2011
BARLET, Daniel
Prépublication numéro : 2011/01
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 \ $[\lambda]-$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.
AMS Classification (2000) : 32 S 25, 32 S 40, 32 S 50.
Key words : Degenerating family of complex manifolds, Relative de Rham cohomology classes, filtered Gauss-Manin connection, Theme, Fresco, (a,b)-module, asymptotic expansion, vanishing period.
BARLET, Daniel
Asymptotics of a vanishing period : the quotient themes of a given fresco.
Prépublication numéro : 2011/02
In this paper we introduce the word "fresco" to denote a \ $[\lambda]-$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 \ $[\lambda]-$primitive case to frescos having a unique
Jordan-H{\"o}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 \ $exp(-2i\pi.\lambda)$ \ produces a \ $[\lambda]-$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.
AMS Classification(2000) : 32 S 25, 32 S 40, 32 S 50.
Key words : Theme, fresco, (a,b)-module, asymptotic expansion, vanishing period.
2010
BARLET, Daniel & MONTEIRO FERNANDES, Teresa
Grauert's theorem for subanalytic open sets in real analytic manifolds
Prépublication numéro : 2010/31
By open neighbourhood of an open subset $\Omega$ of $\mathbb{R}^n$ we mean
an open subset $\Omega'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\cap\Omega'=\Omega.$
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 $\Omega$ is a subanalytic relatively compact open
subset in a real analytic manifold, we show that \ $\Omega$ \ admits a
fundamental system of subanalytic Stein
open neighbourhoods in any of its complexifications.
AMS Classification(2000) : Primary: 32B20, 14P15; Secondary: 32C05, 32C09.
Key words : Subanalytic, Stein Neighbourhood.
BARLET, Daniel
Quasi-proper meromorphic equivalence relations
Prépublication numéro : 2010/13
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.
AMS Classification(2000) : 32 C 25, 32 H 35, 32 H 04.
Key words : Meromorphic equivalence relations, meromorphic quotients, geometric f-flattening, (strongly) quasi-proper.
BARLET, Daniel
Changement de variable pour un thème
Prépublication numéro : 2010/9
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.
AMS Classification (2000) : 32S05, 32S25, 32S40.
Key words : Vanishing period, Berstein module, (a-b)-module, thema, filtered Gauss-Manin system, change of variable.
2009
BARLET, Daniel
Le thème d'une période évanescente
Prépublication numéro : 2009/33
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.
AMS Classification (2000) : 32S05, 32S25, 32S40.
Key words : Vanishing period, Berstein polynomial, filtered Gauss-Manin system, (a,b)-module, Brieskorn module.
BARLET, Daniel & MAIRE, Henri
Oblique poles of $\int_X\vert{f}\vert^{2\lambda}\vert{g}\vert^{2\mu}\square$.
Prépublication numéro : 2009/03
Existence of oblique polar lines for the meromorphic extension of the current valued function $\int |f|^{2\lambda}|g|^{2\mu}\square$ 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 $\exp(-2i\pi u)$ of the monodromy of $f$, and the monodromy of the cover $g|_S$. Two non-trivial examples are completely worked out.
AMS Classification (2000) : 32S40,58K55.
Key words : asymptotic expansion - fibre integrals - meromorphic extension.
BARLET, Daniel
Périodes évanescentes et (a,b)-modules monogènes.
Prépublication numéro : 2009/01
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.
AMS Classification (2000) : 32-S-25, 32-S-40, 32-S-50.
Key words : vanishing cycles, vanishing periods, (a,b)-modules, Brieskorn modul es, Gauss-Manin systems.
2008
BARLET, Daniel
Thom-Sebastiani pour les intégrales-fibres.
Prépublication numéro : 2008/43
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 as
ymptotics 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 \ $g : (\mathb
b{C}^q,0) \to (\mathbb{C},0)$. This reduces to compute the asymptotics of a convolution \ $\Phi_*\Psi$ \ from the asymptotics of \ $\Phi$ \
and \ $\Psi$.
To obtain precise results we have to compute the constants coming from the convolution process. We show that there are given by rational fra
ctions of Gamma factors. This enable us to show that these constants do not vanish.
