The isotriviality of families of canonically-polarized manifolds over a special quasi-projective base
Behrouz Taji
In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.
1310.5391.tex
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\author{Behrouz Taji}

\title[The isotriviality of families of canonically-polarized manifolds]{The isotriviality of families of canonically-polarized manifolds over a special quasi-projective base}



\address{The Department of Mathematics and Statistics, McGill University. Burnside Hall, Room 1031. 805 Sherbrooke W. Montreal, QC, H3A 0B9, Canada}

\email{\href{mailto:behrouz.taji@matil.mcgill.ca}{behrouz.taji@mail.mcgill.ca}}




\date{\today}
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\begin{document}

\begin{abstract} In this paper we prove that a smooth family of canonically polarized manifolds parametrized by a special (in the sense of Campana) quasi-projective variety is isotrivial.
\end{abstract}

\maketitle

\section{introduction}

A conjecture of Shafarevich, settled by Parshin, asks for a smooth family of canonically polarized varieties (canonical bundle is ample) over a non-quasi-hyperbolic curve to be isotrivial. Striving to formulate this statement when the dimension of the base is greater than $1$, Campana has proposed the notion of special quasi-projective varieties (see the definition~\ref{special} below) as higher dimensional analogues of such $1$-dimensional objects and conjectured that similar results should hold over a base of any dimension. 
 
 
 



\begin{conjecture}[The isotriviality conjecture of Campana] \label{iso} Let $Y^{\circ}$ be a smooth quasi-projective variety parametrizing a family of canonically polarized manifolds. If $Y^{\circ}$ is special, then the family is isotrivial. 


\end{conjecture}


\begin{defn}[Special logarithmic pairs]\label{special} Let $(Y,D)$ be a pair consisting of a smooth projective variety $Y$ and a simple normal-crossing reduced boundary divisor $D$. We call $(Y,D)$ special, if for every saturated coherent subsheaf of rank one $L \subseteq \Omega_Y^p \log(D)$ and $p>0$, we have $\kappa(L)<p$. Moreover we shall call a smooth quasi-projective variety $Y^{\circ}$ special, if (Y,D) is special as a logarithmic pair, where $Y$ is a smooth compactification with a simple normal-crossing (snc, for short) boundary divisor $D$.


\end{defn}

The conjecture~\ref{iso} is in fact a generalization of the following celebrated conjecture of Viehweg:

\begin{conjecture}[Viehweg's hyperbolicity conjecture]\label{VC} Let $f^\circ: X^\circ \to Y^\circ$ be a smooth family of canonically-polarized varieties over a quasi-projective variety $Y^\circ$. Assume that Y is a smooth compactification of $Y^\circ$ with snc boundary divisor $D\cong Y\backslash Y^\circ$. If the moduli map $\mu:Y^\circ \to \mathfrak M$ is generically finite, then $(Y,D)$ is of log-general type.
 
\end{conjecture}

\

Remarkably Viehweg and Zuo prove in~\cite[Thm.~1.4]{VZ02} that when such a family is not isotrivial, then there exists a line subbundle  (sometimes referred to as a Viehweg-Zuo subsheaf) $L\subseteq \Sym^N\Omega^1_Y\log (D)$, for some positive integer $N$, such that $\kappa(L) \geq \Var(f^\circ)$. This immediately proves Viehweg's conjecture when $Y^\circ$ is of dimension $1$. Conjecture~\ref{VC} is known up to dimension $3$ by~\cite[Thm.~1.1]{KK08} and recently in all dimensions by~\cite[Sect.~4]{CP13}.

\


The stronger conjecture~\ref{iso} has also been settled when the dimension of the base is at most $3$ by Jabbusch and Kebekus~\cite[Thm.~1.5]{JK11b}. In the final section (section~\ref{last}), after following Campana and Paun's proof of Viehweg's conjecture very closely, we give a proof to the isotriviality conjecture~\ref{iso}. The proof heavily depends on a recent generic semi-positivity result of Campana and Paun (see~\ref{ogsp}), existence of log-minimal models for klt pairs with big boundary divisors established by~\cite[Thm.~1.1]{BCHM10}, and an important refinement of~\cite[Thm.~1.4]{VZ02} given by~\cite[Thm.~1.4]{JK11a}.


\begin{thm}\label{ison} The isotriviality conjecture~\ref{iso} holds in all dimensions. 


\end{thm}

\subsection{Acknowledgements}

The author would like to thank his advisor, S. Lu, for his invaluable help, guidance and support. He also owes a debt of gratitude to F. Campana for his encouragements, generosity, and many fruitful discussions. The author would like to express his sincere thanks to S. Kebekus for his careful reading of the first draft of this paper and many kind and inspiring suggestions. A special thanks is owed to E. Rousseu for his keen interest, specially for drawing the author's attention to a definition of Miyaoka, a variant of which appears in this paper. 


\


\section{Preliminaries}

To approach the isotriviality conjecture~\ref{iso}, it is essential to work with pairs (or the orbifold pairs in the sense of Campana) instead of just logarithmic ones. We refer the reader to~\cite{Ca08} and~\cite{JK11b} for an in-depth discussion of the definitions and background. In the present section we give a brief overview of the key ingredients of this theory that is necessary for our arguments in the rest of the paper. 





\begin{defn}[Normal and smooth pairs]\label{orbipairs} Let $X$ be an $n$-dimensoinal normal quasi-projective variety and $D=\sum d_iD_i$, where $d_i\in \mathbb{Q}\cap [0,1]$, a $\mathbb{Q}$-Weil divisor in $X$ whose irreducible components $D_i$'s are $\mathbb{Q}$-Cartier. We shall call the pair $(X,D)$ a normal pair. If $X$ is smooth and $\supp(D)$ is simple normal-crossing, we say that $(X,D)$ is a smooth pair.

\end{defn}

\

\begin{defn}[$\mathcal{C}$-multiplicity]\label{multiplicity} Let $(X,D)$ be a normal pair as in~\ref{orbipairs}. If $d_i\neq 1$, let $a_i$ and $b_i$ be the positive integers for which the equality $1-\frac{b_i}{a_i}=d_i$ holds. For every $i$, we define the $\mathcal{C}$-multiplicity of the irreducible component $D_i$ of $D$ by 

$$
  m_D(D_i) := \left\{ 
    \begin{matrix}
      \frac{a_i}{b_i}& \text{if $d_i\neq 1$ } \\
      \infty & \text{if $d_i=1.$  }
    \end{matrix}
  \right.
  $$


\end{defn}



\




The following definition requires a notion of pull-back for Weil divisors on normal varieties for which we refer to~\cite[Notation.2.5]{JK11b}.
 

\begin{defn}[Adapted covers]\label{covers} Let $(X,D)$ be a normal pair. Let $Y$ be a normal domain and $\gamma:Y \to X$ a cyclic cover with Galois group $G$ such that if $m_D(D_i)=\frac{a_i}{b_i}<\infty$, then every prime divisor in $Y$ that appears in $\gamma^*(D_i)$ has multiplicity exactly equal to $a_i$. We call $\gamma$ an adapted cover for the pair $(X,D)$, if it additionally satisfies the following properties:  


\begin{enumerate}
\item The branch locus is given by $$\supp(H+\bigcup_{m_D(D_i)\neq \infty} D_i),$$ where H is a general member of a linear system $|L|$ of a very ample divisor $L$ in $X$.
\item $\gamma$ is totally branched over $H$.
\item $\gamma$ is not branched at every general point of $\supp(\lfloor D \rfloor)$.

