Overview
Logarithmic geometry, pioneered by Fontaine–Illusie and developed systematically by Kato, provides a powerful framework for incorporating boundary data and singularities into algebraic geometry in a functorial way. It has become an indispensable tool in modern research, with applications ranging from moduli theory and mirror symmetry to degenerations of varieties and logarithmic Gromov–Witten theory.
This semester-long seminar will introduce the foundations of log geometry and explore its key applications. Each week, a participant will give a talk focusing on a central topic, with suggested references. Our goal is to build familiarity with the language of log structures, understand their use in moduli problems, and glimpse current frontiers of research.
Schedules
We meet every Friday from 1 pm to 2 pm.
(1.5 weeks) Overview and introduction to toric geometry
(2.5 weeks) Monoids and log structure
Charts and morphisms
Differentials and log smooothness
(2 weeks) Log curves and log smooth deformation theory
(1 week) Borne--Vistoli approach and applications to vector bundles
(2 weeks) Tropical geometry
(1 week) Classical Gromov--Witten theory
(3 weeks) Log GW invariants
References
Kato, Fumiharu, Log smooth deformation theory. Tohoku Math. J. (2) 48 (1996), no. 3, 317–354.
Luc Illusie, Deformation Theory, in Fundamental Algebraic Geometry (Grothendieck Festschrift, Seattle 2005), Math. Surveys Monogr. 123, AMS, 2005
Ogus, Arthur Lectures on logarithmic algebraic geometry. Cambridge Stud. Adv. Math., 178 Cambridge University Press, Cambridge, 2018.
Kato, Fumiharu Log smooth deformation and moduli of log smooth curves. Internat. J. Math. 11 (2000), no. 2, 215–232.
Olsson, Martin C. Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 747–791.
Gross, Mark Tropical geometry and mirror symmetry. CBMS Reg. Conf. Ser. Math., 114
Abramovich, Dan; Chen, Qile; Gillam, Danny; Huang, Yuhao; Olsson, Martin; Satriano, Matthew; Sun, Shenghao, Logarithmic geometry and moduli. Handbook of moduli. Vol. I, 1–61. Adv. Lect. Math. (ALM), 24 International Press, Somerville, MA, 2013