De Rham Cohomology of Differential Modules on Algebraic Varieties

1 Regularity in several variables.- 1 Geometric models of divisorially valued function fields.- 2 Logarithmic differential operators.- 3 Connections regular along a divisor.- 4 Extensions with logarithmic poles.- 5 Regular connections: the global case.- 6 Exponents.- Appendix A: A letter of Ph. Robba (Nov. 2, 1984).- Appendix B: Models and log schemes.- 2 Irregularity in several variables.- 1 Spectral norms.- 2 The generalized Poincaré-Katz rank of irregularity.- 3 Some consequences of the Turrittin-Levelt-Hukuhara theorem.- 4 Newton polygons.- 5 Stratification of the singular locus by Newton polygons.- 6 Formal decomposition of an integrable connection at a singular divisor.- 7 Cyclic vectors, indicial polynomials and tubular neighborhoods.- 3 Direct images (the Gauss-Manin connection).- 1 Elementary fibrations.- 2 Review of connections and De Rham cohomology.- 3 Dévissage.- 4 Generic finiteness of direct images.- 5 Generic base change for direct images.- 6 Coherence of the cokernel of a regular connection.- 7 Regularity and exponents of the cokernel of a regular connection.- 8 Proof of the main theorems: finiteness, regularity, monodromy, base change (in the regular case).- Appendix C: Berthelot's comparison theorem on OXDX-linear duals.- Appendix D: Introduction to Dwork's algebraic dual theory.- 4 Complex and p-adic comparison theorems.- 1 Review of analytic connections and De Rham cohomology.- 2 Abstract comparison criteria.- 3 Comparison theorem for algebraic vs.complex-analytic cohomology.- 4 Comparison theorem for algebraic vs. rigid-analytic cohomology (regular coefficients).- 5 Rigid-analytic comparison theorem in relative dimension one.- 6 Comparison theorem for algebraic vs. rigid-analytic cohomology (irregular coefficients).- 7 The relative non-archimedean Turrittin theorem.- Appendix E: Riemann's "existence theorem" in higher dimension, an elementary approach.- References.

"If I had to summarize the differences in the exposition in one sentence, I would say that the authors manage to make the transition from a research monograph to an exposition that reads more like an advanced textbook, while retaining the rigor and the scientific interest; it is an example of what may be called a 'research textbook'." ( Adolfo Quirós, Mathematical Reviews, January, 2023)