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Semisimple holonomic \({\mathcal D}\)-modules. (English) Zbl 0935.32009
Kashiwara, Masaki (ed.) et al., Topological field theory, primitive forms and related topics. Proceedings of the 38th Taniguchi symposium, Kyoto, Japan, December 9-13, 1996 and the RIMS symposium with the same title, Kyoto, Japan, December 16-19, 1996. Boston, MA: Birkhäuser. Prog. Math. 160, 267-271 (1998).
This work gives conjectures and their evidence on semisimple holonomic \({\mathcal D}\)-modules. The precise statements of the conjecturc are the following three conditions. (C1) Let \(f:X\to Y\) be a projective morphism and let \({\mathcal M}\) be a semisimple holonomic \({\mathcal D}_X\)-module. Then \({\mathbf D}f_* ({\mathcal M})\) is isomorphic to \(\oplus_kH^k({\mathbf D}f_*({\mathcal M}))[-k]\) and the summand \(H^k({\mathbf D}f_*({\mathcal M}))[-k]\) is a semisimple holonomic \({\mathcal D}_Y \)-module. Here \({\mathbf D}f_*\) is a push-forward functor \(D^b_h({\mathcal D}_X)\to D^b_h ({\mathcal D}_Y)\).
(C2) Let \(f\) be a regular function on \(X\) and let \({\mathcal M}\) be a semisimple holonomic \({\mathcal D}_X\)-module. Let \(W\) be the weight filtration of the nilpotent part of the monodromy of \(\Psi_f({\mathcal M})\). Then \(\text{Gr}^W (\Psi_f((M)))\) is a semisimple holonomic \({\mathcal D}_X\)-module.
(C3) The hard Lefschetz theorem holds for a semisimple holonomic \({\mathcal D}_X\)-module.
For the entire collection see [Zbl 0905.00081].
Reviewer: M.Muro (Yanagido)

32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)