In this section we prove our semi-positivity results. Here, and also in \autoref{sec:positivity} we use extensively the fiber product notation explained in \autoref{sec:product_notation}.

\subsection{Framework and results}

In this and the next sections we work in the following setup:

\begin{notation}
\label{notation:CM_semi_positivity}
 Let  $f \colon (X,\Delta) \to T$ satisfy the following assumptions:
 \begin{enumerate}
  \item $T$ is a smooth, projective curve, 
  \item $X$ is a normal, projective variety of dimension $n+1$,
  \item $f$ is a projective and surjective morphism with connected fibers, 
  \item $\Delta$ is an effective $\bQ$-divisor on $X$, 
  \item $-(K_X + \Delta)$ is an  $f$-ample $\bQ$-Cartier divisor. 
  \item $(X_t,\Delta_t)$ is klt for general $t \in T$. 
 \end{enumerate}


\end{notation}

The main result of the section is the following, from which the statements of the introduction will follow in a quite straightforward manner.

\begin{theorem}
\label{thm:semi_positivity_curve}
In the situation of \autoref{notation:CM_semi_positivity}, if $\delta\left(X_{\ot},\Delta_{\ot}\right) \geq 1$ for very general geometric points $t \in T$, then $\deg \lambda_{f,\Delta} \geq 0$. 
\end{theorem}


\subsection{Proofs}
\label{sec:semi_pos_proof}


The proof of \autoref{thm:semi_positivity_curve} will be by contradiction with the next proposition.  

\begin{proposition}\label{prop:cont}
In the situation of \autoref{notation:CM_semi_positivity}, let $H$ be an ample $\bQ$-divisor  on $T$. Then, there do not exist $\bQ$-Cartier divisors $\Gamma$ and $\widetilde{\Gamma}$ on $X$ such that:
\begin{enumerate}
 \item\label{class} $\Gamma + \widetilde{\Gamma} \sim_{\bQ} -K_{X/T}-\Delta-f^* H$,
  \item \label{ample} $\widetilde{\Gamma}$ is nef, and 
 \item \label{klt} $(X_t,\Delta_t +\Gamma_t)$ is klt for  $t \in T$ general. 
\end{enumerate}
\end{proposition}
\begin{proof}

Assume that there exist $\Gamma$ and $\widetilde{\Gamma}$ as above. 
Let $a>0$ be a rational number such that $-K_{X/T} - \Delta + af^* H$ is ample. Fix a rational number $\varepsilon > 0$ such that $\varepsilon a - (1-\varepsilon)< 0$. The following computation the yields a contradiction, as \autoref{cor:semi_positivity_engine_upstairs} yields that the right side is nef. 


\begin{equation*}
\underbrace{  (\varepsilon a - (1-\varepsilon)) f^*H}_{\varepsilon a - (1-\varepsilon)<0 \Rightarrow \textrm{ this is not nef}}  

 \sim_{\bQ}
 \underbrace{K_{X/T} + \Delta + (1-\varepsilon) \Gamma }_{(X_t,\Delta_t +(1-\varepsilon) \Gamma_t ) \textrm{ is klt }} 

+ \underbrace{(1-\varepsilon)\widetilde{\Gamma} + \varepsilon\left(- K_{X/T} -\Delta + af^* H \right)}_{\textrm{ample}}  

\end{equation*}
\end{proof}


\begin{proof}[Proof of \autoref{thm:semi_positivity_curve}]
As both the consequences and the conditions of the theorem are invariant under base-extension to another algebraically closed field, we may assume that $k$ is uncountable. In particular, the very general geometric fibers show up also amongst closed fibers.

First, according to \autoref{lem:reducing_reduced_fibers} we may assume that all fibers of $f$ are reduced. This is to guarantee that the $m$-times iterated fiber product $X^{(m)}$ is normal for any integer $m>0$, according to \autoref{lem:normal_stable_pullback_fiber_product}. 

We argue by contradiction, so assume  that $\deg \lambda_{f,\Delta}<0$.  For $m$ big enough, we are going to produce divisors $\Gamma$ and $\widetilde{\Gamma}$ on $\left( X^{(m)},\Delta^{(m)}\right) $ whose existence contradicts \autoref{prop:cont}. 

