Lectures on Logarithmic Algebraic GeometryArthur OgusAugust 25, 20112Contents1Introduction............................71.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Organization . . . . . . . . . . . . . . . . . . . . . . . 141.5 Acknowledgements . . . . . . . . . . . . . . . . . . . . 14I The geometry of monoids 171Basicsonmonoids........................171.1 Limits in the category of monoids . . . . . . . . . . . . 171.2 Monoid actions . . . . . . . . . . . . . . . . . . . . . . 231.3 Integral, fine, and saturated monoids . . . . . . . . . . 271.4 Ideals, faces, and localization . . . . . . . . . . . . . . 311.5 Idealized monoids . . . . . . . . . . . . . . . . . . . . . 362 Finiteness, convexity, and duality . . . . . . . . . . . . . . . . 372.1 Finiteness . . . . . . . . . . . . . . . . . . . . . . . . . 372.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . 482.3 Monoids and cones . . . . . . . . . . . . . . . . . . . . 522.4 Va lu at i ve monoids and valua t i on s . . . . . . . . . . . . 652.5 Simplicial monoids . . . . . . . . . . . . . . . . . . . . 683 Affine to r i c varieties . . . . . . . . . . . . . . . . . . . . . . . 7 03.1 Monoid algebras and monoid schemes . . . . . . . . . . 703.2 Monoid sets and monoid modules . . . . . . . . . . . . 723.3 Faces, orbits, and trajectories . . . . . . . . . . . . . . 783.4 Local geometry of affine toric varieties . . . . . . . . . 823.5 Ideals and Newton polyhedra . . . . . . . . . . . . . . 844Actionsandhomomorphisms...................884.1 Local and logarithmic homomorphisms . . . . . . . . . 8834 CONTENTS4.2 Exact homomorphisms . . . . . . . . . . . . . . . . . . 924.3 Small and Kummer homomorphisms . . . . . . . . . . 984.4 Flat and regula r monoid actions . . . . . . . . . . . . . 1014.5 Integral homomorphisms . . . . . . . . . . . . . . . . . 1144.6 Exactness for idealized monoids . . . . . . . . . . . . . 1244.7 Saturated homomorphisms . . . . . . . . . . . . . . . . 1284.8 Saturation of monoid hom o m or p h i sm s . . . . . . . . . 142II Sheaves of monoids 1471Monoidalspaces..........................1471.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . 1471.2 Monoschemes . . . . . . . . . . . . . . . . . . . . . . . 1541.3 Some universal constructions . . . . . . . . . . . . . . . 1581.4 Quasi-coherent sheaves on monoschemes . . . . . . . . 1611.5 Proj for monoschemes . . . . . . . . . . . . . . . . . . 1651.6 Monoidal transformations . . . . . . . . . . . . . . . . 1701.7 Separated and proper morphisms . . . . . . . . . . . . 1741.8 Monoschemes and toric varieties . . . . . . . . . . . . . 1801.9 Fans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1801.10 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 1841.11 Moment maps . . . . . . . . . . . . . . . . . . . . . . . 1841.12 Resolution of singularities . . . . . . . . . . . . . . . . 1842Chartsandcoherence.......................1842.1 Charts . . . . . . . . . . . . . . . . . . . . . . . . . . 1842.2 Coherent sheaves of ideals . . . . . . . . . . . . . . . . 1862.3 Construction and comparison of charts . . . . . . . . . 1 8 82.4 Constructibility and coherence . . . . . . . . . . . . . . 1962.5 Relative coherence . . . . . . . . . . . . . . . . . . . . 2012.6 Charts for morphisms . . . . . . . . . . . . . . . . . . . 2013logstuffforlater.........................207III Logarithmic schemes 2091 Log str u ct u r es and log schemes . . . . . . . . . . . . . . . . . 2091.1 Log and prelog structures . . . . . . . . . . . . . . . . 2091.2 Direct and inverse images . . . . . . . . . . . . . . . . 2141.3 Coherence of log structures . . . . . . . . . . . . . . . 2161.4 Idealized log schemes . . . . . . . . . . . . . . . . . . . 2201.5 Examples . . . . . . . . . . . . . . . . . . . . . . …