As a corollary we obtain that the roots of the b-function of \ $f\oplus g$ \ are modulo \ $\mathbb{Z}$ \ sum of roots of the b-functions of
\ $f$ \ and \ $g$ \ and conversely.
AMS Classification (2000) : 32-S-25, 32-S-40, 32-S-50.
Key words : Asymptotic expansions, fiber-integrals, Thom-Sebastiani theorem, Bernstein polynomial (b-function).
BARLET, Daniel
Sur les fonctions a singularité de dimension 1.
Prépublication numéro : 2008/42
In our previous paper [B.II] we constructed for a large class of germs of holomorphic functions \ $f : (\mathbb{C}^{n+1}, 0) \to (\mathbb{C} , 0)$ \ with one dimensional singular set \ $S : = \{df = 0 \}$, analytic invariants which generalize the Brieskorn module of an isolated si ngularity germ. In the present article we show that all results obtained in this previous paper are valid for {\bf any holomorphic germ with one dimensional singular locus}. So, these invariants, essentially given by geometric (a,b)-modules (this object is an "abstraction" of a f ormal Brieskorn module) describe the various filtered Gauss-Manin connections associated to such a germ, and the relations between them.
AMS Classification (2000) : 32-S-25, 32-S-40, 32-S-50.
BARLET, Daniel
Two finiteness theorem for (a,b)-modules
Prépublication numéro : 2008/05
We prove the following two results
2007
BARLET, Daniel
Sur les fonctions à singularité de dimension 1
Prépublication numéro : 2007/28
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.
BARLET, Daniel
Finite determination of regular (a,b)-modules
Prépublication numéro : 2007/15
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.
Mots Clés: Brieskorn module, (a, b)-module.
BARLET, Daniel
Sur les germes de fonctions holomorphes a lieu singulier de dimension 1: le cas général.
Prépublication numéro : 2007/01
The main goal of this article is to extend the results of [B.06] to a general holomorphic germ $f$ with a one dimensional singular locus at the origine of $\Bbb C ^{n+1}, n \geq 2$. To obtain this generalization it is enough to prove that some nice properties of the cohomology sheaves of the formal completion "in $f$" of the sub-complex given by holomorphic forms annihilated by $\wedge df$ of the holomorphic de Rham complex, obtained under the assumption (HH) in [B.06] are true in general. We also compute explicitely some examples and show the relationship between the $(a,b)$-connexion introduced previously and integrals "à la Malgrange" on vanishing cycles.
Mots Clés: Hypersurface, Non Isolated Singularity, Vanishing Cycles, (a,b)-modules.
2006
BARLET, Daniel
Sur certaines singularités non isolées d'hypersurfaces II (2e version)
Prépublication numéro : 2006/34
The aim of the present article is to construct analytic invariants for a germ of an holomorphic function having a one dimensionnal critical locus $S$. This is done for a large class of such germs containing for instance any quasi-homogeneous germ at the origine. More precisely, aside the Brieskorn's $(a,b)$-module at the origine and a (locally constant along $S^* := S \ {0})$ sheaf $\hat{H}^n$ n of $(a,b)$-modules associated to the transversal hypersurface singularities along each connected component of $S^*$, we construct also $(a,b)$-modules "with supports" $E_c$ ans $E\prime _...
Mots Clés: Hypersurface, Non Isolated Singularity, Vanishing Cycles, Tangling of Strata, (a - b)-modules
BARLET, Daniel
Reparamétrisation universelle de familles $f$-analytiques de cycles et théorème de $f$-aplatissement géométrique
Prépublication numéro : 2006/10
This article is a new presentation of the main results of D. Mathieu [M.00] on meromorphic equivalence relations. We introduce the space of finite type cycles (closed analytic cycles with finitely many irreducible components) of a given finite dimensional complex space and a natural topology on this space, in order to avoid the "regularity" condition for analytic families of cycles introduced in loc. cit. and also the two notions of "escape to infinity" which are here incoded in a natural way in our framework. Then the results are slightly stronger and much simple to state and to use. This contains, in an other language, a clean and more general version of the works of H. Grauert [G.83] and [G.86] and of B. Siebert [S.93] and [S.94] on meromorphic equivalence relations.
G.83 Grauert, H. Set theoretic equivalence relations, Math. Ann. 265 (1983), p.137-148.