\end{enumerate}

\end{defn}

\

Existence of such covers is guaranteed for smooth pairs by the classical result of Kawamata (see~\cite[Prop.~1.12]{Laz}) and for normal pairs by~\cite[Prop.~2.9]{JK11a} 



\begin{notation} Let $\gamma:Y\to X$ be a cyclic cover adapted to a normal pair $(X,D)$. We shall denote $\gamma^*\lfloor D \rfloor$ by $D_{\gamma}$. Furthermore let $\{D_{ij} \}_{j(i)}$ be the collection of prime divisors in $Y$ that appear in $\gamma^{-1}D_i$. We define a new divisor $D_Y$ in $Y$ by 
$$D_Y:=\sum_{i, m_D(D_i)\neq \infty}\sum_{j(i)}b_iD_{ij},$$ where $b_i$'s are the positive integers defined in~\ref{multiplicity}.

\end{notation}
\





\begin{defn}[$\mathcal{C}$-cotangent sheaf]\label{ocs} Let $(X,D)$ be a normal pair and $\gamma:Y\to X$ a corresponding adapted cover. Let $U$ be the smooth locus of $(X,D)$, i.e. $U\subseteq X$ is the maximal open subset of $X$ such that $(U,D|_U)$ is a smooth pair. Define $\sF$ to be the locally-free $\sO_Y$-module sheaf on $V:=\gamma^{-1}U$, locally generated by the generators of the two locally-free sheaves: $\gamma^*(\Omega_X\log(\lfloor D \rfloor)|_U)$ and $\sE$, where $\sE$ is the kernel of the sheaf morphism $$\rho: \gamma^*(\Omega_X\log(\ulcorner D \urcorner)|_U) \to \sO_{D_Y}|_V,$$ induced by the natural residue map. We define the $\mathcal{C}$-cotangent sheaf $\Omega^{[1]}_{Y^{\partial}}$ of the pair $(X,D)$ by the extension $i_{V_*}\sF$, where $i_V$ is the natural inclusion of $V$ into $Y$, and the $\mathcal{C}$-tangent sheaf $T_{Y^{\partial}}$ by $$T_{Y^{\partial}}:=(\Omega_{Y^{\partial}})^*.$$.




\end{defn}

\


\begin{rem} The $\mathcal{C}$-cotangent sheaf defined in~\ref{ocs} coincides with Campana and Paun's notoin~\cite[Sec.~1.1]{CP13} of the coherent sheaf on $Y$ which they denote by $\gamma^*\Omega^1(X,D)$. It is also identical with the sheaf defined in~\cite[Lem.~4.2]{Lu02}, when the pair is smooth. See also~\cite[Def.~2.13]{JK11b} for an equivalent definition in the classical setting, i.e. when the $\mathcal{C}$-multiplicities are all integral. The definition that we have given in~\ref{ocs} is a generalization of that of Miyaoka's~\cite[pp.~412]{Miy08}.

\end{rem}

\


\begin{rem}[Determinant of the $\mathcal{C}$-cotangent sheaf] Given a normal pair $(X,D)$, let $\gamma:Y \to X$ be an adapted cover of degree $d$, and denote the reflexivization of the determinant sheaf of $\Omega^{[1]}_{Y^{\partial}}$ by $\det(\Omega^{[1]}_{Y^{\partial}})$. There exists a natural isomorphism between $\det(\Omega^{[1]}_{Y^{\partial}})$ and $\sO_Y(\gamma^*(K_X+D)):$
\begin{equation}\label{det}
     \det \Omega_{Y^{\partial}}^{[1]}\cong \sO_Y(\gamma^*(K_X+D)),
         \end{equation}

\noindent which follows from the ramification formula for the adapted cover $\gamma$: 



\begin{align*}
K_Y+D_{\gamma}  &=\gamma^*(K_X+\lfloor D \rfloor)+\sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(a_i-1)D_{ij}+(d-1)\wtilde H\\
     &=\gamma^*(K_X+D)-\gamma^*(D-\lfloor D \rfloor)+\sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(a_i-1)D_{ij}+(d-1)\wtilde H\\
     &=\gamma^*(K_X+D)-\sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(a_i-b_i)D_{ij}+ \sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(a_i-1)D_{ij}+(d-1)\wtilde H \\
     &=\gamma^*(K_X+D)+\sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(b_i-1)D_{ij}+(d-1)\wtilde H,\\
   \end{align*}

\noindent for $\wtilde H:=\gamma^*H$, where $H$ is the very ample divisor defined in~\ref{covers}. As a consequence, we find that~\ref{det} holds by construction: 


\begin{align*}
\det \Omega^{[1]}_{X^{\partial}}  &\cong \sO_Y((K_Y+D_{\gamma})-\sum_{i,m_D(D_i)\neq \infty}\sum_{j(i)}(b_i-1)D_{ij}- (d-1)\wtilde H) && \text{by definition}\\      
        &\cong \sO_Y(\gamma^*(K_X+D))  && \text{by the ramification formula}. \\
    \end{align*}


\end{rem}

\



\begin{defn}[Symmetric $\mathcal{C}$-differential forms, cf. \protect{\cite[Sect.~2.6-7]{Ca08}}]\label{sodf} Let $(X,D)$ be a smooth pair and $V_x$ an analytic neighbourhood of a point $x\in X$ equipped with a coordinate system $z_1,\ldots,z_n$ such that $\supp(D)\cap V_x=\{z_1\cdot \ldots\cdot z_l=0\}$, for a positive integer $1\leq l\leq n$. For every $N\in\mathbb{N}^+$, define the locally-free sheaf $\Sym_{\mathcal{C}}^N\Omega_X^1\log D \subseteq \Sym^N\Omega_X\log(\ulcorner D \urcorner)$ at $x$ by the $\sO_x$-module generated by $$\frac{dz_1^{k_1}}{z_1^{\lfloor (1-1/m_D(D_1))k_1\rfloor}} \cdot \ldots \cdot \frac{dz_l^{k_l}}{z_1^{\lfloor (1-1/m_D(D_l))k_l \rfloor}} \cdot dz_{l+1}^{k_{l+1}} \cdot \ldots \cdot dz_n^{k_n},$$

\noindent where $\sum k_i =N$. When $m_D(D_i)=\infty$, we follow the convention that $\frac{1}{m_D(D_i)}:=0$.

\end{defn}

\

\begin{rem}[An equivalent definition] Following~\cite[Def.~3.5]{JK11b} we give an equivalent description of the sheaf of symmetric $\mathcal{C}$-differential forms: Let $V_x$ be the same analytic neighbourhood as in~\ref{sodf} and take $\gamma:W \to V_x$ to be an adapted cover. Let $\sigma \in \Gamma(V_x,\Sym^{N}\Omega_X(*\ulcorner D \urcorner))$, that is $\sigma$ is a local rational section of $\Sym^{N}\Omega_X$ with poles along $\ulcorner D \urcorner$. Then
 
\begin{equation}\label{alternative}
   \sigma \in \Gamma(V_x,\Sym_{\mathcal{C}}^N\Omega_X^1\log D) \iff  \gamma^{*}\sigma\in    \Gamma(W,\Sym^{N}\Omega^{1}_{W^{\partial}}),
        \end{equation}

\

 \noindent so that in particular $\sigma$ has at worst logarithmic poles along those prime divisors in $W$ that dominate $\lfloor D \rfloor\cap V_x$, and is regular otherwise.
 
