\contentsline {section}{\numberline {1}Introduction}{2}{section.2}
\contentsline {subsection}{\numberline {1.1}Logarithmic structures in algebraic geometry}{2}{subsection.3}
\contentsline {subsection}{\numberline {1.2}Normal crossings and logarithmic smoothness}{3}{subsection.4}
\contentsline {subsection}{\numberline {1.3}The search for a structure}{3}{subsection.5}
\contentsline {subsection}{\numberline {1.4}Organization of this chapter}{4}{subsection.6}
\contentsline {section}{\numberline {2}Definitions and basic properties}{5}{section.7}
\contentsline {subsection}{\numberline {2.1}Logarithmic structures}{5}{subsection.8}
\contentsline {subsection}{\numberline {2.2}The log structure associated to a pre-log structure}{6}{subsection.15}
\contentsline {subsection}{\numberline {2.3}The inverse image}{7}{subsection.20}
\contentsline {subsection}{\numberline {2.4}Charts of log structures}{7}{subsection.24}
\contentsline {subsection}{\numberline {2.5}Fine log structures}{8}{subsection.30}
\contentsline {section}{\numberline {3}Differentials, smoothness, and log smooth deformations}{9}{section.33}
\contentsline {subsection}{\numberline {3.1}Logarithmic differentials}{9}{subsection.34}
\contentsline {subsection}{\numberline {3.2}Logarithmic Smoothness}{11}{subsection.53}
\contentsline {subsection}{\numberline {3.3}Logarithmic smooth deformation}{13}{subsection.69}
\contentsline {section}{\numberline {4}Log smooth curves and their moduli}{14}{section.82}
\contentsline {subsection}{\numberline {4.1}Relative characteristic sheaves}{14}{subsection.83}
\contentsline {subsection}{\numberline {4.2}Log curves}{15}{subsection.88}
\contentsline {subsection}{\numberline {4.3}Moduli}{17}{subsection.95}
\contentsline {subsection}{\numberline {4.4}Back to the big picture}{18}{subsection.97}
\contentsline {section}{\numberline {5}D-semistability and log structures}{19}{section.98}
\contentsline {subsection}{\numberline {5.1}Normal crossings and d-semistability}{19}{subsection.99}
\contentsline {subsection}{\numberline {5.2}Strategy of the Proof}{21}{subsection.114}
\contentsline {section}{\numberline {6}Stacks of logarithmic structures}{23}{section.125}
\contentsline {subsection}{\numberline {6.1}A motivating example}{23}{subsection.126}
\contentsline {subsection}{\numberline {6.2}The stack of log structures}{25}{subsection.134}
\contentsline {subsection}{\numberline {6.3}What is $LOG_S$ good for?}{25}{subsection.137}
\contentsline {section}{\numberline {7}Log deformation theory in general}{27}{section.141}
\contentsline {subsection}{\numberline {7.1}The Log Cotangent Complex}{28}{subsection.143}
\contentsline {subsection}{\numberline {7.2}Basic Properties}{28}{subsection.146}
\contentsline {subsection}{\numberline {7.3}Deformation Theory of Log Schemes in General}{30}{subsection.158}
\contentsline {section}{\numberline {8}Rounding}{31}{section.165}
\contentsline {subsection}{\numberline {8.1}What is rounding?}{31}{subsection.166}
\contentsline {subsection}{\numberline {8.2}The oriented real blowup}{32}{subsection.168}
\contentsline {subsection}{\numberline {8.3}The Kato--Nakayama space}{33}{subsection.169}
\contentsline {subsection}{\numberline {8.4}Relating Kato--Nakayama spaces to oriented blowups}{34}{subsection.171}
\contentsline {subsection}{\numberline {8.5}Topology, cohomology, and the Kato--Nakayama space}{34}{subsection.172}
\contentsline {subsection}{\numberline {8.6}Kato--Nakayama spaces of expanded pairs}{35}{subsection.175}
\contentsline {section}{\numberline {9}Log De Rham and hodge structures}{36}{section.176}
\contentsline {subsection}{\numberline {9.1}Moduli spaces of polarized Hodge structures.}{36}{subsection.177}
\contentsline {subsubsection}{\numberline {9.1.1}The moduli space $M_h=\Gamma \delimiter "026E30F D_h.$}{36}{subsubsection.178}
\contentsline {subsubsection}{\numberline {9.1.2}Variations of Hodge structure.}{37}{subsubsection.179}
\contentsline {subsection}{\numberline {9.2}Logarithmic Hodge structures.}{38}{subsection.181}
\contentsline {subsubsection}{\numberline {9.2.1}The ringed space $X^{\qopname \relax o{log}}.$}{38}{subsubsection.182}
\contentsline {subsubsection}{\numberline {9.2.2}Log variations of polarized Hodge structure.}{39}{subsubsection.183}
\contentsline {subsection}{\numberline {9.3}Kato-Usui spaces.}{40}{subsection.185}
\contentsline {subsubsection}{\numberline {9.3.1}Nilpotent orbits.}{40}{subsubsection.186}
\contentsline {subsubsection}{\numberline {9.3.2}The moduli space $M_{\Sigma }$.}{41}{subsubsection.188}
\contentsline {section}{\numberline {10}The main component of moduli spaces}{42}{section.189}
\contentsline {subsection}{\numberline {10.1}Moduli: compactness and main components}{42}{subsection.190}
\contentsline {subsection}{\numberline {10.2}Example: the toric Hilbert scheme}{42}{subsection.191}
\contentsline {section}{\numberline {11}Twisted curves and log twisted curves}{45}{section.195}
\contentsline {subsection}{\numberline {11.1}Twisted curves}{46}{subsection.196}
\contentsline {subsection}{\numberline {11.2}Log twisted curves}{47}{subsection.206}
\contentsline {subsection}{\numberline {11.3}From twisted curves to log twisted curves}{48}{subsection.211}
\contentsline {subsection}{\numberline {11.4}From log twisted curves to twisted curves}{49}{subsection.215}
\contentsline {section}{\numberline {12}Log stable maps}{50}{section.218}
\contentsline {subsection}{\numberline {12.1}From curves to maps and expansions}{50}{subsection.219}
\contentsline {subsubsection}{\numberline {12.1.1}Curves}{50}{subsubsection.220}
\contentsline {subsubsection}{\numberline {12.1.2}Maps}{50}{subsubsection.221}
\contentsline {subsubsection}{\numberline {12.1.3}Degenerations}{51}{subsubsection.222}
\contentsline {subsubsection}{\numberline {12.1.4}Perfect obstruction and Li's approach}{51}{subsubsection.223}
\contentsline {subsubsection}{\numberline {12.1.5}Expansions}{51}{subsubsection.224}
\contentsline {subsubsection}{\numberline {12.1.6}Predeformability}{52}{subsubsection.226}
\contentsline {subsection}{\numberline {12.2}Logarithmic methods: from Jun Li to Bumsig Kim}{52}{subsection.227}
\contentsline {subsubsection}{\numberline {12.2.1}Predeformable deformations}{52}{subsubsection.228}
\contentsline {subsection}{\numberline {12.3}Unexpanded log maps: from Siebert into the future}{53}{subsection.232}