
\subsection{Logarithmic structures in algebraic geometry}

It can be said that Logarithmic Geometry is concerned with a method of
finding and
using ``hidden smoothness'' in singular varieties. The original
insight comes from consideration of de Rham cohomology. Since
singular varieties naturally occur ``at the boundary'' of many moduli
problems, logarithmic geometry was soon applied in the theory of
moduli. 

The main body of work on logarithmic geometry
has been concerned with deep applications in the  cohomological study
of $p$-adic and arithmetic schemes. This gave the theory an aura of
``yet another extremely complicated theory''.
%, in the same general
%baggage as Fontaine's
%big rings. It is a bit unfortunate that the founders were not inclined
%to pursue simple geometric applications right away to scatter this
%aura. 
The treatments of the the theory are however quite accessible.
We hope to convince the reader here that the theory is simple
enough and useful enough to be considered by anybody interested in
moduli of singular varieties, indeed enough to be included in a
Handbook of Moduli.



\subsection{Normal crossings and logarithmic smoothness} So what is
the original insight? Let $X$ be a
nonsingular complex variety, $S$ a curve with a point $s$ and $f:X \to
S$ a dominant morphism smooth away from $s$, in such a way that
$f^{-1}s=X_s=Y_1\cup\ldots\cup Y_m$ is a reduced simple normal crossings
divisor. Then of course $\Omega_{X/Y}=\Omega_X/f^*\Omega_Y$ fails to be locally free at the
singular points of $f$. But consider instead the sheaves
$\Omega_X(\log(X_s))$ of differential forms with at most logarithmic poles
along the $Y_i$, and similarly $\Omega_S(\log(s))$. Then  there is an
injective sheaf homomorphism $f^*\Omega_S(\log(s))\to
\Omega_X(\log(X_s))$, and {\em the quotient sheaf $\Omega_{f}^{\log}$
  is locally free}. 

So in terms of logarithmic forms, {\em the morphism $f$ is as good as a
smooth morphism.}

There is much more to be said: first, this $\Omega_{f}^{\log}$ can be
extended to a logarithmic de Rham complex, and its hypercohomology,
while not recovering the cohomology of the singular fibers, does give
rise to the limiting Hodge structure. So it is evidently worth
considering. 

Second, the picture is quite a
bit more
general, and can be applied to all toric and toroidal maps between
toric varieties or toroidal embeddings (with a
little caveat about the characteristic of the residue fields). So
there is some flexibility in choosing $X\to S$.


\subsection{The search for a structure}

Since we are considering moduli, then as soon as we consider $X
\to S$ as above we must also consider the normal crossings fiber $X_s
\to \{s\}$.  But what structure should we put on this variety? The
notion of differentials with logarithmic poles along $X_s$ is not
in itself intrinsic to $X_s$. Also
the
normal crossings variety $X_s$ is not in itself toric or 
toroidal, so a new structure is needed to incorporate it into the
picture.

One is tempted to consider varieties which are assembled from nice
variety by some sort of gluing, as normal crossings varieties are. But
already normal crossings varieties do not give a satisfactory
answer in general, because their deformation spaces have ``bad''
components. Here is a classical example: consider a smooth projective  variety $Z$
such that $Pic^0(Z)$  
is nontrivial. Let $L$ be a line bundle on $Z$ and set
$Y=\PP(\cO\oplus L)$, with zero section $Z\subset Y$.  Let $X$ be the
blowing up of $Z\times 0\subset Y \times \AA^1$. We have a flat morphism $f:X
\to \AA^1$ with fiber  $X_0 = f^{-1}(0) \simeq Y\cup Y$, where the two copies
of $Y$ are glued with the zero section of one attached to the $\infty$
section of the other. 

So clearly $X_0$ is a normal crossings variety with a nice smoothing
to a copy of $Y$. But there are other deformations: the variety $Y\cup
Y$ also deforms to $Y \cup Y'$ where $Y' = \PP(\cO\oplus L')$ and $L'$
a deformation of the line bundle $L$. And it is not hard to see that $Y \cup Y'$ does
not have a smoothing. Ideally one really does not want to see this deformation
$Y\cup Y'$ in the picture - and ideally $X_0$ should have a natural
structure whose deformation space excludes $Y\cup Y'$ automatically.