 \begin{explanation} Assume that $\sigma\in \Gamma(V_x,\Sym_{\mathcal{C}}^N\Omega_X^1\log D)$ in the sense of~\ref{alternative}. By the classical result of Iitaka~\cite[Chap.~11]{Iit82}, it follows that $\sigma \in \Gamma(V_x,\Sym^N\Omega_X\log(\ulcorner D \urcorner))$. In particular we find that along the reduced component of $D$ the equivalence between the two definitions trivially holds. So assume, without loss of generality, that $m_D(D_i)\neq \infty$, for all irreducible components $D_i$ of $D$. Furthermore let us assume, for simplicity, that 
  
$$\sigma=f\cdot \frac{dz_1^{k_1}}{z_1^{e_1}}\cdot \ldots \cdot \frac{dz_l^{k_l}}{z_l^{e_l}}\cdot dz_{l+1}^{k_l+1}\cdot \ldots \cdot dz_n^{k_n} \ \ \in \Gamma(V_x,\Sym^{N}\Omega_X\log(\ulcorner D \urcorner)) ,$$ where $f\in \sO_x$ with no zeros along $D_i$'s, is the explicit description of $\sigma$. Since $\gamma^*(\sigma)\in \Sym^N\Omega^1_{W^{\partial}}$, the inequality 
$$k_i\cdot(a_i-1)-a_i\cdot e_i \geq k_i(b_i-1)$$ holds for $1\leq i\leq l$, i.e.

$$e_i \leq k_i(1-\frac{1}{m_D(D_i)}) , \text{\ \ for all}\  1\leq i\leq l .$$ Thus $\sigma$ is also a symmetric $\mathcal{C}$-differential form on $V_x$ in the sense of~\ref{sodf}.
 
 
 
 
 
 \end{explanation}
 
 

\end{rem}

\begin{rem}[Tensorial $\mathcal{C}$-differential forms]\label{tensorial} Similar to the definition~\ref{sodf} and~\ref{alternative}, and for every $N\in\mathbb{N}^+$, we define the sheaf of tensorial $\mathcal{C}$-differential forms $(\Omega_X^1\log D)^{\otimes_{\mathcal{C}}N}$ as a subsheaf of $(\Omega_X\log(\ulcorner D \urcorner))^{\otimes N}$. 


\end{rem}


\

\


As we will see in section~\ref{VZJK} the Viehweg-Zuo subsheaves generically come from the coarse moduli after extending the sheaf of standard differential forms to that of $\mathcal{C}$-differential forms associated to the naturally imposed $\mathcal{C}$-structures or orbifold structures (see~\cite[Sect.~3]{Ca08} or~\cite[Subsect.~5.A]{JK11b} for the definition of this important concept) in the moduli space. But as the usual Kodaira dimension of subsheaves of $\Sym_{\mathcal{C}}^N\Omega_X^1\log D$ is not sensitive to the fractional positivity of those components $D_i$'s of $D$ with $\mathcal{C}$-multiplicity $m_D(D_i)\neq \infty$, a new birational notion is needed to measure the positivity of the Viehweg-Zuo subsheaves in the moduli.



\begin{defn}[$\mathcal{C}$-Kodaira dimension, cf. \protect{\cite[Sect.~2.7]{Ca08}}]\label{okodaira} Let $(X,D)$ be a smooth pair and $\sL\subseteq \Sym_{\mathcal{C}}^r\Omega_X^1\log D$ a saturated coherent subsheaf of rank one. Define the $\mathcal{C}$-product $\Sym_{\mathcal{C}}^m \sL$ of $\sL$, to the order of $m$,  to be the saturation of the image of  $\Sym^{m}\sL$ inside $\Sym_{\mathcal{C}}^{(m\cdot r)}\Omega_X^1\log D$ and define the $\mathcal{C}$-Kodaira dimension of $\sL$ by $$\kappa_{\mathcal{C}}(X,\sL):=\max \{k \; | \; \limsup_{m \to \infty} \frac{h^0(X,\Sym_{\mathcal{C}}^{m}\sL)}{m^k} \neq 0 \}.$$


\end{defn}

\begin{rem} [Comparing Kodaira dimensions] When $D=0$ or when $D$ is reduced the sheaf of symmetric orbifold-differential forms $\Sym_{\mathcal{C}}^r\Omega_X^1\log D$ is equal to $\Sym^{r}\Omega^{1}_X$ and $\Sym^{r}\Omega^{1}_X\log D$, respectively, so that the $\mathcal{C}$-Kodaira dimension of a rank one coherent subsheaf $L$ of $\Sym_{\mathcal{C}}^r\Omega_X^1\log D$ coincides with the usual Kodaira dimension $\kappa(X,\sL)$ of $\sL$.

\end{rem}

\

\


We finish this section by collecting the various notations that we have introduced in the following table.


\begin{table}[htb!]
\caption{Notations}
\centering
\begin{tabular}{c      c}
\hline\hline
$\Omega^{[1]}_{Y^{\partial}}$  &  $\mathcal{C}$-cotangent sheaf~\ref{ocs}\\
$\Sym_{\mathcal{C}}^N\Omega_X^1\log D$   &  Symmetric $\mathcal{C}$-differential forms~\ref{sodf}\\
$(\Omega^1_X\log D)^{\otimes_{\mathcal{C}}N}$   & Tensorial $\mathcal{C}$-differential forms~\ref{tensorial}\\
$\Sym_{\mathcal{C}}^N\sL$                               &  $\mathcal{C}$-product~\ref{okodaira} \\
$\kappa_{\mathcal{C}}(X,\sL)$  & $\mathcal{C}$-Kodaira dimension~\ref{okodaira}\\
\hline
\end{tabular}
\label{notat}
\end{table}



\




\section{The orbifold generic semi-positivity}

According to~\cite{BDPP} Miyaoka's generic semi-positivity result (see~\cite{Miy87} and~\cite{Miy85}) can be interpreted as a characterization of positivity of the canonical bundle by the generic nefness of the cotangent sheaf. This positivity result was achieved by certain characteristic $p$-arguments which cannot be adapted to the context of pairs. In~\cite{CP13} Campana and Paun have overcome this obstacle by deploying an important refinement of Miyaoka's theorem by Bogomolov and McQuillen concerning the algebraicity of leaves of foliations induced by positive subsheaves of the tangent sheaf. This crucial improvement has been formulated for $G$-invraiant quotients of the $\mathcal{C}$-cotangent sheaf. For the definition of the notion of $G$-invariance in this context, we refer the reader to~\cite[Def.~1.2]{CP13}.


\begin{thm}[Generic semi-positivity of $\mathcal{C}$-cotangent sheaf \protect{\cite[Thm.~2.1]{CP13}}]\label{ogsp} Let $(X,D)$ be a lc pair with a corresponding adapted cover $\gamma:Y \to X$ whose Galois group we denote by $G$. If $K_X+D$ is pseudo-effective, then for every collection of $(n-1)$ $\mathbb{Q}$-Cartier ample divisors $H_1,\ldots, H_{n-1}$, there exists $M\in \mathbb{N}^+$ such that for every set of positive integers $m_1,\ldots, m_{n-1}>M$ we have:

\begin{enumerate}
\item The linear system of the ample divisors $|m_iH_i|$ is non-empty and basepoint-free.
\item For any $N\in\mathbb{N}^+$, and for every general member $D_i$ of the linear system $|m_iH_i|$, the curve $\wtilde C:=\gamma^*D_1\cap \ldots \cap \gamma^*D_{n-1}$ is irreducible, smooth and contained in the smooth locus of $\Omega^{[1]}_{Y^{\partial}}$ and that the inequality
$$\deg(\sL|_{\wtilde C})\geq 0$$ holds for every $G$-invariant quotient line bundle $\sL$ of $(\Omega^{p}_{Y^{\partial}})^{\otimes N}|_{\wtilde C}$ , where $p$ is any positive integer verifying $1\leq p \leq n$.