G.86 Grauert, H. On meromorphic equivalence relations, Aspect Math. E9 (1986), p.115-147.
M.00 Mathieu, D.Universal reparametrization..., Ann. Inst. Fourier (Grenoble) t.50 fasc.4 (2000), p.1155-1189
S.93 Siebert, B. Fiber cycles of holomorphic maps I. Local flattening, Math. Ann. 296 (1993), p.328-370.
S.94 Siebert, B. Fiber cycle space and canonical flattening II, Math. Ann. 300 (1994), p.243-271.
Mots Clés: relations d'équivalence analytiques et méromorphes ; cycles ; aplatissement géométrique ; reparamétrisation universelle ; quotients méromorphes
2005
BARLET, Daniel
On the Brieskorn $(a,b)$-module of an isolated hypersurface singularity
Prépublication numéro : 2005/48
We show in this note that for a germ \ $g$ \ of holomorphic function with an isolated singularity at the origin of \ $\mathbb{C}^n$ \ there is a pole for the meromorphic extension of the distribution
\begin{equation*}
\frac{1}{\Gamma(\lambda)} \int_X \vert g \vert^{2\lambda}\bar{g}^{-n} \square \tag{*}
\end{equation*}
at \ $- n - \alpha$ when \ $ \alpha$ \ is the smallest root in its class modulo \ $\mathbb{Z}$ \ of the reduce Bernstein-Sato polynomial of \ $g$. This is rather unexpected result comes from the fact that the self-duality of the Brieskorn (a,b)-module \ $E_g$ \ associated to \ $g$ \ exchanges the biggest simple pole sub-(a,b)-module of \ $E_g$ \ with the saturation of \ $E_g$ \ by \ $b^{-1}a$.
\noindent In the first part of this note, we prove that the biggest simple pole sub-(a,b)-module of the Briekorn (a,b)-module \ $E$ \ of \ $g$ \ is "geometric" in the sense that it depends only on the hypersurface germ \ $\{ g = 0 \}$ \ at the origin in \ $\mathbb{C}^n$ \ and not on the precise choice of the reduced equation \ $g$, as the poles of (*).
By duality, we deduce the same property for the saturation \ $\tilde{E}$ \ of \ $E$. This duality gives also the relation between the "dual" Bernstein-Sato polynomial and the usual one, which is the key of the proof of the theorem.
Mots Clés: isolated hypersurface singularity ; Brieskorn $(a,b)$-module ; Bernstein-Sato polynomial ; dual Bernstein-Sato polynomial
BARLET, Daniel
Sur certaines singularités non isolées d'hypersurfaces II
Prépublication numéro : 2005/42
The aim of the present article is two-fold. First we generalize the
constructions and the finiteness theorem of [B.04 b)] to a much larger
class. Actually every holomorphic germ at the origine of
$\mathbb{C}^{n+1}$ with a one dimensional critical locus $S$ which is
quasi-homogeneous satisfies now our hypothesis.
Moreover it complements our construction of interesting
invariants to compute in order to understand the structure of such germs, by
introducing, aside the Brieskorn's (a,b)-module at the origine and the
(locally constant along $S^* : = S \setminus \{0\}$) sheaf of
(a,b)-modules associated to the transversal singularities along $S^*$, the
(a,b)-modules "with supports" $E_c$ ans $E_{c \cap\, S}$
\footnote{Remark that, even in the case of a germ with an isolated
singularity, the "dual" Brieskorn \ $E_c$ \ was not noticed until the recent
paper [B.05]. In this case \ $E_{c \cap \, S}$ \ co{\'i}ncides with the
usual Brieskorn (a,b)-module \ $E$.}.
An interesting consequence of the local study along $S^*$ is
the corollary showing that for a germ with an isolated
singularity, the largest sub-(a,b)-module having a simple p\^ole in its
Brieskorn-(a,b)-module is independant of the choice of a reduced equation
for the corresponding hypersurface germ. Some consequences of this result
for isolated singularity germs will be given in a forthcoming paper.
We also give precise relations between these various (a,b)-modules via the
exact commutative diagram of corollary.
This is a filtered (a,b)-linear version of the tangling phenomenon for
consecutive strata we have previously studied in the "topological" setting
(see [B.91], [B.02] and [B.04 b)]) for the localized Gauss-Manin system of $f$.
