%%%%% Section 2, log differentials and deformations
\subsection{Logarithmic differentials} %Log differentials

In \cite{EGA_IV} Grothendieck defines a derivation as the difference
of infinitesimal liftings of a  section. We can do the same thing with
logarithmic schemes. First, 
we need a concept of infinitesimal extension, which requires the
following definition.  

\begin{defn} % Strict morphism
A morphism $f: (X,\sM_{X})\rightarrow (Y,\sM_{Y})$ is called {\em strict} if $f^{\flat}: f^{*}\sM_{Y}\rightarrow \sM_{X}$ is an isomorphism. It is called a {\em strict closed immersion} \footnote{The term used in \cite{KKato} is {\em an exact closed immersion}.} if it is strict and the underlying map $X\rightarrow Y$ is a closed immersion in the usual sense.
\end{defn}

%%%% Explaination of Log differentials
Let us consider c commutative diagram of solid arrows:

\[
\xymatrix{
(T_{0},\sM_{T_{0}}) \ar[r]^{\phi} \ar[d]^{j}_{J}& (X,\sM_{X}) \ar[d]^{f}\\
(T_{1},\sM_{T_{1}}) \ar[r]_{\psi} \ar@<1ex>@{.>}[ur]^{g_{1}} \ar@<-1ex>@{.>}[ur]_{g_{2}} & (Y,\sM_{Y})
}
\]
where $j$ is an exact closed immersion defined by an deal $J$ with $J^{2}=0$. Note that $T_{0}$ and $T_{1}$ have the same underlying topological space. Then we have the following commutative diagram of sheaves of algebras given by the underlying maps:

\[
\xymatrix{
\sO_{T_{0}} & \phi^{-1}\sO_{X} \ar[l] \ar@<-1ex>@{.>}[dl]_{g_{1}^{\#}} \ar@<1ex>@{.>}[dl]^{g_{2}^{\#}}\\
\sO_{T_{1}} \ar[u] & \psi^{-1}\sO_{Y} \ar[u] \ar[l]
}
\]
Then $g_{1}^{\#}-g_{2}^{\#}$ is a derivation $\partial_{g_{1}-g_{2}}: \phi^{-1}\sO_{X}\rightarrow J$ in the usual sense. We also have a commutative diagram given by the log structures:

\[
\xymatrix{
\sM_{T_{0}} & \phi^{-1}\sM_{X} \ar[l] \ar@<-1ex>@{.>}[dl]_{g_{1}^{\flat}} \ar@<1ex>@{.>}[dl]^{g_{2}^{\flat}}\\
\sM_{T_{1}} \ar[u] & \psi^{-1}\sM_{Y} \ar[u] \ar[l]
}
\]

Note that we have an ``exact sequence" of mutiplicative
monoids $$\textbf{1}\rightarrow (1+J)\rightarrow
\sM_{T_{1}}\rightarrow \sM_{T_{0}}\rightarrow \textbf{1},$$ by which we mean that the group $1+J$ acts freely on $\cM_{T_1}$ with quotient $\cM_{T_0}$. Hence we
obtain a morphism $D_{g_{1}-g_{2}}:\phi^{-1}\sM_{X}\rightarrow J$ such
that for every $m \in \phi^{-1}\sM_{X}$ we have
$(g^{\flat}_{1}-g^{\flat}_{2})(m) = 1 + D_{g_{1}-g_{2}}m.$ Since $J$
is now viewed as an additive group, we obtain $D_{g_{1}-g_{2}}:
\phi^{-1}\sM_{X}^{gp}\rightarrow J$, and it is not hard to check that
$D_{g_{1}-g_{2}}(m\cdot n)=D_{g_{1}-g_{2}}(m)+D_{g_{1}-g_{2}}(n)$ for
any $m,n \in \phi^{-1}(\sM_{X})$. By the definition of log structures,
we also have  
\begin{enumerate}
 \item $\alpha(m)D_{g_{1}-g_{2}}m = \partial_{g_{1}-g_{2}}(\alpha(m))$, $\forall m\in \phi^{-1}\sM_{X}$;
 \item $D_{g_{1}-g_{2}}|_{\psi^{-1}\sM_{Y}}=0$.
\end{enumerate}

\begin{rem}
 \begin{enumerate}
       \item Since the log structure contains all the invertible
         elements in the strcture sheaf, the map $D_{g_{1}-g_{2}}$
         determines $\partial_{g_{1}-g_{2}}$. 
       \item The above properties show that $D_{g_{1}-g_{2}}$ behaves
         like "$d\log$". This is one of the reasons for the name
         ``logarithmic structure''. 
 \end{enumerate}
\end{rem}