\end{enumerate}

\end{thm}


We now prove an important corollary of~\ref{ogsp} in the setting that we shall need for the proof of~\ref{ison}, that is we will assume that $(X,D)$ is a smooth pair. But we would like to point out that not much more work is needed to similarly prove this corollary for any lc pair.





\begin{cor}\label{invariance} Let $(X,D)$ be a smooth pair. Let $\sL\subseteq (\Omega^1_X\log D)^{\otimes_{\mathcal{C}}N}$ be a line bundle and $L$ a divisor in $X$ such that  $\sO_X(L)\cong \sL$. If $K_X+D$ is pseudo-effective, then for every collection of $(n-1)$ $\mathbb{Q}$-Cartier nef divisors $P_1,\ldots, P_{n-1}$ the following inequality holds:
$$(N(K_X+D)-L)\cdot P_1 \cdot \ldots \cdot P_{n-1}\geq 0.$$



\end{cor}

\begin{proof} Since for any fixed ample divisor $H$ in $X$ the equality 
$$(N(K_X+D)-L) \cdot P_1\cdot \ldots\cdot P_{n-1}= (N(K_X+D)-L) \cdot (P_1+\frac{1}{t}H)\cdot \ldots\cdot (P_{n-1}+\frac{1}{t}H)$$ holds as $t\to \infty$, it suffices to prove that 

\begin{equation}\label{e1}
   (N(K_X+D)-L)\cdot H_1 \cdot \ldots \cdot H_{n-1}\geq 0
       \end{equation}

\noindent for any collection of $(n-1)$ $\mathbb{Q}$-Cartier ample divisors $H_1,  \ldots, H_{n-1}$. 

Take $\gamma:Y\to X$ to be an adapted cover with the corresponding Galois group $G$, and let $D_1,\ldots, D_{n-1}$ be the divisors in $Y$ and $\wtilde C:=\gamma^*D_1\cap\ldots\cap \gamma^*D_{n-1}$ the smooth irreducible curve, also in $Y$, that were introduced in~\ref{ogsp}. Furthermore define the curve $C$ in $X$ by $C:=D_1\cap\ldots \cap D_{n-1}$. To prove the inequality~\ref{e1}, it is enough to show 
\begin{equation}\label{e2}
    \deg(\sO_X(N(K_X+D))\otimes \sL^*|_C)\geq 0.
      \end{equation}
To this end, we fist observe that by~\ref{det} the sheaf isomorphism 

$$(\Omega_{Y^{\partial}}^{1})^{\otimes N}|_{\wtilde C}\cong (\det \Omega_{Y^{\partial}}^{1}\otimes \wedge ^{n-1}T_{Y^{\partial}})^{\otimes N}|_{\wtilde C}$$ reads as 

\begin{equation}\label{e3}
   (\Omega_{Y^{\partial}}^{1})^{\otimes N}|_{\wtilde C}\cong (\sO_Y(\gamma^*(K_X+D))\otimes \wedge ^{n-1}T_{Y^{\partial}})^{\otimes N}|_{\wtilde C}.
       \end{equation}

We know by definition~\ref{sodf} that $(\gamma^*\sL)|_{\wtilde C}$ is a subsheaf of $(\Omega_{Y^{\partial}}^{1})^{\otimes N}|_{\wtilde C}$. Let us for the moment assume that this is a saturated inclusion. As a result $(\gamma^*\sL^*)|_{\wtilde C}$ is a quotient of the dual of the locally-free sheaf given in the right-hand side of the isomorphism~\ref{e3}. Thus we arrive at the following exact sequence of locally free sheaves on $\wtilde C$:

$$
\xymatrix{
(\Omega_{Y^{\partial}}^{n-1})^{\otimes N}|_{\wtilde C} \ar[r] & \gamma^*(\sL^*\otimes \sO_X(N(K_X+D)))|_{\wtilde C}  \ar[r] &  0.
}
$$

Here~\ref{ogsp} applies and we find that  
\begin{equation}\label{e4}
    \deg(\gamma^*(\sL^*\otimes \sO_X(N(K_X+D)))|_{\wtilde C})\geq 0
      \end{equation}

\noindent holds. The required inequality~\ref{e2} now follows by the projection formula.

Finally if $\gamma^*\sL|_{\wtilde C}$ is not saturated inside $(\Omega^{[1]}_{Y^{\partial}})^{\otimes N}|_{\wtilde C}$, define $\overline{\sL}$ to be its saturation. Since $\overline{\sL}$ is not necessarily $G$-invariant anymore, consider $\overline{\sL}^{\otimes r}$, for any positive multiple of the degree of the cyclic cover $\gamma$. Local calculations shows that $\overline{\sL}^{\otimes r}$ is $G$-invariant. On the other hand, it is saturated inside $(\Omega^{[1]}_{Y^{\partial}})^{\otimes (r\cdot N)}|_{\wtilde C}$, as tensorial powers of saturated subsheaves of locally-free sheaves remain saturated. At this point we can argue, as we did in the case of $\gamma^*\sL$, to find that the inequality 

\begin{equation}\label{e5}
    \deg({\overline{\sL}}^* \otimes \sO_Y(\gamma^*(N(K_X+D)))|_{\wtilde C})\geq 0,
       \end{equation}

\noindent holds. Now from~\ref{e5} we can readily establish~\ref{e4}. Then again~\ref{e1} follows from the projection formula, as required.








\end{proof}





\section{Viehweg-Zuo subsheaves in the parametrizing space}\label{VZJK}

The fundamental result of Jabbusch and Kebekus~\cite{JK11b} shows that the symmetric $\mathcal{C}$-differential forms is the correct framework to study the positivity of subsheaves of forms in the coarse moduli space of the canonically-polarized manifolds. In this section we give a brief explanation of how one can then reduce the isotriviality conjecture~\ref{iso} to the problem of showing that existence of rank one subsheaves of the sheaf of symmetric $\mathcal{C}$-differential forms, attached to a smooth pair, with maximal $\mathcal{C}$-Kodaira dimension implies that the given pair is of log-general type (see~\ref{reduction}.\ref{assert}). To prepare the correct setting for this reduction, we introduce a notion that, as far as the author is aware, is originally due to Campana. 


\begin{defn}[Neat model of a pair]\label{neat} Let $(Y,D)$ be a normal logarithmic pair and $h:Y\to Z$ a fibration with connected fibers and a projective base $Z$. We call a smooth pair $(Y_h,D_h)$ a neat model for $(Y,D)$ and h, if there exists a fibration $\wtilde h:Y_h \to Z_h$ that is birationally equivalent to $h$, i.e. there are birational morphisms $\mu:Y_h \to Y$ and $\alpha:Z_h \to Z$ such that the diagram

 $$
  \xymatrix{
     Y \ar[d]_{h}  &Y_h  \ar[l]_{\mu} \ar[d]^{\wtilde h}  \\
     Z   & Z_h \ar[l]_{\alpha}
  }
  $$
\noindent commutes, for which the following conditions are satisfied:

\begin{enumerate}
\item\label{neat1} $(Z_h,\Delta_h)$ is a smooth pair, where $\Delta_h$ is the $\mathcal{C}$-structure on $Z_h$ induced by $D_h$ and $\wtilde h$.
\item\label{neat2} $D_h$ is the extension of the $\mu$-birational transform $\wtilde D$ of $D$ by some reduced $\mu$-exceptional divisor, i.e. $D_h=\wtilde D+E'$, where $E'$ is $\mu$-exceptional.
\item\label{neat3} Every $\wtilde h$-exceptional prime divisor $P$ in $Y_h$, i.e. $P$ verifies the inequality $\codim_{Z_h}(\wtilde h(P))\geq 2$, is contained in $\supp(D_h)$.
\end{enumerate}


