\subsection{Moduli applications}

\begin{definition}
\label{def:functor}
Let $\fX$ be a class of $K$-semi-stable Fano varieties over $k$. A pseudo-functor (or equivalently category fibered in groupoid) $\sM_{\fX}$ is called \emph{a moduli functor for $\fX$}, if for normal test scheme $T$ of finite type  over $k$:
\begin{equation}
    \label{eq:functor}
    \sM_{\sX} (T) = 
    \left\{ \raisebox{16pt}{\xymatrix@R=15pt{ X \ar[d]_f \\ T }} \left| \parbox{26em}{
          \begin{enumerate}
          \item $f$ is a flat morphism,
          \item $K_{X/T}$ is $\bQ$-Cartier,
          \item for each geometric point $\ot \in T$, $X_{\ot} \in \fX$.
          \end{enumerate}} \right.  \right\},
\end{equation}
with arrows being given by Cartesian diagrams. 
\end{definition}

\begin{remark}
We did not require anything in \autoref{def:functor} for the value of $\sM_{\fX}$  on non-normal test schemes to keep the definition open to future developments of the theory.
\end{remark}


\begin{remark}
In higher dimensional moduli theory generally it is required also Koll\'ar's condition, that is, for all integer $m$,            $\omega_{X/T}^{[m]}$ is supposed to be flat and compatible with base-change. Here, the latter compatibility precisely means that $\left( \omega_{X/T}^{[m]}\right)_S \cong \omega_{X_S/S}^{[m]}$ for every base change $S \to T$.

We did not require this condition in \autoref{def:functor} as it is known that over reduced bases in characteristic zero, this condition follows from $K_{X/T}$ being $\bQ$-Cartier \cite[Thm 3.68]{Kollar_Families_of_varieties_of_general_type}
\end{remark}


\begin{definition}
\label{def:moduli}
Let $\fX$ be a class of $K$-semistable Fano varieties over $k$ and $\sM_{\fX}$ a moduli functor for $\fX$  (as in \autoref{def:functor}). We say that a proper algebraic space $M$ is a \emph{moduli space for $\sX$} with the uniform $K$-stable locus $M^u \subseteq M$ being open, if 
\begin{enumerate}
\item $M^u \subseteq M$ is an open sub-algebraic space, 
\item $\mu : \sM_{\fX} \to M$ is universal among maps to algebraic spaces (that is, for any algebraic space $N$, composition with $\mu$ induces a bijection $\Hom\left(\sM_{\fX},N\right)\cong \Hom(M,N)$), 
\item $\mu(k)$ takes exactly the uniformly $K$-stable varieties from $\fX(k)$ to $M^u(k)$ ,
\item $\mu(k)$ is surjective and it is a bijection when restricted to the uniform $K$-stable varieties,  
\item there is a generically finite, proper cover $\pi : Z \to M$ by a scheme, given by a family $f : X \to Z$ in $\sM_{\sX}(Z)$, such that 
\begin{enumerate}
\item $\pi$ is finite over $M^u$, and that 
\item some positive multiple of the CM line bundle $\lambda_{f}$ on $Z$ is numerically equivalent to a line bundle that descends to $M$. 
\end{enumerate}

\end{enumerate}

\end{definition}


\begin{proof}[Proof of \autoref{cor:moduli}]
Consider the generically finite cover $\pi : Z \to M$ given by \autoref{def:moduli}. Also by the assumptions of \autoref{def:moduli}, there is an integer $r>0$ and a line bundle $\sL$ on $M$ such that $\pi^* \sL \equiv r \lambda_f$. Let $\rho : M' \to M$ be the normalization of $M$ and let $\tau : Z \to M'$ be the induced morphism. Set $\sN:= \rho^* \sL$. We note that $\rho^{-1}(M^u)$ is the normalization of $M^u$.