Summarizing the above discussion gives the following definitions:
\begin{defn} \label{LogDer} % Log derivation
Let $X^{\dagger}=(X,\sM_{X})$, $Y^{\dagger}=(Y,\sM_{Y})$ be fine log schemes, and $f: X^{\dagger} \rightarrow Y^{\dagger}$ a morphism of log schemes. Let $I$ be an $\sO_{X}$-module. A {\em log derivation of $X^{\dagger}$ over $Y^{\dagger}$ to $I$} is a pair $(\partial, D)$ where $\partial \in Der_{Y}(X,I)$ and $D:\sM_{X}\rightarrow I$ such that the following conditions hold:
 \begin{enumerate}
   \item $D(ab)=D(a)+D(b)$ for $a,b\in\sM_{X}$;
   \item $\alpha(a)D(a)=\partial(\alpha(a))$, for $a\in\sM_{X}$.
   \item $D(a)=0$, for $a\in f^{-1}\sM_{Y}$.
 \end{enumerate}
The sheaf $Der_{Y^{\dagger}}(X^{\dagger},I)$ of log derivations of $X^{\dagger}$ over $Y^{\dagger}$ to $I$ is the sheaf of germs of pairs $(\partial, D)$. The sheaf $Der_{Y^{\dagger}}(X^{\dagger},\sO_{X})$ is usually denoted by $T_{X^{\dagger}/Y^{\dagger}}$, and is called {\em the logarithmic tangent sheaf of $X^{\dagger}$ over $Y^{\dagger}$}. 
\end{defn} 

As an analogue of derivations of usual schemes, we have the following result:

\begin{prop}\cite[IV.1.1.6]{Ogus} % universal property of Log differentials
There exists an $\sO_{X}$-module $\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$ with a universal derivations $(\partial,D)\in Der_{Y^{\dagger}}(X^{\dagger},\Omega_{X^{\dagger}/Y^{\dagger}}^{1})$, such that for any $\sO_{X}$-module $I$, the canonical map
\[\sH om_{\sO_{X}}(\Omega_{X^{\dagger}/Y^{\dagger}}^{1},I)\rightarrow Der_{Y^{\dagger}}(X^{\dagger},I), \ \ \ u\mapsto (u\circ \partial,u\circ D)\]
is an isomorphism of $\sO_{X}$-modules. In fact, we have the following construction: \[\Omega_{X^{\dagger}/Y^{\dagger}}^{1}=\Omega_{X/Y}\oplus(\sO_{X}\otimes_{\Z}\sM^{gp}_{X})/\mathcal{K}\]
where $\mathcal{K}$ is a $\sO_{X}$-module generated by local sections of the following forms:
 \begin{enumerate}
   \item $(d\alpha(a),0)-(0,\alpha(a)\otimes a)$ with $a\in\sM_{X}$;
   \item $(0,1\otimes a)$ with $a\in Im(f^{-1}(\sM_{Y})\rightarrow\sM_{X})$.
 \end{enumerate}
The universal derivation $(\partial,D)$ is given by $\partial : \sO_{X}\rightarrow\Omega_{X/Y}\rightarrow\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$ and $D: \sM_{X}\rightarrow \sO_{X}\otimes_{\Z}\sM_{X}^{gp}\rightarrow\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$.
\end{prop}

\begin{defn} \label{LodDiff} % Log differential
Given a morphism $f:X^{\dagger}\to Y^{\dagger}$ of log schemes, the $\sO_{X}$-module $\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$ is called {\em the sheaf of logarithmic differentials}. Sometimes we use the short notation $\Omega_{f}^{1}$ for $\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$.