\end{defn}

The interest in the neat models of pairs that are equipped with fibrations, is two-fold. First, the conditions~\ref{neat}.\ref{neat1} and~\ref{neat}.\ref{neat3} ensures that $\wtilde h^*$ defines a pull-back map from symmetric $\mathcal{C}$-differential forms $\Sym_{\mathcal{C}}^N\Omega_{Z_h}^1\log\Delta_h$ attached to $(Z_h,\Delta_h)$ to the sheaf of symmetric logarithmic forms $\Sym^N\Omega_{Y_h}\log D_h$ (see~\cite[Prop.~2.10]{Ca08} for the details). Secondly, by~\ref{neat}.\ref{neat2} the neat model $(Y_h,D_h)$ leaves the birational geometry of the original pair $(Y,D)$ intact. For example if $(Y,D)$ special, then so is $(Y_h,D_h)$ (see~\cite[Cor.~9.5]{JK11b}). These properties will be crucial to the proof of the main result (Thm.~\ref{reduction}) in this section.




\begin{prop}[Construction of neat models, cf. \protect{\cite[Sect.~10]{JK11b}}]\label{construction} Every normal logarithmic pair $(Y,D)$ and a surjective morphism with connected fibers $h:Y \to Z$, where $Z$ is a projective variety, admits a neat model.


\end{prop}

\begin{proof}Let $\alpha_1:Z_1 \to Z$ be a suitable modification of the base of the fibration $h$ such that the normalization of the induced fiber product $Y\times_Z Z_1$, which we denote by $Y_1$, givers rise to an equidimensional fibration $h_1:Y_1 \to Z_1$, i.e. a \emph{flattening} of $h$, and a birational map $\mu_1:Y_1 \to Y$ (see the diagram below). Define $D_1$ to be the maximal reduced divisor contained in the $\supp(\mu_1^{-1}D)$ and let $$D_1=D_1^{\text{ver}}+D_1^{\text{hor}}$$ be the decomposition of $D_1$ into sum of its vertical $D_1^{\text{ver}}$ and horizontal $D_1^{\text{hor}}$ components. Introduce a closed subset in $Z_1$ by $D_{Z_1}:=h_1(D_1^{\text{ver}})$. Let $\Delta_1\subset Z_1$ denote the log-discriminant locus defined by the fibration $h_1$ and the divisor $D_1$. Recall that $\Delta_1$ is the smallest closed subset of $Z_1$ such that $h_1$ is smooth over its complement and that for every $z_1\in Z_1\backslash \Delta_1$, the set-theoretic fiber $h_1^{-1}(z_1)$ is not contained in $D_1$ and that the scheme-theoretic intersection of the fiber ${Y_1}_{z_1}$ with $D_1$ is snc in ${Y_1}_{z_1}$. Now let $\alpha_2:Z_h \to Z_1$ be a desingularization of $Z_1$ such that the maximal reduced divisor in the $\supp(\alpha_2^{-1}\Delta_1 \cup \alpha_2^{-1}D_{Z_1})$ is snc. Set $Y_2$ to be the normalization of the fiber product $Y_1\times_{Z_1}Z_2$, and $\mu_2$ the naturally induced birational morphism. Define $D_2$ in $Y_2$ by the maximal reduced divisor contained in the $\supp(\mu_2^{-1}D_1)$. Finally let $\mu_3:Y_h \to Y_2$ be a log-resolution of $(Y_2,D_2)$ and take $\wtilde h:Y_h \to Z_h$ to be the induced fibration. 


$$
\xymatrix{
    Y   \ar[d]^h && Y_1 \ar[ll]_{\mu_1} \ar[d]^{h_1} &&  Y_2 \ar[ll]_{\mu_2}  \ar[d]^{h_2} && Y_h \ar[ll]_{\mu_3} \ar[dll]^{\wtilde h}     \\
   Z  &&  Z_1 \ar[ll]_{\alpha_1} && Z_h \ar[ll]_{\alpha_2}
 }    
$$





\noindent Now set $\wtilde D_2$ to be the maximal reduced divisor in $\supp(\mu_3^{-1})$. Note that $h_1$ remains equidimensional under the base change of $\alpha_2$, i.e. $h_2$ is also equidimensional. This implies that when we desingularize $Y_2$ by $\mu_3$, every $\wtilde h$-exceptional divisor is $\mu_3$-exceptional. Let $E_3$ be the sum of all $\wtilde h$-exceptional prime divisors in $Y_h$ and define $D_h:=\wtilde D_2+E_3$ to be the extension of $\wtilde D_2$ by $E_3$. We finish by defining the birational morphisms $\mu$ and $\alpha$ in~\ref{neat} by $(\mu_3 \circ \mu_2 \circ \mu_1)$ and $(\alpha_2\circ \alpha_1)$, respectively. Now by construction, the $\mathcal{C}$-structure $\Delta_h$ on $Z_h$ induced by $D_h$ and $\wtilde h$ defines a smooth pair $(Z_h,\Delta_h)$, as required. 



\end{proof}



\begin{thm}[Reduction of the isotriviality conjecture]\label{reduction} The isotriviality conjecture~\ref{iso} holds, if the following assertion is true: 

\begin{enumerate} 
\item\label{assert} Let $(T,B)$ be a smooth pair. If $\Sym_{\mathcal{C}}^N\Omega_T^1\log B$ admits a saturated rank one subsheaf $\sL$ with $\kappa_{\mathcal{C}}(T,\sL)=\dim T$, then $(T,B)$ is of log-general type.

\end{enumerate}


\end{thm}


\begin{proof} Let $f^\circ:X^{\circ} \to Y^{\circ}$ be a smooth family of canonically-polarized manifolds, where $Y^\circ$ is a special quasi-projective variety, and let $Y$ be a smooth compactification with boundary divisor $D$ such that $D \cong Y\backslash Y^{\circ}$ and that the induced map $\wtilde {\mu}: Y \to \overline{\mathfrak M}$ to a compactification of $\mathfrak{M}$ is a morphism. Now if $\wtilde {\mu}$ is generically finite, then thanks to Campana and Paun's solution of the Viehweg's conjecture~\ref{VC} we find that $K_Y+D$ is big, contradicting the assumption that $(Y,D)$ is special. Thus to prove~\ref{iso}, we only need to treat the case where $\wtilde {\mu}$ is not generically finite. So assume that $\wtilde {\mu}:Y\to \overline{\mathfrak{M}}$ is not generically finite, then by the Stein factorization, and for a projective variety $Z$, the morphism $\wtilde \mu$ factors through a fibration with connected fibers $h:Y \to Z$ and a finite map $Z \to \overline{\mathfrak M}$. According to~\ref{construction} we can find a neat model $(Y_h,D_h)$ of the pair $(Y,D)$ and the fibration $h:Y\to Z$. 

$$
  \xymatrix{
    &&  Y  \ar[dll]_{\wtilde \mu}  \ar[d]_{h}  &Y_h  \ar[l]_{\mu} \ar[d]^{\wtilde h}  \\
     \overline{\mathfrak M}  && Z \ar[ll]_{\text{finite}}  & Z_h \ar[l]_{\alpha}
  }
  $$

