arXiv subjects

Daniel Barlet

Publications and source records attributed to Daniel Barlet.

52 records · Page 3Linked to original sources

arXiv2011

The theme of a vanishing period

Let \ $λ\in \mathbb{Q}^{*+}$ \ and consider a multivalued formal function of the type $$ ϕ(s) : = \sum_{j=0}^k \ c_j(s).s^{λ+ m_j}.(Log\, s)^j $$ where \ $c_j \in \C[[s]], m_j \in \mathbb{N}$ \ for \ $j \in [0,k-1]$. The {\bf theme} associated to such a \ $ϕ$ \ is the "minimal filtered differential equation" with generator \ $ϕ$, in a sens which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial fixed. For a given \ $λ$, to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated to the logarithm of the monodromy in the geometric stuation described above. Our purpose is to construct some analytic invariants, for instance in the following situation : Let \ $f : X \to D$ \ be a proper holomorphic function defined on a complex manifold \ $X$ \ with value in a disc \ $D$. We assume that the only critical value is \ $0 \in D$ \ and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \ $f^{-1}(0)$. To a smooth \ $(p+1)-$form \ $ω$ \ on \ $X$ \ such that \ $dω= 0 = df \wedge ω$ \ and to a vanishing \ $p-$cycle \ $γ$ \ choosen in the generic fiber \ $f^{-1}(s_0), s_0 \in D \setminus \{0\}$, we associated a vanishing period \ $ϕ(s) : = \int_{γ_s} \ ω\big/df $ \ which is, when \ $γ$ \ is choosen in the spectral subspace of \ $H_p(f^{-1}(s_0), \C)$ \ for the eigenvalue \ $e^{2iπ.λ}$ \ of the monodromy of \ $f$, of the form above. Here \ $(γ_s)_{s \in D^*}$ is the horizontal multivalued family of \ $p-$cycles in the fibers of \ $f$ \ obtained from the choice of \ $γ$. The result obtained allows, for instance, to associate "natural" holomorphic functions of the parameter space when we have a family of such degenerations depending holomorphically on a parameter.

math.AG↗

arXiv2011

Grauert's theorem for subanalytic open sets in real analytic manifolds

By open neighbourhood of an open subset $Ω$ of $\mathbb{R}^n$ we mean an open subset $Ω'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\capΩ'=Ω.$ A well known result of H. Grauert implies that any open subset of $\mathbb{R}^n$ admits a fundamental system of Stein open neighbourhoods in $\mathbb{C}^n$. Another way to state this property is to say that each open subset of $\mathbb{R}^n$ is Stein. We shall prove a similar result in the subanalytic category, so, under the assumption that $Ω$ is a subanalytic relatively compact open subset in a real analytic manifold, we show that $Ω$ admits a fundamental system of subanalytic Stein open neighbourhoods in any of its complexifications.

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arXiv2011

Asymptotics of a vanishing period : the quotient themes of a given fresco

In this paper we introduce the word "fresco" to denote a $[λ]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the $[λ]-$primitive case to frescos having a unique Jordan-H{ö}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, so to the image of a fresco in the module of asymptotic expansions. For a fixed relative de Rham cohomology class (for instance given by a smooth differential form $d-$closed and $df-$closed) each choice of a vanishing cycle in the spectral eigenspace of the monodromy for the eigenvalue $exp(-2iπ.λ)$ produces a $[λ]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham class itself. So the problem to determine which theme is a quotient of a given fresco is important to deduce possible asymptotic expansions of the various vanishing period integrals associated to a given relative de Rham class when we change the choice of the vanishing cycle. In the appendix we prove a general existence result which naturally associate a fresco to any relative de Rham cohomology class of a proper holomorphic function of a complex manifold onto a disc.

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arXiv2011

Contruction of holomorphic parameters invariant by change of variable in the Gauss-Manin connection of an holomorphic map to a disc

