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 \newpage
 \input{Part_3.tex}
 
-%\newpage
-%\input{Part_4.tex}
+\newpage
+\input{Part_4.tex}
 
 
 
 
 \subsection{Ampleness}
 
+K\"ahler metric
+\begin{equation}
+  \begin{gathered}
+    g_{i\bar j}(z,\bar z) =
+    \partial_i \bar \partial_{\bar j} K(z,\bar z)
+    \\
+    \omega = g_{i\bar j}(z,\bar z) \diff z^i \diff \bar z^{\bar j}
+    = \partial \bar \partial K(z,\bar z)
+    .
+  \end{gathered}    
+\end{equation}
 
+
+K\"ahler potential
+\begin{equation}
+  K(z,\bar z) = \ln
+  \sum_{\alpha, \bar \beta}
+  h^{\alpha \bar \beta}
+  s_\alpha \bar s_{\bar \beta}
+  ,\quad
+  \Span\{s_1,\dots\} = \Gamma(X,\Osheaf(n D))
+  ,~
+  n\gg 1
+\end{equation}
+
+Metric must be positive definite, which turns out to be a constraint
+for the divisor $D$.
 \begin{definition}
   A Cartier divisor $D\in \Pic(\CP_\Sigma)$ with corresponding
   function $\varphi_D\in SF(\Sigma,N)$ is ample if and only if
   $\varphi_D$ is strictly convex.
 \end{definition}
 
+A strictly convex support function does not always exist! One
+tautological case where it does exists is when the cones of the fan
+are the linear regions of a strictly convex support function.
+
+In particular, the resolution of a singularity $\Xhat \to X$
+corresponding to a subdivision of the fan may fail to be K\"ahler even
+if the singular variety $X$ is. This is so because subdividing a cone
+can change the strictly convex condition on the facets of the cone,
+for example the conifold.
+
+The existence of a K\"ahler resolution is guaranteed if the
+subdivision of each generating cone $\sigma$ is induced from a
+strictly convex support function that is equal to zero on the facets
+$\partial \sigma$.
+
 
 \subsection{The Canonical Bundle}
 
 
 \subsection{Kahler and Mori Cone}
 
+For a 3-dimensional variety, the K\"ahler cone is the subset of the
+$J\in H^{1,1}(X)$ satisfying
+\begin{equation}
+  \int_C J >0, 
+  \qquad 
+  \int_S J \wedge J >0, 
+  \quad \dots, 
+  \int_X J \wedge \cdots \wedge J >0 
+  .
+\end{equation}
 
-Complicated example~\cite{Berglund:1995gd}