 We note that since $Y_h\backslash D_h$ is isomorphic to an open subset of $Y^{\circ}$, it also parametrizes a smooth family of canonically polarized manifolds. Thus by~\cite[Thm.~1.4]{VZ02}, for some positive integer $N$, we can find a line subbundle $\sL\subseteq \Sym^N\Omega_{Y_h}\log D_h$ such that $\kappa(Y_h,\sL)\geq \dim Z_h$. Moreover by~\cite[Thm.~1.4]{JK11a}, we know that the Viehweg-Zuo subsheaf $\sL$ generically comes form the coarse moduli space. More precisely $\sL\subseteq\Sym^N\sB$, where $\sB$ is the saturation of the image of 

$$d\wtilde h: {\wtilde h}^*(\Omega_{Z_h}) \to \Omega_{Y_h}\log D_{h}.$$

Let us now collect the various properties of the pairs $(Y_h,D_h)$ and $(Z_h,\Delta_h)$, and the fibration $\wtilde h:Y_h\to Z_h$, where $\Delta_h$ is the $\mathcal{C}$-structure on $Z_h$ induced by $\wtilde h$ and $D_h$, that we have found so far:


\begin{enumerate}

\item $(Y_h,D_h)$ and $(Z_h,\Delta_h)$ are both smooth pairs, by definition~\ref{neat}.\ref{neat1}.
\item $D_{h}$ contains all $\wtilde h$-exceptional prime divisors, again by definition~\ref{neat}.\ref{neat3}.
\item There exists a saturated rank one subsheaf $\sL \subseteq \Sym^N \sB$, for some positive integer $N$, such that $\kappa(Y_h,\sL)\geq \dim Z_h$.

\end{enumerate}

With these conditions, we can apply~\cite[Cor.~5.8]{JK11a} to find a saturated rank one subsheaf $\sL_{Z_h}\subseteq \Sym_{\mathcal{C}}^N\Omega_{Z_h}^1\log \Delta_h$ such that 
\begin{equation}\label{sub}
     \kappa_{\mathcal{C}}(Z_h,\sL_{Z_h})= \kappa(Y_h,\sL)\geq \dim(Z_h).
        \end{equation}





Now if the statement~\ref{reduction}.\ref{assert} holds, then $(Z_h,\Delta_h)$ is of log-general type. On the other hand by the definition of a neat model~\ref{neat}.\ref{neat2}, for every $1\leq p \leq n$, we can pull-back line subbundles of $\Omega^p_{Y_h}\log D_{h}$ to subbundles of $\Omega^p_Y\log D$ (see~\cite[Cor.~9.5]{JK11b} for more details). This implies that $(Y_h,D_{h})$ is also special, which is a contradiction to the fact that $K_{Z_h}+\Delta_{h}$ is big.





\end{proof}





\

\section{The Isotriviality conjecture: The approach of Campana and Paun}\label{last}

In this section we prove the statement~\ref{reduction}.\ref{assert} in the previous section. The isotriviality conjecture will then follow from~\ref{reduction}. The proof is completely based on the solution of~\cite[Sect.~4]{CP13} to the Viehweg's hyperbolicity conjecture~\ref{VC}. 

For the ease of notation we replace the pair $(T,B)$ by $(X,D)$ with the warning that $D$ should not be confused with the boundary divisor of the compactification of $Y^{\circ}$ that was introduced in the previous sections. 


\begin{prop}\label{cp} Let $(X,D)$ be a smooth pair and $\sL\subseteq \Sym_{\mathcal{C}}^N\Omega_X^1\log D$ a  saturated rank one subsheaf with $\kappa_{\mathcal{C}}(X,\sL)=\dim X$. There exists a constant $c\in \mathbb{Q}^+(\sL)$ such that for every ample divisor $A$ in $X$, the inequality 

\begin{equation}\label{volume}
   vol(K_X+D+G)\geq c\cdot vol (A),
     \end{equation}


\noindent holds for every $\mathbb{Q}$-Cartier divisor $G$ satisfying the following properties:
\begin{enumerate}
\item\label{prop1} $D+G \sim_{\mathbb{Q}} P$, for some $\mathbb{Q}$-Cartier big divisor $P$ such that $\lfloor P \rfloor =0$.
\item\label{prop2} $(X,D+G)$ and $(X,P)$ are both smooth pairs.
\item\label{prop3} $K_X+D+G$ is pseudo-effective. 

\end{enumerate}


\end{prop}

\begin{proof} Let $L$ be a divisor in $X$ such that $\sO_X(L)\cong \sL$. We first observe, by Kodaira's lemma, that for any ample divisor $A$ in $X$ we have $A\leq F$, where $F$ is a divisor such that $\O_X(F)\cong \Sym_{\mathcal{C}}^m\sL$ (see~\ref{okodaira} for the definition) and $m$ is a sufficiently  large positive integer. With no loss of generality we may assume that $m=1$, so that $A\leq L$. 

\subsubsection{Step 1: A minimal model for $(X,P)$}\label{s1}

Since $(X,P)$ is smooth (property~\ref{cp}.\ref{prop2}), $P$ is big and $\lfloor P \rfloor =0$      (peroperty~\ref{cp}.\ref{prop1}), according to~\cite[Thm.~1.1]{BCHM10} $(X,P)$ admits, after a finite sequence of divisorial contractions or log-flips, either a log-minimal model or a Mori fiber space. But $K_X+D$ is pseudo-effective (property~\ref{cp}.\ref{prop3}), therefore $(X,P)$ has a log-minimal model $(X',P')$. Let $\pi: X \dashrightarrow X'$ be the corresponding birational map which can be then resolved by a suitable modification $\mu:\wtilde X \to X$ such that the resolution $\wtilde \pi:\wtilde X \to X'$ is a morphism and that $\supp(\Exc(\mu)\cup \wtilde D\cup \wtilde G)$ is an snc divisor in $\wtilde X$.


$$
  \xymatrix{
X \ar@{.>}[d]_{\pi}  &  & \wtilde X  \ar[ll]_{\mu}  \ar[dll]^{\wtilde {\pi}}  \\
 X'
}
$$

Now by construction $\pi^{-1}$ doesn't contract any divisors. As a result we can find an open subset $U\in X'$ such that $\codim_{X'}(X'\backslash U)\geq 2$ and that $\pi^{-1}|_{U}$ and ${\wtilde {\pi}}^{-1}|_{U}$ are isomorphisms. 

Fix an ample divisor $H$ in $X'$ and take $r$ to be any positive integer. Let $l_r$ be a sufficiently large positive integer such that the linear system $|l_r(K_{X'}+P'+\frac{1}{r}H)|$ is non-empty and basepoint-free and that every general member $B_r\in |l_r(K_{X'}+P'+\frac{1}{r}H)|$ cuts out a smooth irreducible curve $$C_r:=\underbrace{B_r\cap \ldots\cap B_r}_{(n-1)-\text{times}}$$ in $X'$ that is contained in $U$.




