V - Betti and de Rham Cohomology

Summary

Chapter V is devoted to the topology of log schemes. It begins with adiscussion of log analytic spaces and their “Betti realizations,” as invented by Kato and Nakayama, which reveal the geometric effect of adding a log structure to a log analytic space.TheBetti realization of a log analytic space is home to the “logarithmic exponential map” and carries a sheaf of rings in which can be found solutions to differential equations with log poles and unipotent monodromy.Then the de Rham complex of a log scheme is discussed, along with some of its canonical filtrations which come from the combinatorics of the underlying log structure. After a discussion of the Cartier operator, the chapter concludes with some comparison and finiteness theorems for analytic, algebraic, and Betti cohomologies.