Fix a closed point $t$ in $T$ such that $X_t$ is normal, $X_t \not\subseteq \Supp \Delta_t$, $(X_t,\Delta_t)$ is klt and $\delta(X_t, \Delta_t) \geq 1$ (using \autoref{prop:delta_general_fiber}).  Let $H$ be an ample line bundle on $T$. Fix rational numbers $a,\varepsilon>0$ and $0<c<1$ and an integer $q>0$, such that:
\begin{enumerate}[itemsep=4pt]
 \item the intersection product inequality $(-K_{X/T} -\Delta- \varepsilon f^* H)^{n+1}>0$ holds. This is possible because \autoref{def:CM_log}
and the assumption $\deg \lambda_{f,\Delta}<0$ implies that $ (-K_{X/T} - \Delta)^{n+1}>0$. Set $M:= -K_{X/T} - \Delta - \varepsilon f^* H$.
 \item  $D:=-K_{X/T}-\Delta+af^*H$ is ample.
 \item \label{negative}
$c<\frac{\varepsilon}{a+\varepsilon}$.
\item \label{Cartier} $qM$ is Cartier, which is possible, as $M$ is $\bQ$-Cartier.
\item \label{higher_cohomology} $R^if_*\sO_X(qM)=0$ for all $i>0$, which is possible, as $M$ is $f$-ample.
\item \label{degree} $\deg\left(f_*\sO_X(qM)\right)>0$, using \autoref{lem:Mumford_line_bundles_over_curve}.
\item \label{delta} $\delta_q(X_t,\Delta_t)>1-c$, using \autoref{thm:BJ}.
\end{enumerate}
From now on, let $\sE:=f_*\sO_X(qM)$. Remark that according to \cite[Lemma 3.6]{Kovacs_Patakfalvi_Projectivity_of_the_moduli_space_of_stable_log_varieties_and_subadditvity_of_log_Kodaira_dimension} for every integer $m>0$,
$$
\sE^{\otimes m}=f_*^{(m)} \sO_{X^{(m)}} \left( q M^{(m)} \right) \cong f_*^{(m)} \sO_{X^{(m)}}\left( q\left( -  K_{X^{(m)}/T} -\Delta^{(m)}-  m\varepsilon \left(f^{(m)}\right)^* H\right)\right),
$$
and by item \autoref{higher_cohomology}, the following base change holds
$$
\sE^{\otimes m}_t=H^0\left(X_t^{(m)}, q\left( -  K_{X_t^{(m)}} -\Delta^{(m)}_t \right)  \right).
$$
In general, it is not possible to lift a basis of $\sE_t$ to sections of $\sE$. However, thanks to \autoref{prop:vectP2}, we can choose a basis $e_i$ of $\sE_t$, an integer $m>0$, and $\ell$ global sections $s_i$ of $\sE^{\otimes m}$ so that the sections $s_i$, when restricted over $t$, are linearly independent pure tensor in the $e_i$, and furthermore
\begin{equation}\label{bound}
\frac{\ell}{h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta_t^{(m)}\right)\right)}

>

\underbrace{\frac{1-c}{\delta_q(X_t,\Delta_t)}}_{\parbox{57pt}{\tiny $<1$ according to assumption \autoref{delta}}} \,.
\end{equation}
We are now ready to construct $\Gamma$ and $\widetilde{\Gamma}$ on $X^{(m)}$ as in \autoref{prop:cont}. We let
$$ \Gamma:=(1-c)\frac{1}{q\ell}\sum_{i=1}^{\ell}\{s_i=0\}\,,$$
and
$$ \widetilde{\Gamma}:=cD^{(m)}  \,.$$ 
To complete the proof of \autoref{thm:semi_positivity_curve}, we have to prove that $\Gamma$ and $\widetilde{\Gamma}$ are as in \autoref{prop:cont}, with $f$ replaced by $f^{(m)}$. To check item \autoref{class}, remark that
$$\Gamma+\widetilde{\Gamma} \sim_{\bQ} -K_{X^{(m)}/T}-\Delta^{(m)}+ m\left( c a - (1 - c) \varepsilon \right) \left(f^{(m)}\right)^*H.$$
Furthermore,  because of assumption \autoref{negative}, $c a - (1 - c) \varepsilon <0$ holds; so, we may apply \autoref{prop:cont} replacing $H$ by $-m\left( c a - (1 - c) \varepsilon \right) H$. Item \autoref{ample} of \autoref{prop:cont} follows from the ampleness of $D$.

