Morsification of \({\mathcal D}\)-modules
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)].
scientific article; zbMATH DE number 1240997