We denote the dual sheaf of $\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$ by $T_{X^{\dagger}/Y^{\dagger}}$.
\end{defn}

\begin{rem} % Log differentials of fine log schemes form coherent sheaves 
If we consider only fine log structures, and assume that $Y$ is locally noetherian and $X$ locally of finite type over $Y$, then both $Der_{Y^{\dagger}}(X^{\dagger},I)$ and $\Omega_{X^{\dagger}/Y^{\dagger}}^{1}$ in the definitions above are coherent sheaves. The proof of this can be found in \cite[IV.1.1]{Ogus}
\end{rem}

\begin{eg}\label{NCsmooth}
Let $R=k[x_{1},\cdots,x_{n}]/(x_{1}\cdots x_{r})$, where $k$ is a field. Denote $X = \Spec R$, and $\sM_{X}$ to be the log structure on $X$ given by $\N^{r}\rightarrow R$, $e_{i}\mapsto x_{i}$, where $e_{i}$ is the standard generator of the monoid $\N^{r}$. Let $(Y,\sM_{Y})=$Spec$(\N\rightarrow k)$ be the logarithmic point described in \ref{Logpt}. Now we can define a morphism $f:(X,\sM_{X})\rightarrow (Y,\sM_{Y})$ by the following diagram:
\[
\xymatrix{
\N^{r} \ar[r] & R \\
\N \ar[r] \ar[u]^{\Delta} & k \ar[u]
}
\]
where $\Delta : e \mapsto e_{1}+\cdots + e_{r}$, and $e$ is the standard generator of $\N$. Then it is easy to see that $Der_{(Y,\sM_{Y})}((X,\sM_{X}),\sO_{X})$ is a free $\sO_{X}$-module generated by $x_{1}\frac{\partial}{\partial x_{1}},\cdots,x_{r}\frac{\partial}{\partial x_{r}},\frac{\partial}{\partial x_{r+1}},\cdots,\frac{\partial}{\partial x_{n}}$, with a relation $x_{1}\frac{\partial}{\partial x_{1}}+\cdots+x_{r}\frac{\partial}{\partial x_{r}}=0$. The sheaf $\Omega_{f}^{1}$ is a free $\sO_{X}$-module generated by the logarithmic differentials: $\frac{dx_{1}}{x_{1}},\cdots,\frac{dx_{r}}{x_{r}},x_{r+1},\cdots,x_{n}$, with a relation $\frac{dx_{1}}{x_{1}}+\cdots+\frac{dx_{r}}{x_{r}}=0$.
\end{eg}

\begin{eg}
Let $h: Q \rightarrow P$ be a morphism of fine monoids. Denote $(X,\sM_{X})=\ $Spec$(P\rightarrow\Z[P])$ and $(Y,\sM_{Y})=$ Spec$(Q\rightarrow\Z[Q])$. Then we have a morphism $f: (X,\sM_{X})\rightarrow (Y,\sM_{Y})$ induced by $h$. A direct calculation shows that $\Omega_{f}^{1}=\sO_{X}\otimes \textbf{Cok} (h^{gp})$.
\end{eg}

%%%%%%%%%%Log smooth
\subsection{Logarithmic Smoothness}

Let us go back to the following diagram:

\begin{equation}\label{diag:smooth}
\xymatrix{
(T_{0},\sM_{T_{0}}) \ar[r]^{\phi} \ar[d]^{j}_{J}& (X,\sM_{X}) \ar[d]^{f}\\
(T_{1},\sM_{T_{1}}) \ar[r]_{\psi}  & (Y,\sM_{Y})
}
\end{equation}
where $j$ is an exact closed immersion defined by $J$ with $J^{2}=0$. As in the usual case, we can define log smoothness by the existence of lifting sections.

\begin{defn}%%%%Definition for log smooth
A morphism $f:(X,\sM_{X})\rightarrow (Y,\sM_{Y})$ of fine log schemes is called smooth (resp. \'etale) if the underlying morphism $X\rightarrow Y$ is locally of finite presentation and for any commutative diagram as (\ref{diag:smooth}), there exists \'etale locally on $T_{1}$ a (resp. unique) morphism $g:(T_{1},\sM_{T_{1}})\rightarrow (X,\sM_{T_{1}})$ such that $\phi=g\circ j$ and $\psi=f\circ g$.
\end{defn}

We have the following useful criterion for smoothness from \cite[Theorem 3.5]{KKato}.

