The positivity theorem, the comparison theorem and Riemann’s existence theorem.
(Le théorème de positivité, le théorème de comparaison et le théorème d’existence de Riemann.)

*(French. English summary)*Zbl 1082.32006
Elements of the theory of geometric differential systems. Papers from the C.I.M.P.A summer school, Séville, Spain, September 1996. Paris: Société Mathématique de France (ISBN 2-85629-151-1/pbk). Séminaires et Congrès 8, 165-310 (2004).

This is a very interesting set of lectures, delivered at the CIMPA School at Seville in 1996, with a somehow provocative title, which explains the author’s approach (which does not require to involve Hironaka’s result on desingularization) to the Comparison Theorem. As explained in a detailed introduction, indispensable to any student, but very useful to anyone interested in this area, the motivations and the main ideas of the original approach developed in these lectures are neatly put in evidence and in the proper perspective. It is out of question to review this work in detail.

After the mentiond introduction, it begins with a review of known facts and definitions concerning the theory of \(\mathcal{D}_X\)-modules; let us mention the relation between the category of crystals over an infinitesimal site on a scheme \(X/ S\) over another one which allows a simple proof of the comparison theorem in the case of zero characteristic, and of regular schemes. In Chapter 3 the author introduces, for a holomorphic system and a hypersurface, a complex, whose characteristic cycle is positive and whose multiplicities generalize the known Fuchs numbers in differential equations with singular points in dimension one [see also Z. Mebkhout, Prog. Math. 88, 83–132(1990; Zbl 0731.14007); Publ. Math., Inst. Hautes Étud. Sci. 69, 47–89 (1989; Zbl 0709.14015)].

More precisely, let \(X\) be an analytic complex manifold, \(Z\) a closed analytic subset, \(i : Z \to X\), \(j : U = X \backslash Z \to X\) the canonical inclusions, \(\mathcal{M}\) a holomorphic \(\mathcal{D}_X\) module. Then the irregularity complex \(\text{Irr}(\mathcal{M})\) along \(Z\) is defined by \[ \text{Irr} (\mathcal{M}) = DR (R \Gamma_2(*Z)) \mathop{\to}\limits^{\sim} R\Gamma_2 (DR(\mathcal{M}) (*Z)) \] and \(\text{Irr}(\mathcal{M}) [+1]\) is in fact the cone of the natural morphism \[ a_Z(\mathcal{M}) : DR(R \mathcal{M} (*Z_+)) \to DR(Rj_* j^{-1}(\mathcal{M})) \] which, in the case of the trivial bundle \(\mathcal{O}_X\) and of a hypersurface \(Z\), is exactly the obstruction in the local Comparison Theorem of Grothendieck: Here \(\mathcal{M}(*Z) = \varinjlim_k \mathcal{H}_{\mathcal{O}_X} (J_Z^k, \cdot \mathcal{M})\), \(DR(\mathcal{M}) = R \mathcal{H}_{\mathcal{D}_X} (O_X, \mathcal{M})\) and \(R\) means the right derived functor in the derived category. \(\text{Irr}_Z(\mathcal{M})\) is constructible and has good properties. The complex \(\text{Irr}_Z^*(\mathcal{M})\) is defined to be \(i^{-1} S(R\mathcal{M}(*Z))\), and (if \(\mathcal{M}\) is holonomic) \(\text{Irr}_Z (\mathcal{M})\), \(\text{Irr}_Z^*(\mathcal{M})\), are two exact functors of triangulated categories which are in duality. When \(X\) is a Riemann surface, \(\mathcal{M}\) a holomorphic \(\mathcal{D}_X\)-module, then the set of singular points of \(\mathcal{M}\) consists of the isolated points of \(Z\) and \(\text{irr}_Z(\mathcal{M})\) is punctual, and its dimension in every singular point equals the Fuchs number of the point. An important result is the criterion of regularity (Th. 4.3-1) which says that if \(\mathcal{M}\) is regular on \(U\) then \(\text{Irr}_Z(\mathcal{M})\) vanishes iff \(\dim \sup \mathcal{M}\) is strictly smaller than \(\dim Z\). A more general result is given by Theorem 4.3-6.

