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Logarithmic structure of the generalized bifurcation set. (English) Zbl 0868.58014

The author considers the following generalized bifurcation sets. Let \(G: \mathbb{C}^n\times \mathbb{C}^r\to \mathbb{C}\) be a holomorphic family of functions. If \(\Lambda\subset \mathbb{C}^n\times \mathbb{C}^r\) is an analytic variety then \[ Q_\Lambda(G)=\{(x,u)\in \mathbb{C}^n\times\mathbb{C}^r\mid G(\cdot,u) \text{ has a critical point in }\Lambda\cap\pi_r^{-1}(u)\} \] is a natural generalization of the bifurcation variety of \(G\), where \(\pi_r:\mathbb{C}^n\times \mathbb{C}^r\to\mathbb{C}\) is the canonical projection. The author investigates the local structure of \(Q_\Lambda(G)\) for locally trivial deformations of \(\Lambda_0=\Lambda\cap \pi^{-1}_r(0)\). In particular, he constructs an algorithm for determining logarithmic stratifications provided \(G\) is versal.

MSC:

58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
58A14 Hodge theory in global analysis
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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