\subsection{From curves to maps and expansions}
\subsubsection{Curves}
In section \ref{Satriano} we discussed how prestable curves can be encoded as log smooth curves, and how in particular the stack of Deligne--Mumford stable curves can be interpreted as a logarithmic stack, representing log smooth stable curves over the category of fine  and saturated log schemes. Stability in this situation just means that $\Omega^1_{(C,\cM_C)}$ is an ample line bundle. This is the same as saying that $\omega_C(D)$ is ample, where $D$ is the divisor of markings.

\subsubsection{Maps}
Kontsevich \cite{Kontsevich} introduced the moduli  of {\em stable maps} of prestable curves into a projective target variety $X$. This is a proper Deligne--Mumford stack $\ocM_\Gamma(X)$ having projective coarse moduli space, where $\Gamma = (g,n,\beta)$ is the relevant numerical data: genus, number of markings and homology class of the image curve.
It parametrizes maps $f: C \to X$, where this time stability means that $\omega_C$ is $f$-ample. 

Kontsevich's moduli space has the property that it carries a {\em perfect relative obstruction theory} giving rise to a {\em virtual fundamental class} $[\ocM_\Gamma(X)]^{vir}$, see \cite{Li-Tian, Behrend-Fantechi}. This is a key ingredient in defining Gromov--Witten invariants, with their applications in enumerative geometry and theoretical physics. The simplest GW invariants are 
$$\langle\gamma_1\cdots \gamma_n\rangle := 
\int_{[\ocM_\Gamma(X)]^{vir}} e_1^*\gamma_1\cdots e_n^*\gamma_n,$$ 
where $\gamma_i\in H^*(X,\QQ)$ and $e_i: \ocM_\Gamma(X) \to X$ are the natural {\em evaluation maps} at the $n$ markings.

So far no logarithmic structures are necessary.

\subsubsection{Degenerations}

Among the methods of computing GW invariants, the {\em degeneration formula} is among the most powerful ones. It was introduced by A.-M. Li and Y. Ruan in symplectic geometry \cite{Li-Ruan}, see also Ionel-Parker \cite{Ionel-Parker1,Ionel-Parker2}. For our purposes, Jun Li's treatment in algebraic geometry \cite{Li,Li-degeneration} is most relevant.

We are interested in the invariants of the smooth projective variety  $X$. Since these are deformation invariant, it is natural to consider a degeneration of  $X$, with smooth total space, into a union $X_0 = Y_1\sqcup_D Y_2$ of two smooth varieties $Y_i$ meeting transversally along a smooth divisor $D$. One wants to show that invariants of $X$ coincide with suitably defined invariants of $X_0$, and these in turn can be computed in terms of appropriately defined invariants of $Y_i$ relative to $D$.

This is where logarithmic structures begin to show up, but there is still some way to go.

%\subsection{Expanded degenerations}
\subsubsection{Perfect obstruction and Li's approach}
The difficulty with the degeneration is precisely the fact that the variety $X_0$ is singular, and therefore the natural obstruction theory on the moduli space of stable maps is not perfect in general.  A similar situation occurs when considering the pair $(Y_i,D)$, but we will not get into this discussion.

The problem occurs when a component of the source curve $C$ maps entirely into $D$.
 
 Jun Li's approach uses 
  {\em expanded degenerations}. There are similar ideas in  \cite{Li-Ruan}, but the symplectic approach builds in deformations of Cauchy-Riemann equations and has, at least on the surface, a significantly different flavor.

The idea is, that just as in stable pointed curves, if a marking travels towards a node one sprouts a new component of the curve, Jun Li says that when a component of $C$ travels into  $D$ we can let $X_0$ sprout a new component. 

\subsubsection{Expansions}
Here is how the new component looks like. 
Denote by $N_{D\subset Y_{i}}$ the normal bundle of $D$ in $Y_{i}$. Since the total space is smooth, we have $N_{D\subset Y_{1}}\cong N_{D\subset Y_{2}}^{\vee}$. Let $\PP=\PP roj_{D}(1_{D}\oplus N_{D\subset Y_{1}})$. We have $\PP\cong \PP roj_{D}(N_{D\subset Y_{2}}\oplus 1_{D})$ so we can denote by $D^{+}$ and $D^{-}$ the smooth divisors in $\PP$ which correspond to the normal bundle $N_{D\subset Y_{1}}$ and $N_{D\subset Y_{2}}$ respectively. Note that the divisor $D^{+}$ and $D^{-}$ are canonically isomorphic to $D$. 