In Chapter 5 one finds the global comparison theorem for the de Rham cohomology. These results were obtained by transcendental means, but in the algebraic case the theory makes sense in purely algebraic terms, with the condition to take into account the divisor at infinity. Let \(X\) be a regular affine complex algebraic variety, non singular, \(\mathcal{M}\) a holonomic module \(j : X \to \mathbb{P}^N\) an immersion, and \(j_*^d \mathcal{M}\) the direct image (in the sense of \(\mathcal{D}_X\)-modules) of \(\mathcal{M}\). If \(Z\) is a subvariety locally closed in \(\mathbb{P}^N\) the irregularity complex of \(\mathcal{M}\) along \(Z\), denoted by \(\text{Irr}(j_*^d \mathcal{M})\) is the complex \(\text{Irr}_{Z^{an}}(j_*^d \mathcal{M})^{a_n}\) (here “an” means the associated analytic object). It is important that the regularity of \(\mathcal{M}\) does not depend on the immersion.

After explaining why he uses the somewhat provocative expression “Existence theorems of Riemann type”, he states the Riemann existence theorem in the following manner: any finite covering of an algebraic, non singular complex curve is algebraic, and using extension results of coherent analytic sheaves, the author proves the Riemann type existence theorem first in the case of modules analytically constructible. To pass from the local situation to the global one, two steps are necesary: the first is the glueing of objects, the second the glueing of morphisms. To achieve this one uses the formalism of perverse sheaves, then the analogous result for modules is algebraically constructible. We have (Th. 10.6.-1) that the exact functor of triangulated categories \(\mathcal{M}\mapsto DR(\mathcal{M})\) from \(\mathcal{D}_{hr}^b (\mathcal{D}_X) \to \mathcal{D}_c^b (\mathbb{C}_X)\) is an equivalence of categories.

Finally, the author proves an existence theorem of Frobenius type for modules of infinite order. Many interesting remarks and examples are given, which make the text more accessible. All results are proved, except a few which were proved in other courses of the Seville School.

The author has made a remarkable effort to explain the underlying ideas and to point out the real difficulties.

For the entire collection see [Zbl 1050.32001].

After the mentiond introduction, it begins with a review of known facts and definitions concerning the theory of \(\mathcal{D}_X\)-modules; let us mention the relation between the category of crystals over an infinitesimal site on a scheme \(X/ S\) over another one which allows a simple proof of the comparison theorem in the case of zero characteristic, and of regular schemes. In Chapter 3 the author introduces, for a holomorphic system and a hypersurface, a complex, whose characteristic cycle is positive and whose multiplicities generalize the known Fuchs numbers in differential equations with singular points in dimension one [see also Z. Mebkhout, Prog. Math. 88, 83–132(1990; Zbl 0731.14007); Publ. Math., Inst. Hautes Étud. Sci. 69, 47–89 (1989; Zbl 0709.14015)].

