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mastersub-rev - Logarithmic Geometry and Moduli Dan...

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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 d-semistability 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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2 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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