zbMATH — the first resource for mathematics

Generalized local and global Sebastiani-Thom type theorems. (English) Zbl 0744.32020
Let \(g: (\mathbb{C}^ n,0)\to(\mathbb{C},0)\) resp. \(h: (\mathbb{C}^ m,0)\to(\mathbb{C},0)\) be analytic germs with Milnor fiber \(G\) resp. \(H\). Let \(p: (\mathbb{C}^ 2,0)\to(\mathbb{C},0)\) be an analytic germ in two variables (\(c\) and \(d\)) with Milnor fiber \(P\). In this paper, we suppose that \(P\) is connected. In this case \(P\) has the homotopy type of a bouquet of \(\mu_ p=1-\chi_ p\) circles. We define \(n_ c=0\) if \(c\) is a factor of \(p\) and \(n_ c\)= the intersection multiplicity \(m_ 0(p,c)\) otherwise. Symmetrically we define \(n_ d\). Then \(n_ c\) is the number of points of the intersection \(P\cap\{c=0\}\).
Theorem. The Milnor fiber \(F\) of the analytic germ \(f=p(h,g): (\mathbb{C}^ n\times\mathbb{C}^ m,0)\to(\mathbb{C},0)\) defined by \(f(x,y)=p(g(x),h(y))\) has the homotopy type of a space obtained from the total space of a fiber bundle with base space \(P\) and fiber \(G\times H\) by gluing with the natural applications to a fiber \(G\times H\) \(\hbox{n}_ c\) copies of \(\hbox{Con} G\times H\) and \(\hbox{n}_ d\) copies of \(G\times \hbox{Con}H\).
Theorem. The zeta function of \(f\) is determined by \[ \zeta_ f(\lambda)=\zeta_ g(\lambda^{n_ d})\cdot\zeta_ h(\lambda^{n_ c})\cdot\prod_ g\det \Delta(\lambda^{m_ 1}E_{q,1},\lambda^{m_ 2}E_{q,2},\lambda^{m_ 3}\cdot I,\dots,\lambda^{m_ r}\cdot I)^{(-1)^ g}. \]

32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
Full Text: Numdam EuDML
[1] Demazure, M. : Classification des germs à points critiques isolés et à nombres de modules 0 ou 1 (d’après V. I. Arnold) , Séminaire Bourbaki, 26-e année, 1973/74, 443, Février 1974. · Zbl 0359.57012 · numdam:SB_1973-1974__16__124_0 · eudml:109844
[2] Eisenbud, D. and Neumann, W. : Three-dimensional link theory and invariants of plane curve singularities , Annals of Math. Studies, Princeton Univ. Press., 110 (1985). · Zbl 0628.57002 · doi:10.1515/9781400881925
[3] Gabrielov, A.M. : Intersection matrices for certain singularities , Funktional’nyi Analiz i ego Prilozheniya, 1973, 7:3, 18-32; FAA (Eng. Tranl. of FAP) 7, 182-193. · Zbl 0288.32011 · doi:10.1007/BF01080695
[4] Massey, D.B. : The Lê Varieties, II : submitted. · Zbl 0727.32015 · doi:10.1007/BF01245068 · eudml:143876
[5] Milnor, J. : Singular points of complex hypersurfaces , Annals of Math. Studies, Princeton Univ. Press, 61 (1968). · Zbl 0184.48405 · doi:10.1515/9781400881819
[6] Neumann, W.D. , Rudolf, L. : Unfoldings in knot theory . Math. Ann. 278, 409-439 (1987) and Corrigendum , ibid. 282, 349-351 (1988). · Zbl 0675.57011 · doi:10.1007/BF01456981
[7] Neumann, W.D. : Complex algebraic plane curves via their links at infinity . Invent. Math. 98, 445-489 (1989). · Zbl 0734.57011 · doi:10.1007/BF01393832 · eudml:143738
[8] Némethi, A. : The Milnor fiber and the zeta function of the singularities of type f = P(h, g) ; to appear in Compositio Math. · Zbl 0724.32020 · numdam:CM_1991__79_1_63_0 · eudml:90099
[9] Némethi, A. : Global Sebastiani-Thom theorem for polynomial maps ; to appear in J. Math. Soc. Japan. · Zbl 0739.32034 · doi:10.2969/jmsj/04320213
[10] Némethi, A. , Zaharia, A. : The topology of some affine hypersurfaces (preprint, INCREST, 65/1986).
[11] Oka, M. : On the homotopy types of hypersurfaces defined by weighted homogeneous polynomials , Topology, 12, 19-32 (1973). · Zbl 0263.32005 · doi:10.1016/0040-9383(73)90019-0
[12] Sakamoto, K. : Milnor fiberings and their characteristic maps , Proc. Intern. Conf. on manifolds and Related Topics in Topology, Tokyo (1973). · Zbl 0321.32010
[13] Sakamoto, K. : The Seifert matrices of Milnor fiberings defined by holomorphic functions , J. Math. Soc. Japan, 26, 714-721 (1974). · Zbl 0286.32010 · doi:10.2969/jmsj/02640714
[14] Sebastiani, M. , Thom, R. : Un résultat sur la monodromie , Invent. Math. 1971, 13: 1-2, 90-96. · Zbl 0233.32025 · doi:10.1007/BF01390095 · eudml:142085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.