More precisely, let \(X\) be an analytic complex manifold, \(Z\) a closed analytic subset, \(i : Z \to X\), \(j : U = X \backslash Z \to X\) the canonical inclusions, \(\mathcal{M}\) a holomorphic \(\mathcal{D}_X\) module. Then the irregularity complex \(\text{Irr}(\mathcal{M})\) along \(Z\) is defined by \[ \text{Irr} (\mathcal{M}) = DR (R \Gamma_2(*Z)) \mathop{\to}\limits^{\sim} R\Gamma_2 (DR(\mathcal{M}) (*Z)) \] and \(\text{Irr}(\mathcal{M}) [+1]\) is in fact the cone of the natural morphism \[ a_Z(\mathcal{M}) : DR(R \mathcal{M} (*Z_+)) \to DR(Rj_* j^{-1}(\mathcal{M})) \] which, in the case of the trivial bundle \(\mathcal{O}_X\) and of a hypersurface \(Z\), is exactly the obstruction in the local Comparison Theorem of Grothendieck: Here \(\mathcal{M}(*Z) = \varinjlim_k \mathcal{H}_{\mathcal{O}_X} (J_Z^k, \cdot \mathcal{M})\), \(DR(\mathcal{M}) = R \mathcal{H}_{\mathcal{D}_X} (O_X, \mathcal{M})\) and \(R\) means the right derived functor in the derived category. \(\text{Irr}_Z(\mathcal{M})\) is constructible and has good properties. The complex \(\text{Irr}_Z^*(\mathcal{M})\) is defined to be \(i^{-1} S(R\mathcal{M}(*Z))\), and (if \(\mathcal{M}\) is holonomic) \(\text{Irr}_Z (\mathcal{M})\), \(\text{Irr}_Z^*(\mathcal{M})\), are two exact functors of triangulated categories which are in duality. When \(X\) is a Riemann surface, \(\mathcal{M}\) a holomorphic \(\mathcal{D}_X\)-module, then the set of singular points of \(\mathcal{M}\) consists of the isolated points of \(Z\) and \(\text{irr}_Z(\mathcal{M})\) is punctual, and its dimension in every singular point equals the Fuchs number of the point. An important result is the criterion of regularity (Th. 4.3-1) which says that if \(\mathcal{M}\) is regular on \(U\) then \(\text{Irr}_Z(\mathcal{M})\) vanishes iff \(\dim \sup \mathcal{M}\) is strictly smaller than \(\dim Z\). A more general result is given by Theorem 4.3-6.

In Chapter 5 one finds the global comparison theorem for the de Rham cohomology. These results were obtained by transcendental means, but in the algebraic case the theory makes sense in purely algebraic terms, with the condition to take into account the divisor at infinity. Let \(X\) be a regular affine complex algebraic variety, non singular, \(\mathcal{M}\) a holonomic module \(j : X \to \mathbb{P}^N\) an immersion, and \(j_*^d \mathcal{M}\) the direct image (in the sense of \(\mathcal{D}_X\)-modules) of \(\mathcal{M}\). If \(Z\) is a subvariety locally closed in \(\mathbb{P}^N\) the irregularity complex of \(\mathcal{M}\) along \(Z\), denoted by \(\text{Irr}(j_*^d \mathcal{M})\) is the complex \(\text{Irr}_{Z^{an}}(j_*^d \mathcal{M})^{a_n}\) (here “an” means the associated analytic object). It is important that the regularity of \(\mathcal{M}\) does not depend on the immersion.

After explaining why he uses the somewhat provocative expression “Existence theorems of Riemann type”, he states the Riemann existence theorem in the following manner: any finite covering of an algebraic, non singular complex curve is algebraic, and using extension results of coherent analytic sheaves, the author proves the Riemann type existence theorem first in the case of modules analytically constructible. To pass from the local situation to the global one, two steps are necesary: the first is the glueing of objects, the second the glueing of morphisms. To achieve this one uses the formalism of perverse sheaves, then the analogous result for modules is algebraically constructible. We have (Th. 10.6.-1) that the exact functor of triangulated categories \(\mathcal{M}\mapsto DR(\mathcal{M})\) from \(\mathcal{D}_{hr}^b (\mathcal{D}_X) \to \mathcal{D}_c^b (\mathbb{C}_X)\) is an equivalence of categories.

Finally, the author proves an existence theorem of Frobenius type for modules of infinite order. Many interesting remarks and examples are given, which make the text more accessible. All results are proved, except a few which were proved in other courses of the Seville School.

The author has made a remarkable effort to explain the underlying ideas and to point out the real difficulties.

For the entire collection see [Zbl 1050.32001].

Reviewer: Gheorghe Gussi (Bucureşti)