We now glue things up.
Let $\PP_{i}$ for $i\in \N$ be copies of $\PP$. We can glue $Y_{1}$ and $\PP_{1}$ along $D$ and $D^{-}$ respectively, $\PP_{i}$ and $\PP_{i+1}$ along $D^{+}$ and $D^{-}$ respectively, and $\PP_{n}$ and $Y_{2}$ along $D^{+}$ and $D$ respectively. We denote the resulting gluing by 
\begin{equation}
X_{0}[n]= Y_{1} \bigsqcup_{D_{1}} \PP_{1} \bigsqcup_{D_{2}} \cdots \bigsqcup_{D_{n}} \PP_{n} \bigsqcup_{D_{n+1}}Y_{2}, 
\end{equation}
where $D_{1},\cdots, D_{n+1}$ are the disjoint singular loci of $X_{0}[n]$.

Such a beast is known as an {\em expanded degeneration}, or by the more folksy name {\em an $n$-accordion}.

This led Jun Li to define {\em degeneration stable maps with target $X_0$}  as {\em nondegenerate maps $C \to X_0[n]$:} a map is nondegenerate if no component of $C$ maps into any of the $D_i$.

\subsubsection{Predeformability}


But here Jun Li meets another phenomenon, already present in the space of admissible covers of Harris--Mumford \cite{Harris-Mumford}: nondegenerate maps to $C \to X_0[n]$ have many redundant components which have nothing to do with maps to the generic fiber $X$. Near a singular point of $X_0[n]$ which looks like $\{xy=0\}$, a curve map locally deforms to a smoothing of $X_0$ if and only if the curve looks like $\{uv=0\}$, and the map given by $x=u^m, y=v^m$. 
Such nice maps are called {\em predeformable}. But predeformable maps are clearly not open in the space of maps - they are actually closed among nondegenerate maps. This means that the restricted obstruction theory on them is ``wrong" - definitely not perfect

Of course the virtual fundamental class of Gromov--Witten theory has no problem dealing with the total moduli space with its extra components, but these extra components do get in the way of decomposing invariants of $X_0$ in terms of $(Y_i,D)$. So we really do want to stick by predeformable maps. 

\subsection{Logarithmic methods: from Jun Li to Bumsig Kim}
\subsubsection{Predeformable deformations}

At this point Jun Li's reasoning arrives at a point where a new obstruction theory on predeformable maps is needed. Having read this chapter, the reader will immediately recognize that 
\begin{enumerate} 
\item nodal curves are log-smooth, 
\item $n$-accordions are d-semistable and admit a canonical log-smooth structure, and
\item predeformble maps can be viewed as log maps from the log-smooth curve to the d-semistable target.
\end{enumerate}

Jun Li also recognized this fact, as did Shin Mochizuki before him \cite{Mochizuki} when he in his turn revisited the space of admissible covers of Harris--Mumford. What he lacked at the time was a formalism for logarithmic deformation theory of singular spaces, such as the moduli space itself: a perfect obstruction theory is a two-term complex mapping to the cotangent complex of the moduli space, but the moduli space is, as usual highly singular, even taking its logarithmic structure into account.

So Jun Li resorted to an ad-hoc construction of his perfect obstruction theory. This is the most difficult part of his work.

In section \ref{LogDef} we saw how log deformation theory works in the necessary generality. This is where Bumsig Kim's paper \cite{Kim} comes in: he provides a correct formalism for nondegenerate logarithmic stable maps into expanded degenerations, and shows that it carries a perfect obstruction theory. The degeneration formula in this formalism has been worked out by one of us (Q. Chen), and should appear as part of a larger project indicated below.