  
To prove of item \autoref{klt} of \autoref{prop:cont}, we compute the log canonical threshold. We first remark that, since the sections $s_i$ restricted to $X_t^{(m)}$ are linearly independent pure tensors in the $e_i$, we have that
$$
 \frac{\ell}{h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta^{(m)}_t\right)\right)} \Gamma_t \leq (1-c)P
$$
for the $q$-product basis type divisor  $P$  on $X_t^{(m)}$ associated to $\{e_i\}$, as in \autoref{def:prod_basis} and \autoref{rem:prod_basis}. Using \autoref{prop:prod_basis}, we obtain that $\lct\left(X^{(m)}_t,\Delta^{(m)}_t;P_t\right)\geq \delta_q(X_t,\Delta_t)$. This yields 
$$
\lct\left(X_t^{(m)},\Delta_t^{(m)};\Gamma_t\right) 

\geq 

\underbrace{\frac{\delta_q(X_t,\Delta_t)\ell}{(1-c)h^0\left(X_t^{(m)},-q\left(K_{X_t^{(m)}}+\Delta^{(m)}_t\right)\right)} >1}_{\textrm{by assumption \autoref{bound}}}.
$$
Hence, all assumptions of \autoref{prop:cont} are verified, implying that $\Gamma$ and $\widetilde{\Gamma}$ cannot exist. Therefore, we obtained a contradiction with our initial assumption that $\deg \lambda_{f,\Delta} <0$. 
\end{proof}

\begin{proof}[Proof of \autoref{thm:semi_positive_boundary}]
\emph{The proof of point \autoref{itm:semi_positive_boundary:pseff}:}
As at the beginning of the proof of \autoref{thm:semi_positivity_curve}, we may assume that $k$ is uncountable. 
 According to \cite[Thm 0.2]{Boucksom_Demailly_Paun_Peternell_The_Pseudo_effective_cone_of_a_compact_Kahler_manifold_and_varieties_of_negative_Kodaira_dimension}, it is enough to show that $\lambda_{f,\Delta} \cdot C \geq 0$ for every morphism $\iota: C\to X$ from a smooth projective curve  such that $C \to \iota(C)$ is the normalization and  $\iota(C)$ is a very general curve in a  family covering $T$.   In particular,  for a very general closed point $t \in \iota(C)$, $X_t$ is normal, $(X_t, \Delta_t)$ is klt and $\delta\left(X_t, \Delta_t\right) \geq 1$. Let $Z \to X_C$ be the normalization, $g : Z \to C$ the induced morphism and $\Delta_Z$ the boundary induced by $\Delta$ on $Z$ as explained in \autoref{sec:base_change_relative_canonical_smooth_base}. According to \autoref{prop:relative_canonical_base_change_normal}.\autoref{itm:relative_canonical_base_change:base_change}, $g : (Z, \Delta_Z) \to C$ satisfies the assumptions of \autoref{thm:semi_positivity_curve}. Hence the following computation concludes the proof of point \autoref{itm:semi_positive_boundary:pseff}:
\begin{equation*}
0 \leq  

\underbrace{\deg \lambda_{g, \Delta_Z}}_{\textrm{\autoref{thm:semi_positivity_curve}}}

=

\underbrace{C \cdot \lambda_{f, \Delta}}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:smooth_base}}}.
\end{equation*}


\emph{The proof of point \autoref{itm:semi_positive_boundary:nef}:}
In this case for each finite morphism $C \to T$ from a smooth projective curve, according to \autoref{sec:base_change_relative_canonical_normal_fibers}, $f_C : (X_C, \Delta_C) \to C$ satisfy the assumptions of \autoref{thm:semi_positivity_curve}. So:
\begin{equation*}
0 \leq  

\underbrace{\deg \lambda_{f_C, \Delta_C}}_{\textrm{\autoref{thm:semi_positivity_curve}}}

=

\underbrace{C \cdot \lambda_{f, \Delta}}_{\textrm{\autoref{prop:CM_base_change}.\autoref{itm:CM_base_change:normal_fiber}}}.
\end{equation*}
\end{proof}

\begin{proof}[Proof of points \autoref{itm:semi_positive_no_boundary:pseff} and \autoref{itm:semi_positive_no_boundary:nef} of \autoref{thm:semi_positive_no_boundary}]
These are special cases of \autoref{thm:semi_positive_boundary}. 
\end{proof}
{"config": "arxiv", "file": "1806.07180/proof.tex"}