When we consider a proper holomorphic map \ $\tilde{f}: X \to C$ \ of a complex manifold \ $X$ \ on a smooth complex curve \ $C$ \ with a critical value at a point \ $0$ \ in \ $C$, the choice of a local coordinate near this point allows to dispose of an holomorphic function \ $f$. Then we may construct, using this function, an (a,b)-modules structure on the cohomology sheaves of the formal completion (in \ $f$) \ of the complex of sheaves \ $(Ker\, df^{\bullet},d^{\bullet})$. These (a,b)-modules represent a filtered version of the Gauss-Manin connection of \ $f$. The most simple example of this construction is the Brieskorn module (see [Br.70]) of a function with an isolated singular point. See [B.08] for the case of a 1-dimensional critical locus. But it is clear that this construction depends seriously on the choice of the function \ $f$ \ that is to say on the choice of the local coordinate near the critical point \ $0$ \ in the complex curve \ $C$. The aim of the present paper is to study the behaviour of such constructions when we make a change of local coordinate near the origin. We consider the case of \ $[λ]-$primitive frescos, which are monogenic geometric (a,b)-modules corresponding to a minimal filtered differential equation associated to a relative de Rham cohomology class on \ $X$ \ (see [B.09-a] and [B.09-b]). An holomorphic parameter is a function on the set of isomorphism classes of frescos which behave holomorphically in an holomorphic family of frescos. In general, an holomorphic parameter is not invariant by a change of variable, but we prove a theorem of stability of holomorphic families of frescos by a change of variable and it implies that an holomorphic parameter gives again an holomorphic parameter by a change of variable. We construct here two different kinds of holomorphic parameters which are (quasi-)invariant by change of variable. The first kind is associated to Jordan blocks of the monodromy with size at least two. The second kind is associated to the semi-simple part of the monodromy and look like some "cross ratio" of eigenvectors. They allow, in the situation describe above, to associate to a given (vanishing) relative de Rham cohomology class some numbers, which will depend holomorphically of our data, and are independant of the choice of the local coordinate near \ $0$ \ to study the Gauss-Manin connection of this degeneration of compact complex manifolds.

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arXiv2010

Quasi-proper meromorphic equivalence relations

The aim of this article is to complete results of [M.00] and [B.08] and to show that they imply a rather general existence theorem for meromorphic quotient of strongly quasi-proper meromorphic equivalence relations. In this context, generic equivalence classes are asked to be pure dimensionnal closed analytic subset with finitely many irreducible components. As an application of these methods we prove a Stein factorization theorem for a strongly quasi-proper map

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arXiv2010

Changements de variable pour un th`eme.

We study the behaviour of the notion of "thema", introduced in our previous article [B.09b], by a change of variable. We show not only that the fundamental invariants of such a thema, corresponding to the Bernstein polynomial, are stable by a change of variable, but also other numerical invariants called principal parameters. \\ We show on a rank 3 example that nevertheless the isomorphism class of a thema is not stable in general by a change of variable. We conclude in proving that a change of variable transforms an holomorphic family of thema in an holomorphic family. This implies that non principal parameters change holomorphically.

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arXiv2009

Le thème d'une période évanescente

In this article we study holomorphic deformations of the filtered Gauss-Manin systems associated to a vanishing period integral. For that purpose we introduce a new sub-class of the class of monogenic (a,b)-modules (Brieskorn modules) which was studied in our previous article [B. 09]. We show that these new objects, called ?themes?, have good functorial properties and that there exists a canonical order on the roots of the corresponding Bernstein polynomial. We construct, for given fundamental invariants, a finite dimensional versal holomorphic family and we show that, when all themes with these fundamental invariants are ?stable?, this versal family is in fact universal. We also give a sufficient condition on the roots of the Bernstein polynomial in order that the previous condition is satisfied. We show with an example that a universal family may not exist for some values of the fundamental invariants.

math.CV↗

arXiv2009

Un théorème à la "Thom-Sebastiani" pour les intégrales-fibres

The aim of this article is to prove a Thom-Sebastiani theorem for the asymptotics of the fiber-integrals. This means that we describe the asymptotics of the fiber-integrals of the function $f \oplus g : (x,y) \to f(x) + g(y)$ \ on $(\mathbb{C}^p\times \mathbb{C}^q, (0,0))$ in term of the asymptotics of the fiber-integrals of the holomorphic germs $f : (\mathbb{C}^p,0) \to (\mathbb{C},0)$ and $g : (\mathbb{C}^q,0) \to (\mathbb{C},0)$. This reduces to compute the asymptotics of a convolution $Φ_*Ψ$ from the asymptotics of $Φ$ and $Ψ$ modulo smooth terms. To obtain a precise theorem, giving the non vanishing of expected singular terms in the asymptotic expansion of $f\oplus g$, we have to compute the constants coming from the convolution process. We show that they are given by rational fractions of Gamma factors. This enable us to show that these constants do not vanish.

math.CV↗

arXiv2009

Oblique poles of $\int_X| {f}| ^{2λ}| {g}|^{2μ} \square$

Existence of oblique polar lines for the meromorphic extension of the current valued function $\int |f|^{2λ}|g|^{2μ}\square$ is given under the following hypotheses: $f$ and $g$ are holomorphic function germs in $\CC^{n+1}$ such that $g$ is non-singular, the germ $S:=\ens{\d f\wedge \d g =0}$ is one dimensional, and $g|_S$ is proper and finite. The main tools we use are interaction of strata for $f$ (see \cite{B:91}), monodromy of the local system $H^{n-1}(u)$ on $S$ for a given eigenvalue $\exp(-2iπu)$ of the monodromy of $f$, and the monodromy of the cover $g|_S$. Two non-trivial examples are completely worked out.