Let $E$ be the maximal reduced divisor contained in the support of the $\Exc({\mu})$ and define $\wtilde \sL:=\mu^*(\sL) \subseteq \Sym_{\mathcal{C}}^N\Omega_{\wtilde X}^1\log(\wtilde D+\wtilde G+E)$, where $\wtilde D$ and $\wtilde G$ are $\mu$-birational transforms of $D$ and $G$. From the ramification formula for $\mu$ we have the equality $K_{\wtilde X}+\wtilde D+\wtilde G={\mu}^*(K_X+D+G)+E'$, for an effective $\mu$-exceptional divisor $E'$. Now since $K_X+D+G$ is pseudo-effective (property~\ref{cp}.\ref{prop3}) then so is $K_{\wtilde X}+\wtilde D+\wtilde G$. This implies that $K_{\wtilde X}+\wtilde D+\wtilde G+E$ is also pseudo-effective. As a result we can apply~\ref{invariance} to find that for every $r\in \mathbb{N}^+$, the inequality 
\begin{equation}\label{eq1}
   N(K_{\wtilde X}+\wtilde D+\wtilde G+E)\cdot \wtilde \pi^*B_r^{n-1}\geq \wtilde L \cdot \wtilde \pi^* B_r^{n-1}
      \end{equation}
\noindent holds, where $\wtilde L$ is a divisor in $\wtilde X$ such that $\sO_{\wtilde X}(\wtilde L)\cong \wtilde \sL$.


\subsubsection{Step 2: The volume of $(K_{X'}+P')$} As $B_r^{n-1}$ cuts out an irreducible curve in $U$, we can rewrite the inequality~\ref{eq1} as 


\begin{equation}\label{eq2}
    N(K_{X'}+P') \cdot B_r^{n-1}\geq \wtilde L\cdot \wtilde {\pi}^*B_r^{n-1},\ \forall r\in \mathbb{N}^+
         \end{equation}

\noindent by using the assumption that $(K_X+D+G) \sim_{\mathbb{Q}}(K_X+P)$ (property~\ref{cp}.\ref{prop1}). Moreover the inequality~\ref{eq2} implies that 

\begin{equation}\label{eq3}
  (N(K_{X'}+P'+\frac{1}{r}H))\cdot B_r^{n-1} \geq \wtilde L\cdot \wtilde {\pi}^*B_r^{n-1}, \ \forall r\in \mathbb{N}^+
       \end{equation}


\noindent also holds, where the left-hand side of~\ref{eq3} is equal to $N\cdot l_r^{n-1} \cdot vol(K_{X'}+P'+\frac{1}{r}H)$. On the other hand since $L-A$ is effective (up to linear equivalence) and $\wtilde \pi^*B_r$ is nef, the inequality $$\mu^*(L-A)\cdot  \wtilde \pi^* B_r^{n-1}\geq 0,\ \forall r\in \mathbb{N}^+,$$ holds, i.e. 
\begin{equation}\label{eq4}
   \wtilde L\cdot \wtilde \pi^* B_r^{n-1}\geq \mu^*A\cdot \wtilde \pi^* B_r^{n-1}=l_r^{n-1}\cdot \mu^*A\cdot \wtilde \pi^*(K_{X'}+P'+\frac{1}{r}H)^{n-1}, \ \forall r\in \mathbb{N}^+.
      \end{equation}


Now by the Tiessier's inequality~\cite[Thm.~1.6.1]{Laz} the right-hand side of~\ref{eq4} verifies the inequality 

\begin{equation}\label{eq5}
   l_r^{n-1}\cdot \mu^*A\cdot \wtilde \pi^*(K_{X'}+P'+\frac{1}{r}H)^{n-1}\geq l_r^{n-1}\cdot vol(A)^{\frac{1}{n}}\cdot vol(K_{X'}+P'+\frac{1}{r}H)^{\frac{n-1}{n}},
\end{equation}


\noindent for all $r\in \mathbb{N}^+$. By combining the inequalities~\ref{eq3},~\ref{eq4} and~\ref{eq5} we find that the inequality 

\begin{equation}\label{eq6}
  vol(K_{X'}+P'+\frac{1}{r}H)\geq \frac{1}{N^n} \cdot vol(A), \ \forall r\in \mathbb{N}^+
    \end{equation}
    holds. After taking the limit $r\to \infty$, the continuity of $vol(.)$ implies that
 
 \begin{equation}\label{eq7}
     vol(K_{X'}+P')\geq \frac{1}{N^n}\cdot vol(A).
       \end{equation} 
    
    
\subsubsection{Step 3: The volume of $(K_X+P)$} By the negativity lemma in the minimal model theory, we know that $H^0(X,m(K_X+P))\cong H^0(X',m(K_{X'}+P'))$, for all $m\in \mathbb{N}^+$.  In particular the equality $vol(X,K_X+P)=vol(X',K_{X'}+P')$ holds. The required inequality~\ref{volume} now follows form~\ref{eq7} in the previous step and property~\ref{cp}.\ref{prop1}.


\end{proof}

\



\begin{thm}\label{final} Let $(X,D)$ be a smooth pair and $\sL\subseteq \Sym_{\mathcal{C}}^N\Omega_X\log D$ a saturated rank one subsheaf. If $\kappa_{\mathcal{C}}(X,\sL)=\dim X$  , then $K_X+D$ is big.


\end{thm}

\begin{proof} Let $H$ be a very ample divisor such that $H-D$ is ample, and let $r$ be the smallest positive integer for which $r(H-D)$ is very ample. Define the sufficiently general hyperplane section $B_{D}$ to be a general member of the linear system $|r(H-D)|$. From construction it now follows that, for every integer $M>r$, the $\mathbb{Q}$-divisor $D+\frac{1}{M}B_{D}$ is $\mathbb{Q}$-linearly equivalent to an snc divisor, which we denote by $P_M$, with no reduced components:

\begin{align*}
D+\frac{1}{M}B_{D} &\sim_{\mathbb{Q}}D+\frac{1}{M}(r(H-D))\\
     &=(1-\frac{r}{M})D+\frac{r}{M}H=:P_M.\\
  \end{align*}

\begin{subclaim}\label{claim} $K_X+P_M$ is pseudo-effective, for all integers verifying the inequality $M>r$.

\end{subclaim}

Let us for the moment assume that the claim holds. Define the $\mathbb{Q}$-Cartier divisor in~\ref{cp} by $G:=\frac{1}{M}B_{D}$. As the conditions~\ref{cp}.\ref{prop1}.~\ref{cp}.\ref{prop2} and~\ref{cp}.\ref{prop3} in~\ref{cp} are all verified, it follows from the inequality~\ref{volume} that  

\begin{equation}\label{eq8}
   vol(K_X+D+\frac{1}{M}B_{D}) \geq c\cdot vol(A),\ \   \forall M\in \mathbb{N} \text{\ such that \  } M>r
       \end{equation}

 Thus, after taking $M\to \infty$, the continuity property of $vol(.)$ implies that $K_X+D$ is big.

It now remains to prove the claim~\ref{claim}.

\

\emph{Proof of claim~\ref{claim}}. Aiming to extract a contradiction, suppose that $K_X+P_M$ is not pseudo-effective for some $M>r$. Let $H'$ be a suitably chosen very ample divisor such that the effective log-threshold given by 

$$\epsilon:=min \{ t\in \mathbb{R}^+: K_X+P_M+tH' \textrm{\;\ is pseudo-effective}\},$$ is smaller than $1$. According to~\cite[Cor. 1.1.7]{BCHM10} $\epsilon$ is rational. Now by applying~\ref{cp} to the pair $(X,D)$ with $G=\frac{1}{M}B_{D}+\epsilon H'$, we find that $K_X+P_M+\epsilon H'$ is big. But as the big cone forms the interior of the cone of pseudo-effective $\mathbb{Q}$-Cartier classes, for sufficiently small $\delta$, $K_X+D_M+(\epsilon-\delta)H'$ is also pseudo-effective, contradicting the minimality condition on $\epsilon$.



\end{proof}


The isotriviality conjecture~\ref{iso} now follows from~\ref{final} together with~\ref{reduction}.



