Introduction to algebraic theory of linear systems of differential equations.

*(English)*Zbl 0841.14014
Maisonobe, Philippe (ed.) et al., \(D\)-modules cohérents et holonomes. Cours d’été du CIMPA ’Éléments de la théorie des systèmes différentiels’, août et septembre 1990, Nice, France. Paris: Hermann. Trav. Cours. 45, 1-80 (1993).

This set of notes is an introduction to the theory of linear differential equations of a complex variable with coefficients in the ring \(A\), where \(A = \) the polynomial ring \(\mathbb{C} [x]\), the formal power series ring \(\mathbb{C} [[x]]\) or the convergent power series ring \(\mathbb{C} \{x\}\). This classical subject is presented here from the algebraic point of view identifying a linear differential system to a finite type \(D\)-module, \(D = A \langle \partial_x \rangle\) being the quotient algebra of the free algebra generated by \(A\) and the element \(\partial_x = \partial/ \partial_x\). The exposition is clear, self-contained and with minimal prerequisites. There is also a good bibliography with topic-wise references for an interested reader.

The first chapter ‘Algebraic methods’ is concerned with the algebraic study of \(D\)-modules. It deals with elementary properties of \(D\)-modules, their characteristic varieties, holonomic \(D\)-modules and their localisations etc. The connection between localised holonomic \(D\)-modules and meromorphic connections is emphasized and their structure is studied in detail over \(A = \mathbb{C} [[x]]\), a normal form is also obtained. Keeping in mind generalization to higher dimensions (i.e. several complex variables), the notions of nearby and vanishing cycles are stressed upon.

The second chapter extends the results of the previous chapter to the case \(A = \mathbb{C} \{x\}\) and hence is an analytic study of \(D\)-modules and meromorphic connections. The main results of this section are the theorem of existence of Jordan normal form of the matrix of \(x \partial_x\) in a meromorphic connection and the local index theorem for holonomic \(D\)-modules.

The final chapter deals with the global study of linear differential equations on the Riemann sphere \(\mathbb{P}^1 (\mathbb{C})\). Many results on global properties of holonomic \(D\)-modules proved in this chapter are valid for compact Riemann surfaces also. The author has preferred to restrict himself to the case of Riemann sphere. For a holonomic \(D_{\mathbb{P}^1}\)-module \(M\), the de Rham complex \(\Omega^*_{\mathbb{P}^1} \omega M\) is studied. Let \(U\) denote an affine open subset of \(\mathbb{P}^1\). The comparison of algebraic de Rham cohomology of \(\Omega^*_p \otimes M\) with hypercohomology \(H^* (U,DR^{an} (M^{an}))\) and global index theorem are given (without proof).

For the entire collection see [Zbl 0824.00033].

The first chapter ‘Algebraic methods’ is concerned with the algebraic study of \(D\)-modules. It deals with elementary properties of \(D\)-modules, their characteristic varieties, holonomic \(D\)-modules and their localisations etc. The connection between localised holonomic \(D\)-modules and meromorphic connections is emphasized and their structure is studied in detail over \(A = \mathbb{C} [[x]]\), a normal form is also obtained. Keeping in mind generalization to higher dimensions (i.e. several complex variables), the notions of nearby and vanishing cycles are stressed upon.

The second chapter extends the results of the previous chapter to the case \(A = \mathbb{C} \{x\}\) and hence is an analytic study of \(D\)-modules and meromorphic connections. The main results of this section are the theorem of existence of Jordan normal form of the matrix of \(x \partial_x\) in a meromorphic connection and the local index theorem for holonomic \(D\)-modules.

The final chapter deals with the global study of linear differential equations on the Riemann sphere \(\mathbb{P}^1 (\mathbb{C})\). Many results on global properties of holonomic \(D\)-modules proved in this chapter are valid for compact Riemann surfaces also. The author has preferred to restrict himself to the case of Riemann sphere. For a holonomic \(D_{\mathbb{P}^1}\)-module \(M\), the de Rham complex \(\Omega^*_{\mathbb{P}^1} \omega M\) is studied. Let \(U\) denote an affine open subset of \(\mathbb{P}^1\). The comparison of algebraic de Rham cohomology of \(\Omega^*_p \otimes M\) with hypercohomology \(H^* (U,DR^{an} (M^{an}))\) and global index theorem are given (without proof).

For the entire collection see [Zbl 0824.00033].

Reviewer: U.N.Bhosle (Bombay)