\begin{thm}\label{KatoStrThm} %%%%%%%%%Kato's Structure theorem
(K.Kato) Let $f:(X,\sM_{X})\rightarrow(Y,\sM_{Y})$ be a morphism of fine log schemes. Assume we have a chart $Q\rightarrow \sM_{Y}$, where $Q$ is a finitely generated integral monoid. Then the following are equivalent:
 \begin{enumerate}
  \item $f$ is log smooth (resp. \'etale);
  \item \'etale locally on $X$, there exists a chart $(P_{X}\rightarrow \sM_{X},Q_{Y}\rightarrow \sM_{Y},Q\rightarrow P)$ extending the chart $Q_{Y}\rightarrow\sM_{Y}$, satisfying the following properties.
    \begin{enumerate}
      \item The kernel and the torsion part of the cokernel (resp. the kernel and the cokernel) of $Q^{gp}\rightarrow P^{gp}$ are finite groups of order invertible on $X$.
      \item The induced morphism from $X\rightarrow Y\times_{Spec \ \Z[Q]} Spec \ \Z[P]$ is \'etale in the classical sense.
    \end{enumerate}
 \end{enumerate}
\end{thm}

\begin{rem}
 \begin{enumerate}
   \item We can require $Q^{gp}\rightarrow P^{gp}$ in (a) to be injective, and replace the \'etaleness of $X\rightarrow Y\times_{Spec\ \Z[Q]} Spec\ \Z[P]$ in (b) by the smoothness without changing the conclusion of the theorem \ref{KatoStrThm}. 
   \item The arrow in (b) shows that a log smooth morphism is ``locally toric'' relative to the base. If we consider the case $Y=$ Spec $\C$ with the trivial log structure, and $X=$ Spec$(P\rightarrow \C[P])$ where $P$ is a fine, saturated and torsion free monoid. Then $X$  is a toric variety with the action of Spec $\C[P^{gp}]$. According to the theorem, $X$ is log smooth relative to $Y$, though the underlying space might be singular. These singularities are called toric singularities in the sense of \cite{ToricSing}.
 \end{enumerate}
\end{rem}

\begin{eg} %% normal crossing are log smooth
Using the theorem, we can check directly that the morphism $f$ in example \ref{NCsmooth} is log smooth, but the underlying map has normal crossing singularities. We will see later that one of the major advantages of log structure is to deal with the normal crossing singularites.
\end{eg}

Let $(X,\sM_{X})\stackrel{f}{\rightarrow}(Y,\sM_{Y})\stackrel{g}{\rightarrow}(Z,\sM_{Z})$ be morphisms of fine log schemes. Consider the sheaves of log differentials $\Omega_{g}^{1}$ and $\Omega_{g\circ f}^{1}$, with their universal derivations $(\partial_{g},D_{g})$ and $(\partial_{g\circ f},D_{g\circ f})$ respectively. We have a canonical map $f^{*}\Omega_{g}^{1}\rightarrow \Omega_{g\circ f}^{1}$ induced by 
\[f^{*}(\partial_{g}u)\mapsto \partial_{g\circ f}f^{*}(u) \mbox{\ \ and\ \ }f^{*}(D_{g}v) \mapsto D_{g\circ f}f^{\flat}(v),\]
where $u\in \sO_{Y}$ and $v\in \sM_{Y}$. Denote by $(\partial_{f},D_{f})$ the universal derivation associated to $\Omega_{f}^{1}$. Similarly, we have a canonical map $\Omega_{g\circ f}^{1}\to \Omega_{f}^{1}$ induced by 
\[\partial_{g\circ f}u'\mapsto \partial_{f}u' \mbox{\ \ and\ \ }D_{g\circ f}v' \mapsto D_{f}v',\]
where $u'\in \sO_{X}$ and $v'\in \sM_{X}$. For log smooth morphisms, the following proposition shows that log differentials behave like the usual differentials of smooth morphisms.

\begin{prop} % properties of log differentials
 \begin{enumerate}
  \item The sequence $f^{*}\Omega_{g}^{1} \rightarrow \Omega_{g\circ f}^{1}\rightarrow \Omega_{f}^{1}\rightarrow 0$ is exact.
  \item If $f$ is log smooth, then $\Omega_{f}^{1}$ is a locally free $\sO_{X}$-module, and we have the following exact sequence: $0\rightarrow f^{*}\Omega_{g}^{1} \rightarrow \Omega_{g\circ f}^{1}\rightarrow \Omega_{f}^{1}\rightarrow 0$.
  \item If $g\circ f$ is log smooth and the sequence in (2) is exact and splits locally, then $f$ is log smooth.
 \end{enumerate}
\end{prop}

A proof can be found in \cite[Chapter IV]{Ogus}.


