# zbMATH — the first resource for mathematics

Multidimensional local residues and holonomic $$D$$-modules. (English) Zbl 0944.32008
Let $$X$$ be a complex manifold with the structure sheaf $${\mathcal O}_X$$, $$Y$$ a subvariety of $$X$$ and $${\mathcal I}_Y$$ be the sheaf of defining ideal of $$Y$$ in $$X$$.
The description of the sections of the local cohomology group ${\mathcal H}^k_{[Y]}({\mathcal O}_X):=\lim_{l\to\infty}{\mathcal E}xt_{{\mathcal O}_X}({\mathcal O}_X/{\mathcal I}^l_Y; {\mathcal O}_X)$ is the main theme of this paper. The authors prove that each section of $${\mathcal H}^k_{[Y]} ({\mathcal O}_X)$$ is characterized as a solution to a holonomic system on $$X$$ and give some examples of the calculation of the section $$m$$ of $${\mathcal H}^k_{[Y]} ({\mathcal O}_X)$$ by computing a holonomic system to $$m$$ explicitly. The examples of computations are carried out by using the computer algebra system “kan” by N. Takayama.
Reviewer: M.Muro (Yanagido)

##### MSC:
 32C36 Local cohomology of analytic spaces 32C38 Sheaves of differential operators and their modules, $$D$$-modules 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 32-04 Software, source code, etc. for problems pertaining to several complex variables and analytic spaces 32A27 Residues for several complex variables
##### Keywords:
$$D$$-module; Gröbner basis; Weyl algebra