Barlet, Daniel Monodromie et pôles du prolongement méromorphe de \(\int _{X}| f| ^{2\lambda}\square\). (Monodromy and poles of the meromorphic extension of \(\int _{X}| f| ^{2\lambda}\square)\). (French) Zbl 0652.32010 Bull. Soc. Math. Fr. 114, 247-269 (1986). Let \(\tilde f:\) (\({\mathbb{C}}^{n+1},0)\to ({\mathbb{C}},0)\) be a germ of non- constant holomorphic function such that the singularities of \(\tilde f=0\) may not be isolated. Let f: \(X\to D=\{z\in {\mathbb{C}}|\) \(| z| <\epsilon \}\) be a Milnor representation of \(\tilde f.\) Namely, (1) X is a contractible Stein manifold, \((2)\quad f: X-f^{-1}(0)\to D-\{0\}\) is a locally trivial \(C^{\infty}\) fibre bundle, (3) for any non-negative integer p and a point \(s\in D-\{0\},\) \(\dim_{{\mathbb{C}}}H\) \(p(X(s_ 0),{\mathbb{C}})<+\infty\), where \(X(s_ 0)=f^{-1}(s_ 0)\). By Milnor [J. Milnor, Singular point of complex hypersurfaces (1968; Zbl 0184.484)] there always exists such a representation f of \(\tilde f.\) Suppose the monodromy \(T_ p\) on H \(p(X(s_ 0),{\mathbb{C}})\) of the fibration f has a Jordan block of type (k,k) with eigenvalue \(\lambda =e^{2\pi \sqrt{- 1}u}\), \(0\leq u<1\), that is there are linearly independent elements \(e_ 1,...,e_ k\) of H \(n(X(s_ 0),{\mathbb{C}})\) such that \[ T_ pe_ 1=e_ 1,\quad T_ pe_ j=e_ j+e_{j-1},\quad 2\leq j\leq k. \] The main result of the present paper is the following Theorem. Theorem. Under the notation and the assumption above, the distribution \(\int_{X}| f|^{2\lambda}\square\) associated with f has a pole of order at least k at the point -p-u. Moreover, the support of the pole part of order \(\geq k\) at -p-u is not contained in the closed analytic subset of X of codimension \(\geq p+2\). Reviewer: K.Ueno Cited in 6 Documents MSC: 32S05 Local complex singularities 32Sxx Complex singularities Keywords:Milnor fiber; monodromy; distribution Citations:Zbl 0184.484 PDFBibTeX XMLCite \textit{D. Barlet}, Bull. Soc. Math. Fr. 114, 247--269 (1986; Zbl 0652.32010) Full Text: DOI Numdam EuDML References: [1] Développements asymptotiques des fonctions obtenues par intégration dans les fibres , Inv. Math., vol. 68, 1982 , p. 129-174. · Zbl 0508.32003 [2] Contribution effective de la monodromie aux développements asymptotiques , Ann. scient. Éc. Norm. Sup., 4e série, t. 17, 1984 , p. 293-315. Numdam | MR 86i:32013 | Zbl 0542.32003 · Zbl 0542.32003 [3] Contribution du cup-produit de la fibre de Milnor aux pôles de | f |2\? , Annales de l’Institut Fourier, t. 34, fasc. 4, 1984 , p. 75-107. Numdam | MR 86h:32012 | Zbl 0525.32007 · Zbl 0525.32007 [4] Contribution effective dans le réel , Comp. Math., vol. 56, 1985 , p. 351-359. Numdam | MR 87f:32019 | Zbl 0581.32016 · Zbl 0581.32016 [5] BJORK (J. E.) . - Rings of differential operators , North Holland, 1979 . MR 82g:32013 | Zbl 0499.13009 · Zbl 0499.13009 [6] HAMM (H.) . - Zür analytischen und algebraichen Beschreibung der Picard-Lefchetz Monodromie , Habilitationsschrift, Göttingen, 1974 . [7] HAMM (H.) et LE DŨNG TRANG . - Un théorème de Zariski de type Lefschetz , Ann. Scient. Éc. Norm. Sup., 4e série, t. 6, 1973 , p. 317-366. Numdam | MR 53 #5582 | Zbl 0276.14003 · Zbl 0276.14003 [8] KASHIWARA (M.) . - B-Functions and holonomic systems , Inv. Math., vol. 38, 1976 , p. 33-53. MR 55 #3309 | Zbl 0354.35082 · Zbl 0354.35082 [9] KATO (M.) et MATSUMOTO (Y.) . - On the connectivity of the Milnor fiber of a holomorphic function at a critical point , Manifold Tokyo, 1973 , University of Tokyo Press, 1975 . Zbl 0309.32008 · Zbl 0309.32008 [10] LE DŨNG TRANG et TEISSIER (B.) . - Cycles évanescents, sections planes et conditions de Whitney II , Proceedings of Symposia in Pure Math., vol. 40, part. 2, 1983 , p. 65-103. MR 86c:32005 | Zbl 0532.32003 · Zbl 0532.32003 [11] LOJASIEWICZ (S.) . - Triangulation of semi-analytic sets , Ann. Scuola Norm. Sup. Pisa Sc. Fist. Mat., sér. 3, vol. 18, fasc. 4, 1964 , p. 449-474. Numdam | MR 30 #3478 | Zbl 0128.17101 · Zbl 0128.17101 [12] MALGRANGE (B.) . - Intégrales asymptotiques et monodromie , Ann. scient. Éc. Norm. Sup., 4e série, t. 7, 1974 , p. 405-430. Numdam | MR 51 #8459 | Zbl 0305.32008 · Zbl 0305.32008 [13] MALGRANGE (B.) . - Polynômes de Bernstein-Sato et cohomologie évanescente , Analyse et topologie sur les espaces singuliers, Astérisque, 101-102, 1983 , p. 230-242. MR 86f:58148 | Zbl 0528.32007 · Zbl 0528.32007 [14] MILNOR (J.) . - Singular points of complex hypersurfaces , Ann. of Math. Studies, n^\circ 61, Princeton, 1968 . MR 39 #969 | Zbl 0184.48405 · Zbl 0184.48405 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.