Morsification of \({\mathcal D}\)-modules
type: Item
instance of: scholarly article
title: Morsification of \({\mathcal D}\)-modules
zbMATH Open document ID: 0938.32014
author: Dũng Tráng Lê
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2000-06-15T00:00:00Z
review text: Let \(M\) be a holonomic \(\mathcal D\)-module over a complex analytic manifold \(X\). Denote by \(CCh(M)=\sum_\alpha m_\alpha V_\alpha\) the characteristic cycle of \(M\) (where the \(V_\alpha\) are the irreducible components of the characteristic variety \(V\) of \(M\)). Let \(x\in X\) and \(f: U \rightarrow {\mathbb C}\) a complex analytic function defined on an open neighborhood \(U\) of \(x\). Suppose the point \((x,df_x)\) isolated in \(\text{Im} df\cap V.\) The author proves the following result (conjectured by Deligne): Theorem. The intersection number of the characteristic cycle \(CCh(M)\) and \(\text{Im} df\) at the point \((x,df_x)\) equals the complex dimension of the space of vanishing cycles of \(M\) relatively to \(f\) at \(x\). The proof uses the notion of Morse unfolding (or morsification) of the \(\mathcal D\)-module \(M\) along \(f\), which is a morsification of \(f\) by a suitable and carefully chosen linear form. This morsification is used to estimate the Euler-Poincaré characteristic \(\chi(H^*(F,\text{Sol}(M)))\) of the hypercohomology of global sections of the complex \(Sol(M)\) of holomorphic solutions of \(M\) on the general fiber \(F\) of \(f\) at \(x\). The same result was proved by \textit{C. Sabbah} [Astérisque 130, 161-192 (1985; Zbl 0598.32011)] and by \textit{V. Ginsburg} [Invent. Math. 84, 327-403 (1986; Zbl 0598.32013)].
Mathematics Subject Classification ID: 32L10
Mathematics Subject Classification ID: 32C38
Mathematics Subject Classification ID: 32S60
zbMATH DE Number: 1240997
zbMATH Keywords: holonomic D-module
zbMATH Keywords: morsification
zbMATH Keywords: vanishing cycles
zbMATH Keywords: characteristic variety
reviewed by: Francisco J. Castro-Jiménez
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1240997