Such a structure was proposed by Friedman in \cite{Friedman}, where
the notion of {\em d-semistable varieties} was introduced. This structure
is somewhat subtle, and while it solves the issue in this case, it is
not quite as flexible as one could wish. As we will see in Section
\ref{D-SS}, logarithmic structures subsume d-semistability and
do provide an appropriate flexibility.  


\subsection{Organization of this chapter} 
The purpose of this chapter is to briefly describe logarithmic structure and
to indicate where they can be useful in the study of moduli spaces.   
Section \ref{Qile1} gives the basic definitions of  logarithmic
structures, and section \ref{Qile2} discusses logarithmic
differentials and log smooth deformations, which are important in
considering moduli spaces.

 Section \ref{Satriano}  
gives the first example where logarithmic geometry fits well with
moduli spaces: the moduli space of stable curves is the moduli space
of log smooth curves. The issue of d-semistability does not arise
since a nodal curve is automatically d-semistable. So the theory for
curves is simple.   
Turning to higher dimensions, Section \ref{D-SS} shows how
d-semistability can be described within logarithmic structures. 

If one is to enlarge algebraic geometry to include logarithmic
structures, the task of generalizing the techniques of algebraic
geometry to logarithmic structure can certainly seem daunting. In
section \ref{LogStacks} we show how to encode logarithmic structure in
terms of certain algebraic stacks. This allows us to reduce various
constructions to the case of algebraic stacks. (One can argue that the
theory of stacks is not simple either, but at least in the theory of
moduli they have come to be accepted, with some exceptions
\cite{Kollar-companion}.)  

In section \ref{LogDef} we make use of logarithmic stacks to describe
the complexes which govern deformations and obstructions for
logarithmic structures even in the non-smooth case. This comes in
handy later. For instance, even when studying moduli of log-smooth
schemes, the moduli spaces tend to be singular, and their cotangent
complexes are a necessary ingredients in constructing virtual
fundamental classes. 

Section \ref{Rounding} describes a beautiful construction, similar to
polar coordinates, in which families of complex log smooth varieties
give rise canonically to families of topological
manifolds. Differential geometers have used polar coordinates on nodal
curves to ``make space" for monodromy to act by Dehn twists. Rounding
(using Ogus's terminology) is a magnificent way to generalize this. 

The immediate implications of logarithmic structures for De Rham
cohomology and Hodge structures are described in  Section
\ref{DeRham}. 

We conclude by describing three applications, where logarithmic
structures serve as the proverbial ``magic powder" (term suggested by
Kato and Ogus) to clarify or remove unwanted behavior from moduli
spaces. 

Section \ref{MainComp} describes a number of cases where the
main irreducible component of a moduli space can be separated from
other ``unwanted" components by sprinkling the objects with a bit of
logarithmic structures. 

In Section \ref{Roots} we introduce twisted curves, a central object of orbifold stable maps, and show how logarithmic structures  give a palatable way to construct the moduli stack of twisted curves. 

Section \ref{Stablemaps} gives background for the work of B. Kim,
in which Jun Li's moduli space of relative stable maps, with its
obstruction theory and virtual fundamental class, is beautifully
simplified using logarithmic structures. 

%\section{Log smooth curves and their moduli}\label{Satriano}
%\section{D-semistability and log structures}\label{D-SS}
%\section{Stacks of logarithmic structures}\label{LogStacks}
%\section{Log deformation theory in general}\label{LogDef}
%\section{Rounding}\label{Rounding}
%\section{Log De Rham and hodge structures}\label{DeRham}
%\section{The main component of moduli spaces}\label{MainComp}
%\section{Log twisting and root constructions}\label{Roots}
%\section{Log stable maps}\label{Stablemaps}



