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\(L ^{2}\)-Betti numbers of plane algebraic curves. (English) Zbl 1184.14034

Let \(D\) be a reduced curve in the complex affine plane \(\mathbb{C}^2\) and consider the complement \(X(D)=\mathbb{C}^2 \setminus \nu D\), with \(\nu D\) a regular neighborhood of \(D\) inside \(\mathbb{C}^2\).
Let \(\widetilde{ X(D)}\) be the infinite cyclic cover of \(X(D)\) associated to the kernel of the natural group homomorphism \(\pi_1(X(D)) \to Z\).
The main results of this paper give information on the \(L^2\)-Betti numbers of \(X(D)\) and \(\widetilde { X(D)}\) twisted by group homomorphisms of the fundamental group of \({ X(D)}\) and respectively \(\widetilde{ X(D)}\) to a countable group \(\Gamma\), under the assumption that \(D\) is transversal to the line at infinity.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14H30 Coverings of curves, fundamental group
14H50 Plane and space curves
32S20 Global theory of complex singularities; cohomological properties
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References:

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