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)

“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)