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De Rham cohomology of differential modules on algebraic varieties. (English) Zbl 0995.14003

Progress in Mathematics (Boston, Mass.). 189. Basel: Birkhäuser. vii, 214 p. (2001).
In the general introduction the authors briefly discuss the classical approach to computing the cohomology of a manifold by means of differential forms in the algebraic, analytic and smooth settings originated with E. Cartan, G. de Rham, A. Grothendieck, P. Deligne and others. In fact, the main idea is to set up an isomorphism between hypercohomologies of appropriate De Rham complexes or, in other words, to prove the so-called comparison theorem. In the book under review the authors give a purely algebraic proof of some known comparison theorems based on the modern theory of differential modules. This enables them to avoid Hironaka’s resolution of singularities, monodromy arguments and some other standard constructions of differential-analytic character. As a result they obtain a comparison theorem in the non-archimedean case and they describe some useful properties of connections in the \(p\)-adic setting.
The book is divided into 4 chapters and 5 appendixes. The first chapter provides a self-contained exposition of the algebraic theory of regularity in several variables. Let \(X\) be an algebraic variety over an algebraically closed field of characteristic zero, let \(Z\) be a divisor with strict normal crossings in \(X,\) and let \(\nabla\) be an algebraic integrable connection on \(X\setminus Z.\) There are known at least \(4\) different conditions under which the connection \(\nabla\) is regular singular along the divisor \(Z.\) The classical analysis of regularity conditions is based essentially on transcendental arguments and Hironaka’s resolution of singularities [see P. Deligne, “Equations différentielles à points singuliers réguliers”, Lect. Notes Math. 163 (1970; Zbl 0244.14004)]. In contrast with earlier works the authors prove the equivalence of these conditions making use of the theory of \(D\)-modules and of simple geometric and algebraic considerations.
The second chapter deals with the notion of irregularity studied in a similar manner. First the authors remark that one of the characterization of regularity along a divisor can be formulated as follows: Logarithmic differential operators of increasing order act with poles of bounded order at the generic point of \(Z.\) Thus a natural notion of a generalized Poincaré-Katz rank of irregularity of \(\nabla\) at \(Z\) arises. Then the stratification of the singular divisor \(Z\) by Newton polygons is introduced. The authors study the variation of Newton polygons, prove a semicontinuity theorem and obtain a formal decomposition of an integrable connection at a singular divisor. The last section of this chapter contains a very useful and less known material concerning properties of the indicial polynomial and Turrittin exponents of a connection.
The next chapter develops a new approach to the study of direct images of connections with respect to a smooth morphism \(X \rightarrow S\) between smooth algebraic varieties in characteristic zero or, in other words, to the study of direct images of the relative de Rham complex over \(X/S\) with coefficients, endowed with the Gauss-Manin connection [N. M. Katz, Publ. Math., Inst. Hautes Étud. Sci. 39, 175–232 (1970; Zbl 0221.14007)]. The authors prove the generic and fundamental finiteness, regularity, monodromy and base change theorems for direct images of regular algebraic connections with using neither resolution of singularities, nor the holonomy theory. The main tool of their proofs is the dévissage method from Artin’s theory of elementary fibrations [cf. Y. André and F. Baldassarri in: Arithmetic geometry, Proc. Symp., Cortona 1994, Symp. Math. 37, 1–22 (1997; Zbl 0936.14014)]. As a result the problem reduces to the simplest case of an ordinary differential operator in one variable.
In the last chapter an effective proof of the classical Grothendieck-Deligne comparison theorem is presented. Since this proof does not rely on resolution of singularities, does not make use of moderate growth conditions and properties of the monodromy, the authors’ arguments remain valid also in the \(p\)-adic setting. This leads to a simple proof of the Kiehl-Baldassarri theorem [F. Baldassarri, Math. Ann. 280, No. 3, 417–439 (1988; Zbl 0651.14012)]. Among other things the authors explain how the comparison theorem extends to the case of an irregular connection in the non-archimedean setting.
Five appendixes contain some additional interesting material. In particular, a non-published original proof of Berthelot’s comparison theorem between two different notions of the dual of a differential module is reproduced and a natural interpretation of Dwork’s algebraic dual theory in terms of the relative algebraic de Rham cohomology with compact supports is given.
The book is written in a clear and concise style, almost all key topics are followed by examples, non-formal remarks and comments. The authors underline that they use neither the language of derived categories nor the general theory of holonomic modules. Thus, a major part of the book is accessible to non-specialists and graduate students while the main results should be of interest to specialists of \(D\)-modules and differential equations as well as to algebraic and arithmetic-algebraic geometers.

MSC:

14F40 de Rham cohomology and algebraic geometry
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
13N05 Modules of differentials
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
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