\begin{eqnarray*}
p(t) & = & a_0+a_1s+\sum_{i\geq 2}a_i t^i \\
& \geq & a_0+Gt-G(d-1)t^2.
\end{eqnarray*}

So it it enough to prove that $q(t):=a_0+Gt-H't^2$ takes a strictly positive value on $(0,l/2)$, with $H':=H(d-1)$. First consider the special case where $H=0$. Then $a_0>-Gl/4$, so $q(l/3)>Gl/12>0$. From now assume that $H>0$. We have to show that $q(t)$ has a real positive root $t_0\in (0,l/2)$ such that $q'(t_0)>0$. Real roots exist if

\begin{equation}\label{eqn: Condition I}
\end{equation}

Assume <ref> holds. Then the smallest positive root of $q(t)$ is

Note that

$$q'(t_0)=G-2H't_0=\sqrt{G^2+4a_0H'}> 0.$$

Hence we just have to ensure that $t_0<l/2$. This condition is equivalent to

This inequality is trivially satisfied if $G-lH'<0$. If $G-lH'\geq 0$,  then it is equivalent to

which is implied by the stronger condition

\begin{equation}\label{eqn: Condition II}
\end{equation}

as $-G+lH'\leq 0$. Therefore if <ref> and <ref> are satisfies, then $q(t)$ takes a strictly positive value in $(0,l/2)$, as desired.

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