A generalization of Milnor's formula
In the article a generalization of the classical notion of the Milnor number \(\mu _{f}(0)\) of a holomorphic function \(f :(\mathbb{C}^{n} ,0) \rightarrow (\mathbb{C} ,0)\) with an isolated critical point at \(0\) (= an isolated singularity) is given in the following setting: 1. \((X ,0) \subset (\mathbb{C}^{n} ,0)\) is a singular reduced analytic space, 2. \((\mathcal{S}_{\alpha })\) is the canonical Whitney stratification of \((X ,0) ,\) 3. \(f :(\mathbb{C}^{n} ,0) \rightarrow (\mathbb{C} ,0)\) is a holomorphic function such that \(f\vert (X ,0)\) has an isolated singularity in the stratified sense at 0 (it means \(f\vert \mathcal{S}_{\alpha }\) has only regular points in a small vicinity of \(0 \in \mathcal{S}_{\alpha }\)). The author defines this generalization for any strata \(\mathcal{S}_{\alpha }\). The Milnor number \(\mu _{f}(\alpha ;X ,0)\) is the number of Morse critical points on \(\mathcal{S}_{\alpha \text{}}\) in a morsification of \(f\vert (X ,0)\) (a morsification is an unfolding \((f_{t})\) of \(f\) such that for small \(t\) and any \(\mathcal{S}_{\alpha \text{}}\) the restriction \(f_{t}\vert \mathcal{S}_{\alpha \text{}}\) has only classical Morse critical points and at these points the differential \(df_{t}\) does not annihilate any limiting tangent space from other adjacent strata \(\mathcal{S}_{\beta }\)). He shows that the numbers \(\mu _{f}(\alpha ;X ,0)\) generalize the classical known characterizations of the Milnor number \(\mu _{f}(0)\).
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