One aspect that deserves mention is Kim's notion of {\em minimal log strutures} on maps.  Recall that the log stack $\mgnb$ can be constructed as a stack over the category of fine  saturated log schemes, whose objects over $S = (\underline S,M)$ are log smooth stable pointed curves over $S$. But in order to exhibit the underlying stack, one needs to use the {\em canonical} log structure on $\underline S$, which is initial among all possible ones carrying the log smooth curve. 

Kim describes his stack similarly - given a predeformable map $\underline C \to \underline X$ of underlying schemes over $\underline S$,  it amounts to describing what he calls {\em minimal} log structure $S$ on $\underline S$ carrying a log map $C \to X$. Kim does this by explicitly describing the combinatorial structure of such log structures.

\subsection{Unexpanded log maps: from Siebert into the future} 
A very different approach was proposed in a 2001 lecture by Bernd Siebert, but lay dormant for almost a decade.

The point is this. If one embraces logarithmic structures, and logarithmic maps from log smooth curves to some logarithmic scheme, then expansions are no longer necessary. Defined correctly, the space of log maps automatically has a perfect log-obstruction theory, which in view of sections \ref{LogStacks} and \ref{LogDef} can be viewed as an obstruction theory relative to the stack $LOG$. This automatically results in a virtual fundamental class.  

With this way of thinking, one can consider much more general logarithmic stable maps, gaining access to invariants of much more general degenerations of varieties. This has been a desired goal for a number of years.


So what is the correct definition? Consider a fine and saturated log scheme $X$. Following the work of F. Kato \cite{FKato} as discussed in section \ref{Satriano}, one  comes up with a definition of a category $\ocM_\Gamma(X)$ fibered over the category $LSch_{fs}$ of fine and saturated log schemes: an object over a fine saturated log scheme $S$ is a log smooth curve $C \to S$ and a log map $f:C \to X$. We further require it to be {\em stable}: the line bundle  of logarithmic differentials $\Omega^1_{C/S}$ is required to be $f$-ample. This is tantamount to requiring  the map of underlying schemes to be a stable map. 

So the main claim is: $\ocM_\Gamma(X)$ is represented by a logarithmic Deligne--Mumford stack with projective coarse moduli space. In fact this stack is proper and quasi-finite over the usual stack of stable maps $\ocM_\Gamma(\underline X)$ of the underlying scheme. As in the discussion of Kim's work, the underlying stack can be viewed as a moduli of log maps with {\em minimal} log structure.

This is the subject of current work - of Gross and Siebert on the one hand and of two of us (mainly Chen, and to a lesser extent Abramovich) on the other, so it would not be appropriate to get into details until definite results appear.   

Let us instead  put this in a larger context. Consider logarithmic schemes $Z \to B$ and $X$,  and assume we are given a morphism of underlying schemes $\underline f: \underline Z \to \underline X$. We can define a category $\Lift_{\underline f}$ fibered over $LSch_{fs}$ whose objects over a fine saturated log $B$-scheme $S\to B$ are lifts $f_S:Z_S \to X$ of the morphism of underlying schemes $\underline f_{\underline S}:\underline{Z}_{\underline S} \to \underline X$. 

One can ask the following general questions: 
\begin{question}
\begin{enumerate}
\item Under what conditions is $\Lift_{\underline f}$ a log stack  locally of finite type over $B$?
\item What natural numerical data cut out a substack of finite type?
\item Under what conditions is the result proper?
\end{enumerate}
\end{question}

We want to stress our belief that this question is natural, important  and quite tractable. For instance, the case where $B = \Spec \CC$ with trivial structure and $Z = X = \Spec (\NN \to \CC)$ the result is a countable union of components. It  is similar in nature to an inertia stack. 
The more general case where 
$B =X$, $Z = X\times \Spec (\NN \to \CC)$ and $\underline f$ is the diagonal, is the relevant analogue of the inertia stack of $X$. It is  important for Gromov--Witten theory - up to $\CC^*$ action the result is the natural target for evaluation maps associated to log smooth curves. Its components account for the contact orders of relative stable maps. Further examples of a similar nature govern gluing of nodes of log-smooth curves. 









 