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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

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

MSC:

32S05 Local complex singularities
32Sxx Complex singularities

Citations:

Zbl 0184.484
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References:

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