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The Milnor fibre signature is not semi-continuous. (English) Zbl 1225.32030

Cogolludo-Agustín, José Ignacio (ed.) et al., Topology of algebraic varieties and singularities. Invited papers of the conference in honor of Anatoly Libgober’s 60th birthday, Jaca, Spain, June 22–26, 2009. Providence, RI: American Mathematical Society (AMS); Madrid: Real Sociedad Matemática Española (ISBN 978-0-8218-4890-6/pbk). Contemporary Mathematics 538, 369-376 (2011).
Let \((X,o)\) be a \(2\)-dimensional isolated hypersurface singularity. Let \(I\) be the intersection form for the vanishing cycles on the Milnor fiber \(X_{\varepsilon}\) (i.e., a quadratic form on \(H:=H_2(X_{\varepsilon},{\mathbb Z})\)). Let \(\mu\) be the Milnor number (\(=\) the rank of \(H\)). Let \(\mu_+(X)\) (resp. \(\mu_-(X)\)) be the number of the positive (resp. negative) eigenvalues of \(I\). A. Durfee (1978) asked the following question: is the signature \(\text{sign\,}(X)\) (\(:=\mu_+(X) - \mu_+(X)\)) non-decreasing under degenerations?
In this paper, the authors give a counterexample whose Newton diagram is non-degenerate. Namely, they consider
\[ X_t=f_t^{-1}(0), \quad f_t=txyz+xyz(x+y+z)+x^4y+y^4z+z^ 4x, \] and prove that \(\text{sign\,}(X_t)=-35\) and \(\text{sign\,}(X_0)=-32\). Using Kouchnirenko’s formula they prove some formulae (Proposition 3.1) on \(\mu_+(X)\) and \(\mu_-(X)\) from the Newton diagram. From this result, using the computer software Singular, they get the example. Also, they give a class of degenerate families which have the same property (Proposition 4.1).
For the entire collection see [Zbl 1210.14004].

MSC:

32S05 Local complex singularities

Software:

SINGULAR
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