According to \cite[Thm 6.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds} to prove that $\rho^{-1}(M^u)$ is quasi-projective, we have to show that $\sN$ is nef and for all closed, irreducible subspaces $V \subseteq \rho^{-1}(M^u)$, $c_1(\sN)^{\dim V} >0$. However, these are immediate, as:
\begin{enumerate}
 \item $\tau^* \sN \equiv r \lambda_f$ is nef according to \autoref{thm:semi_positive_no_boundary}.\autoref{itm:semi_positive_no_boundary:nef}, hence so is $\sN$.
 \item $\tau^{-1}V$ is a closed subspace of $Z$ intersecting non-trivially $\pi^{-1}(M^u)$. Let $\iota : W \to Z$ be the resolution of a component of $\tau^{-1}V$ dominating $V$. By Nagata's theorem we may assume that $W$ is projective (as opposed to just proper). Then $\iota(W)$ has to intersect $\pi^{-1}(M^u)$, and as $\tau|_{\pi^{-1}(M^u)}$ is finite, $\dim W = \dim V$. By the assumption $\tau^* \sN \equiv r \lambda_f$, it is enough to prove that $\lambda_f|_{\tau^{-1}V}^{\dim W}>0$. As $\lambda_f$ is nef, for this in turn it is enough to show that $\lambda_f|_{W}=\lambda_{f_W}$ is big, where the latter equality is given by \autoref{prop:CM_base_change}. However, $f_W$ is of maximal variation by the finiteness of $\tau|_{\pi^{-1}(M^u)}$. So, we are done by \autoref{thm:semi_positive_no_boundary}.\autoref{itm:semi_positive_no_boundary:big} applied to $f_W$.
\end{enumerate}

\end{proof}

\begin{proof}[Proof of \autoref{cor:bigger_open_set}]
The proof is a verbatim copy of the above proof of \autoref{cor:moduli} with two differences:
\begin{itemize}
 \item One does not have to assume the existence of the moduli space $M_{\sX}$, as its semi-normalization is known to exist by \cite[Thm 1.3]{Li_Wang_Xu_Degeneration} and \cite[Thm 1.1]{Odaka_Compact_moduli_space_of_Kahler_Einstein_Fano_varieties}. As we work on the normalization this is enough for our purposes because the normalization maps to the semi-normalization. 
 \item The argument eventually boils down to showing that if $g : Y \to V$ is a family of $K$-semi-stable klt Fanos of maximal variation such that the very general fiber is either smooth or uniformly $K$-stable, then $\lambda_g$ is big. In the uniformly $K$-stable case this is \autoref{cor:proper_base}, and in the smooth case this is \cite[Thm 1.1]{Li_Wang_Xu_Qasi_projectivity_of_the_moduli_space_of_smooth_Kahler_Einstein_Fano_manifolds}. 
 
\end{itemize}


\end{proof}


\subsection{Applications to anticanonical volumes}

\begin{proof}[Proof of \autoref{cor:bounding_volume}]
We have
\begin{multline*}
\vol(-K_X-\Delta) 

=
\underbrace{(-K_X - \Delta)^{\dim X}}_{\textrm{$-K_X - \Delta$ is ample}}

= \left((-K_{X/\bP^1} - \Delta) - f^* K_{\bP^1}\right)^{\dim X}

\\ = (-K_{X/\bP^1} - \Delta)^{\dim X} + (\dim X) 2 \vol (-K_F-\Delta_F)
= - \deg \lambda_{f, \Delta} + (\dim X) 2 \vol (-K_F-\Delta_F)

\\ \leq 

\underbrace{(\dim X) 2 \vol (-K_F-\Delta_F)}_{\textrm{\autoref{thm:semi_positive_boundary}}}
\end{multline*}
For the second claim, if $F$ is smooth, we apply the bound on the volume of K-semi-stable Fano varieties obtained in \cite[Thm 1.1]{Fujita_Optimal_bounds_for_the_volumes_of_Kahler-Einstein_Fano_manifolds}  to $F$; if $F$ is singular, we apply  \cite[Thm 3]{Liu_The_volume_of_singular_Kahler-Einstein_Fano_varieties}.

\end{proof}
{"config": "arxiv", "file": "1806.07180/applications.tex"}