Logarithmic Geometry and Moduli
Dan Abramovich, Qile Chen, Danny Gillam, Yuhao Huang,
Martin Olsson, Matthew Satriano, and Shenghao Sun
Abstract.
We discuss the role played by logarithmic structures in the theory
of moduli.
Contents
1
Introduction
1
2
Definitions and basic properties
4
3
Differentials, smoothness, and log smooth deformations
8
4
Log smooth curves and their moduli
14
5
Friedman’s concept of dsemistability and log structures
20
6
Stacks of logarithmic structures
24
7
Log deformation theory in general
30
8
Rounding
34
9
Log de Rham and Hodge structures
39
10
The main component of moduli spaces
47
11
Twisted curves and log twisted curves
51
12
Log stable maps
55
1. Introduction
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, where logarithmic differentials
can reveal such hidden smoothness.
Since singular varieties naturally occur “at
the boundary” of many moduli problems, logarithmic geometry was soon applied
in the theory of moduli.
Foundations for this theory were first given by Kazuya Kato in [
27
], following
ideas of Fontaine and Illusie.
The main body of work on logarithmic geometry
2000
Mathematics Subject Classification.
Primary 14A20; Secondary 14Dxx.
Key words and phrases.
moduli, logarithmic structures.
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Logarithmic Geometry and Moduli
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”. The treatments of 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.
Normal crossings and logarithmic smoothness
So what is the original insight? Let
X
be a nonsingular irreducible complex
variety,
S
a smooth curve with a point
s
and
f
:
X
→
S
a dominant morphism
smooth away from
s
, in such a way that the fiber
f

1
s
=
X
s
=
Y
1
∪
. . .
∪
Y
m
is
a reduced simple normal crossings divisor.
Then of course Ω
X/S
= Ω
X
/f
*
Ω
S
fails to be locally free at the singular points of
f
.
But consider instead the
sheaves Ω
X
(log(
X
s
)) of differential forms with at most logarithmic poles along
the
Y
i
, and similarly Ω
S
(log(
s
)). Then there is an injective sheaf homomorphism
f
*
Ω
S
(log(
s
))
→
Ω
X
(log(
X
s
)), and
the quotient sheaf
Ω
X
(log(
X
s
))
/
Ω
S
(log(
s
))
is
locally free
.
So in terms of logarithmic forms,
the morphism
f
is as good as a smooth
morphism.
There is much more to be said: first, this Ω
X
(log(
X
s
))
/
Ω
S
(log(
s
)) 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.
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 Spring '08
 Grunbaum,F
 Math, Geometry, Algebraic geometry, Sheaf, Moduli, log structure

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