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Grassmann duality for \(\mathcal D\)-modules. (English) Zbl 0947.32004

If \(V\) is a complex \(n\)-dimensional vector space and \(G\) is the Grassmann manifold of \(p\)-dimensional linear subspaces of \(V\), then let \(G^*\) be the dual manifold of \((n-p)\)-subspaces. The intent of this work is to extend properties of projective duality (the natural correspondence between complex projective space and its dual) to the more general setting of Grassmann duality. (For examples of the kinds of properties extended see A. D’Agnolo and P. Schapira, Duke Math. J. 84, 453-496 (1996; Zbl 0879.32011).)
Let \(\Omega= \{(x,y)\in G\times G^*: x\cap y=\{0\})\) and let \(S\) be the complex hypersurface complementary to \(\Omega\). In the case of projective duality \((p=1)\) \(S\) is smooth, but in general this is no longer the case. However, as the author remarks, \(S\) has a Whitney stratification given by \(S_j= \{(x,y)\in G\times G^*: \dim(x \cap y)-j\}\). The lack of smoothness of \(S\) is the main feature that has to be dealt with in trying to generalize the known results.
There are a good number of examples and remarks illustrating the results or showing how they relate to simpler known cases and ample references.

MSC:

32C38 Sheaves of differential operators and their modules, \(D\)-modules
32C37 Duality theorems for analytic spaces
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials

Citations:

Zbl 0879.32011
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References:

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