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Characteristic varieties and vanishing cycles. (English) Zbl 0598.32013
The main theme of this paper is a systematical study on characteristic variety of a holonomic system with regular singularity on a complex manifold X, especially extensions of index theorems on characteristic cycles. A holonomic system of \({\mathcal D}_ X\)-modules is originally introduced by Sato-Kawai-Kashiwara in 1972, which is a \({\mathcal D}_ X\)- module \({\mathcal M}\) whose dimension of its characteristic variety SS(\({\mathcal M})\) coincides with the dimension of X. For two holonomic systems \({\mathcal N}\) and \({\mathcal M}\) with regular singularity, the author of this paper proves a global index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})=I(SS({\mathcal M}),SS({\mathcal N}))\) where I(A,B) is the multiplicity of the intersection of the cycles A and B. Moreover the author proves the local index theorem: \(\Sigma_ k(- 1)^ k\cdot \dim (Ext^ k_{{\mathcal D}_ X}({\mathcal N},{\mathcal M})_ X=I_ X(SS({\mathcal M}),SS({\mathcal N})),\) and the microlocal index theorem: \(\Sigma_ k(-1)^ k\cdot \dim (Ext^ k_{{\mathcal E}_ X^{{\mathbb{R}}}}({\mathcal N}^{{\mathbb{R}}},{\mathcal M}^{{\mathbb{R}}})_{\xi}=I_{\xi}(SS({\mathcal M}),SS({\mathcal N})).\)
The present work arise from the author’s interest to characteristic varieties of the highest-weight modules over complex semi-simple Lie algebras. The detailed treatment of that subject appears in the next paper [Adv. Math. 61, 1-48 (1986)].
Reviewer: M.Muro

MSC:
32Q99 Complex manifolds
32C25 Analytic subsets and submanifolds
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
14E05 Rational and birational maps
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