\subsection{Logarithmic smooth deformation}%%%%%%%%Log smooth deformation

After we have the log smoothness, a natural thing to do is to develop the log smooth deformation. In many cases, we would require this to be a flat deformation for the underlying space. Unfortunately log smoothness does not imply flatness, so we need the following definition.

\begin{defn} 
A map of fine monoids $h:Q\rightarrow P$ is called {\em integral} if the induced map on monoid algebra $\Z[Q]\rightarrow\Z[P]$ is flat.
\end{defn}

\begin{defn} %%%% integral morphism
A morphism $f:(X,\sM_{X})\rightarrow(Y,\sM_{Y})$ of integral log schemes is called \emph{integral} if for every geometric point $\bar{x} \in X$, the map of characteristic monoids $h:f^{-1}(\overline{\sM}_{Y})_{\bar{x}}\rightarrow\overline{\sM}_{X,\bar{x}}$ is integral.
\end{defn}

\begin{rem}
 \begin{enumerate}
  \item If $f$ is integral, then \'etale locally we have a chart $(P_{X}\rightarrow\sM_{X},Q_{Y}\rightarrow\sM_{Y},Q\stackrel{h}{\rightarrow}P)$ such that $h$ is integral.
  \item If $h:Q\rightarrow P$ is integral of integral monoids, then for any integral monoid $Q^{'}$, the push-out of $P\leftarrow Q\rightarrow Q^{'}$ in the category of monoids is integral. Thus integral morphisms are stable under base change by integral log schemes.
  \item Given a morphism $h: Q\rightarrow P$ of integral monoids, there is an explicit criterion, which looks complicated, but sometimes is useful for checking integrality of $h$ directly: if $a_{1},a_{2}\in Q$, $b_{1},b_{2}\in P$ and $h(a_{1})b_{1}=h(a_{2})b_{2}$, then there exist $a_{3},a_{4}\in Q$ and $b\in P$ such that $b_{1}=h(a_{3})b,b_{2}=h(a_{4})b$ and $a_{1}a_{3}=a_{2}a_{4}$. This is essentially coming from the equational criterion for flatteness.  
 \end{enumerate}
\end{rem}

Now we have the following fact from \cite[4.5]{KKato}.

\begin{prop} %%%integrality and flatness
If $f$ is a log smooth and integral morphism of fine log schemes, then $\underline{f}$ the underlying map is flat in the usual sense.
\end{prop}

Now let us consider the following deformation problem. We are given a log smooth integral morphism $f_{0}:X^{\dagger}_{0}\rightarrow B^{\dagger}_{0}$ of fine log schemes, and a strict closed immersion $j:B^{\dagger}_{0}\rightarrow B^{\dagger}$ defined by an ideal $J$ with $J^{2}=0$. We want to find a log smooth lifting $f:X^{\dagger}\rightarrow B^{\dagger}$ fitting into the following cartesian diagram:
\[
\xymatrix{
X^{\dagger}_{0} \ar@{^{(}->}[r] \ar[d] & X^{\dagger} \ar[d]\\
B^{\dagger}_{0} \ar@{^{(}->}[r] & B^{\dagger}.
}
\]

\begin{rem}
Denote by $\sM_{X}$ and $\sM_{X_{0}}$ the log structures of $X^{\dagger}$ and $X^{\dagger}_{0}$ respectively. Since $f_{0}$ is integral, and $\sM_{X}/(1+J) \cong \sM_{X_{0}}$, it is not hard to show that the lifting $f$ is automatically integral and hence flat.
\end{rem}

We have the following theorem for log smooth deformations.

\begin{thm} %%%% Log smooth deformation
With the notation as above, we have:
 \begin{enumerate}
  \item There is a canonical obstruction $\eta\in H^{2}(X_{0},T_{X^{\dagger}_{0}/B^{\dagger}_{0}}\otimes J)$ such that $\eta=0$ if and only if there exists a log smooth lifting.
  \item If $\eta=0$, then the set of log smooth deformations form a torsor under $H^{1}(X_{0},T_{X^{\dagger}_{0}/B^{\dagger}_{0}}\otimes J)$.
  \item The automorphism of any deformation is isomorphic to $H^{0}(X_{0},T_{X^{\dagger}_{0}/B^{\dagger}_{0}}\otimes J)$.
 \end{enumerate}
\end{thm}

The theorem can be proved in a manner similar to the case of usual deformation theory as in \cite[Expos\'e 3]{SGA_I}. Another proof using the logarithmic cotangent complex can be found in \cite[Thm 5.6]{LogCot}, which we will discuss later.