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arXiv2009

Périodes évanescentes et $(a,b)$-modules monogènes

In order to describe the asymptotic behaviour of a vanishing period in a one parameter family we introduce and use a very simple algebraic structure : regular geometric (a,b)-modules generated (as left $\A-$modules) by one element. The idea is to use not the full Brieskorn module associated to the Gauss-Manin connection but a minimal (regular) differential equation satisfied by the period integral we are interested in. We show that the Bernstein polynomial associated is quite simple to compute for such (a,b)-modules and give a precise description of the exponents which appears in the asymptotic expansion which avoids integral shifts. We show a couple of explicit computations in some classical (but not so easy) examples.

math.AG↗

arXiv2008

Two finiteness theorem for $(a,b)$-module

We prove the following two results 1. For a proper holomorphic function $ f : X \to D$ of a complex manifold $X$ on a disc such that $\{df = 0 \} \subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric (a,b)-module $E^p$ \ associated to the (filtered) Gauss-Manin connexion of $f$. This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module $E$ we give an integer $N(E)$, explicitely given from simple invariants of $E$, such that the isomorphism class of $E\big/b^{N(E)}.E$ determines the isomorphism class of $E$. This second result allows to cut asymptotic expansions (in powers of $b$) \ of elements of $E$ without loosing any information.

math.AG↗

arXiv2007

Finite determination of regular (a,b)-modules

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal completion of the Brieskorn-module of an isolated singularity. The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some ''finite order jet'' of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply \ $b^{-1}.a \simeq \partial_z.z$ in order to reach a simple pole (a,b)-module. The second invariant is the ''width'' which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of $b^{-1}.a$ (the logarithm of the monodromy). In the computation of examples this theorem is quite helpfull because it tells you at which power of $b$ in the expansions you may stop without loosing any information.

math.CV↗

arXiv2007

Sur les fonctions à singularité de dimension 1

In this article we show that all results proved for a large class of holomorphic germs $f : (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ with a 1-dimension singularity in [B.II] are valid for an arbitrary such germ.

math.CV↗

arXiv2006

Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities

We study the Brieskorn modules associated to a germ of holomorphic function with non-isolated singularities, and show that the Brieskorn module has naturally a structure of a module over the ring of microdifferential operators of nonpositive degree, and that the kernel of the morphism to the Gauss-Manin system coincides with the torsion part for the action of $t$ and also with that for the action of the inverse of the Gauss-Manin connection. This torsion part is not finitely generated in general, and we give a sufficient condition for the finiteness. We also prove a Thom-Sebastiani type theorem for the sheaf of Brieskorn modules in the case one of two functions has an isolated singularity.

math.CV↗

arXiv2006

Sur certaines singularites non isolees d'hypersurfaces I

The aim of this fisrt part is to introduce, for a rather large class of hypersurface singularities with 1 dimensionnal locus, the analog of the Brieskorn lattice at the origin (the singular point of the singular locus). The main results are the finitness theorem for the corresponding (a,b)-module obtained via Kashiwara's constructibility theorem, and non torsion results for a plane curve singularity (not nessarily reduced) and for the suspension of such non torsion cases with an isolated singularity.

math.AG↗

arXiv2003

Singularites reelles isolees et developpements asymptotiques d'integrales oscillantes

Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated singularity at 0. Let (f_R,0) : (X_R, 0) --> (R,0) a real analytic germ with an isolated singularity at 0, such that its complexification f_C vanishes on the singular set S of X_C. We also assume that X_R-[0] is orientable. To each $ A \in H^{0}(X_{\mathbb{R}} - \lbrace 0 \rbrace ,\mathbb {C}) $ we associate a $n-$cycle $ Γ(A) $ ("explicitly " described) in the complex Milnor fiber of $f_{\mathbb{C}}$ at 0 such that the non trivial terms in the asymptotic expansions of the oscillating integrals $ \int_{A} e^{iτf(x)} ϕ(x) $ when $ τ\to \pm \infty $ can be read from the spectral decomposition of $Γ(A) $ relative to the monodromy of $f_{\mathbb{C}}$ at 0 .

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