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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)

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