nLab holonomic D-module

Idea

(Solutions of) holonomic systems of differential equations are formalized in the notion of a holonomic D-module. A D-module $M$ on a smooth complex analytic variety $X$ of dimension $n$ is holonomic if its characteristic variety is of dimension $n$. It follows that the characteristic variety of a holonomic D-module is conic and lagrangian.

Holonomicity of D-modules is important also in geometric representation theory.

Related concepts

• coherent D-module

• arithmetic D-module

References

Lecture notes include

• Joseph Bernstein, Lecture 2, section 11 of Algebraic theory of D-modules (pdf, ps, dvi)

See also

• Masaki Kashiwara, On the holonomic systems of linear differential equations. II, Invent. Math. 49 (1978), no. 2, 121–135, doi

• Bernard Malgrange, On irregular holonomic D-modules, Séminaires et Congrès 8, 2004, p. 391–410, pdf; Équations différentielles à coefficients polynomiaux, Progress in Math. 96, Birkhäuser 1991. vi+232 pp.

• P. Maisonobe, C. Sabbah, D-modules cohérents et holonomes, Hermann, Paris 1993.

• V. Ginsburg, Characteristic varieties and vanishing cycles, Inv. Math. 84, 327–402 (1986)