\



\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
% \MRhref is called by the amsart/book/proc definition of \MR.
\providecommand{\MRhref}[2]{%
  \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
\begin{thebibliography}{BCHM10}


\bibitem[BCHM10]{BCHM10}
Caucher Birkar, Paolo Cascini, Christopher D.Hacon, and James McKernan, \emph{Existence of minimal models for varieties of log general type}, Journal of the AMS 23 (2010), 405-468.


\bibitem[BDPP04]{BDPP}
S{\'e}bastien Boucksom, Jean-Pierre Demailly, Mihai Paun, and Thomas Peternell, \emph{The pseudo-effective cone of a compact {K}\"ahler manifold and varieties of negative Kodaira dimension}, J. Algebraic. Geom. \textbf{22} (2013), 201-248. doi:10.1090/S1056-3911-2012-00574-8.	
     
     
  
\bibitem[Cam04]{Ca04} 
 Fr{\'e}d{\'e}ric Campana, \emph{Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble)}, Vol54, no. 3, 499-630 (2004)


\bibitem[Cam08]{Ca08}
 Fr{\'e}d{\'e}ric Campana, \emph{Orbifoldes sp\'ciales et classification bim\'eromorphe des vari\'et\'es {K}\"ahl\'eriennes compactes}, arXiv:0705.0737v5 (2008).



\bibitem[CP13]{CP13}
Fr{\'e}d{\'e}ric Campana, and Mihai Paun, \emph{Orbifold generic semi-positivity: an application to families of canonically polarized families}, version~1, arXiv:1303.3169.



\bibitem[Iit82]{Iit82} Shigeru iitaka, \emph{Algebraic geometry}, Springer-Verlag, New York, 1977, Graduate texts in Mathematics, No. 52. 57. Number 3116.


\bibitem[JK11a]{JK11a}
Kelly Jabbusch, and Stefan Kebekus, \emph{Positive sheaves of differentials coming from coarse moduli spaces}, Annales de l'institut Fourier, 61 no. 6 (2011), p. 2277-2290.


\bibitem[JK11b]{JK11b}
\bysame, \emph{Families over special base manifolds and a conjecture of Campana}, Mathematische Zeitschrift: Volume 269, Issue 3 (2011), Page 847-878.



\bibitem[KK08]{KK08}
Stefan Kebekus, and S\'andor Kov\'acs, \emph{The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties}, Duke Math. J. vol.~155, Number 1, pp.~1-33, 2010.


\bibitem[Laz]{Laz}
Robert Lazarsfeld, \emph{Positivity in Algebraic Geometry I} , Springer, Vol. 48.


\bibitem[Lu02]{Lu02}
Steven S. Y. Lu, \emph{A refined Kodaira dimension and its canonical fibration}, arXiv:math/0211029.


\bibitem[Miy87a]{Miy87}
Yoichi Miyaoka, \emph{The {C}hern classes and {K}odaira dimension of a minimal
  variety}, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol.~10,
  North-Holland, Amsterdam, 1987, pp.~449--476. \MR{89k:14022}


\bibitem[Miy87b]{Miy85}
\bysame, \emph{Deformations of a morphism along a foliation and applications},
  Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos.
  Pure Math., vol.~46, Amer. Math. Soc., Providence, RI, 1987, pp.~245--268.
  \MR{MR927960 (89e:14011)}


\bibitem[Miy08]{Miy08}
\bysame, \emph{The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem}, Publ. RIMS, Kyoto University, vol.~44, pp.~403-417, 2008.



\bibitem[VZ02]{VZ02}
Eckart Viehweg and Kang Zuo, \emph{Base spaces of non-isotrivial families of smooth minimal models}, Complex geometry (G\"ottingen, 2000), Springer, Berlin, 2002, pp. 279-328.




\end{thebibliography}
 
 
 
 
 



\end{document}







1310.5391.extracted_bib
\begin{thebibliography}{BCHM10}


\bibitem[BCHM10]{BCHM10}
Caucher Birkar, Paolo Cascini, Christopher D.Hacon, and James McKernan, \emph{Existence of minimal models for varieties of log general type}, Journal of the AMS 23 (2010), 405-468.


\bibitem[BDPP04]{BDPP}
S{\'e}bastien Boucksom, Jean-Pierre Demailly, Mihai Paun, and Thomas Peternell, \emph{The pseudo-effective cone of a compact {K}\"ahler manifold and varieties of negative Kodaira dimension}, J. Algebraic. Geom. \textbf{22} (2013), 201-248. doi:10.1090/S1056-3911-2012-00574-8.	
     
     
  
\bibitem[Cam04]{Ca04} 
 Fr{\'e}d{\'e}ric Campana, \emph{Orbifolds, special varieties and classification theory, Ann. Inst. Fourier (Grenoble)}, Vol54, no. 3, 499-630 (2004)


\bibitem[Cam08]{Ca08}
 Fr{\'e}d{\'e}ric Campana, \emph{Orbifoldes sp\'ciales et classification bim\'eromorphe des vari\'et\'es {K}\"ahl\'eriennes compactes}, arXiv:0705.0737v5 (2008).



\bibitem[CP13]{CP13}
Fr{\'e}d{\'e}ric Campana, and Mihai Paun, \emph{Orbifold generic semi-positivity: an application to families of canonically polarized families}, version~1, arXiv:1303.3169.



\bibitem[Iit82]{Iit82} Shigeru iitaka, \emph{Algebraic geometry}, Springer-Verlag, New York, 1977, Graduate texts in Mathematics, No. 52. 57. Number 3116.


\bibitem[JK11a]{JK11a}
Kelly Jabbusch, and Stefan Kebekus, \emph{Positive sheaves of differentials coming from coarse moduli spaces}, Annales de l'institut Fourier, 61 no. 6 (2011), p. 2277-2290.


\bibitem[JK11b]{JK11b}
\bysame, \emph{Families over special base manifolds and a conjecture of Campana}, Mathematische Zeitschrift: Volume 269, Issue 3 (2011), Page 847-878.



\bibitem[KK08]{KK08}
Stefan Kebekus, and S\'andor Kov\'acs, \emph{The structure of surfaces and threefolds mapping to the moduli stack of canonically polarized varieties}, Duke Math. J. vol.~155, Number 1, pp.~1-33, 2010.


\bibitem[Laz]{Laz}
Robert Lazarsfeld, \emph{Positivity in Algebraic Geometry I} , Springer, Vol. 48.


\bibitem[Lu02]{Lu02}
Steven S. Y. Lu, \emph{A refined Kodaira dimension and its canonical fibration}, arXiv:math/0211029.


\bibitem[Miy87a]{Miy87}
Yoichi Miyaoka, \emph{The {C}hern classes and {K}odaira dimension of a minimal
  variety}, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol.~10,
  North-Holland, Amsterdam, 1987, pp.~449--476. \MR{89k:14022}


\bibitem[Miy87b]{Miy85}
\bysame, \emph{Deformations of a morphism along a foliation and applications},
  Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos.
  Pure Math., vol.~46, Amer. Math. Soc., Providence, RI, 1987, pp.~245--268.
  \MR{MR927960 (89e:14011)}


\bibitem[Miy08]{Miy08}
\bysame, \emph{The orbibundle Miyaoka-Yau-Sakai inequality and an effective Bogomolov-McQuillan theorem}, Publ. RIMS, Kyoto University, vol.~44, pp.~403-417, 2008.



\bibitem[VZ02]{VZ02}
Eckart Viehweg and Kang Zuo, \emph{Base spaces of non-isotrivial families of smooth minimal models}, Complex geometry (G\"ottingen, 2000), Springer, Berlin, 2002, pp. 279-328.




\end{thebibliography}