De Rham Cohomology of Differential Modules on Algebraic Varieties

“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)

• 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.