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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)

##### MSC:
 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)
##### Keywords:
holonomic systems; $${\mathcal D}$$-modules