Finally we show that in our situation there exists a non-degenerate (a,b)-sesquilinear pairing
$$h : E \times E_{c\,\cap\, S} \longrightarrow \vert \Xi' \vert^2 $$
where \ $\vert \Xi' \vert^2$ \ is the space of formal asymptotic expansions
at the origine for fiber-integrals.
This generalizes the canonical hermitian form defined in [B.85] for the
isolated singularity case (for the (a,b)-module version see [B.05]). Its
topological analogue (for the eigenvalue one of the monodromy) is the
non-degenerate sesquilinear pairing
$$ \mathcal{H} : H^n_{c\,\cap\,S}(F, \mathbb{C})_{=1} \times H^n(F,
\mathbb{C})_{=1} \to \mathbb{C} $$
defined in [B.04 b)] for an arbitrary germ with a one dimensional critical
locus (and more generally for a germ such that the eigenvalue one of the
monodromy acting on the reduced cohomology of the Milnor' fibers only
appears along a curve).
Mots Clés: hypersurface singularity ; 1 dimensional singular locus ; Brieskorn module ; (a,b)-module ; formal microlocal operators
BARLET, Daniel
Quelques résultats sur certaines fonctions à lieu singulier de dimension 1
Prépublication numéro : 2005/07
This next is a survey on my recent work [B.04] on some holomorphic germs having a one dimensional singular locus. An analoguous of the Brieskorn module of an isolated singularity is defined and a finitness theorem is proved using Kashiwara's constructibility theorem. A bound for the (finite dimensional) torsion is also obtained. Non existence of torsion is proved for curves (reduced or not) and this property is stable by "Thom-Sebastiani" adjunction of an isolated singularity. This provides a lot of examples in any dimension.
Mots Clés: Hypersurface singularity ; 1 dimensional singular locus ; Brieskorn module ; (a,b)-module ; formal microlocal operators.
BARLET, Daniel
Modules de Brieskorn et formes hermitiennes pour une singularité isolée d'hypersurface. (3ème version)
Prépublication numéro : 2005/06
This article intends to give a synthetic survey about the canonical hermitian form of a germ of holomorphic function $f$ with an isolated singularity at the origin in $\Bbb C^{n+1}$. The link between this canonical hermitian form, the hermitian Poincaré duality on the Milnor' fiber of $f$ and the variation map, proved by F. Loeser in [Lo.86], is presented in the (a,b)-module setting with purely local arguments.
Mots Clés: (a,b)-Module ; Brieskorn Module ; Variation ; Canonical Hermitian Form ; Isolated Singularity
2004
BARLET, Daniel & SAITO, Morihiko
Brieskorn modules and Gauss-Manin systems for non isolated hypersurface singularities
Prépublication N° 2004/54
Abstract: We study the Brieskorn modules associated to a function with non isolated singularities, and show that the kernel of the morphism to the Gauss-Manin system coincides with the torsion part for the action of $t$ and also for 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 Brieskorn modules in the case one of two functions has an isolated singularity.
Mots Clés: Brieskorn Modules ; non isolated hypersurface singularitie
BARLET, Daniel
Interaction de strates consécutives pour les cycles évanescents III : le cas de la valeur propre 1
Résumé: Ce texte étudie le cas manquant de notre article [B.91], à savoir le cas de la valeur propre 1. C'est évidemment un cas plus compliqué que celui d'une valeur propre $\neq$ 1 puisque la présence de la strate des points lisses de l'hypersurface $\{f = 0\}$ oblige à considérer trois strates pour les cycles proches. Et l'on sait déjà que cette strate lisse est toujours "emmêlée" en présence d'une autre strate (voir [B.84b] et l'introduction de [B.03]). Le phénomène nouveau est le rôle joué ici par un "nouveau" groupe de cohomologie, noté $H^n_{c \cap S}(F)_{= 1}$, de la fibre de Milnor de $f$ à l'origine, qui est de même dimension que $H^n(F)_{=1}$ et $H^n_c(F)_{=1}$ et qui donne lieu à une factorisation non triviale de l'application canonique $\it can$ : $H^n_c(F)_{=1} \longrightarrow H^n(F)_{=1}$, et à un isomorphisme monodromique de variation $\it var$ : $H^n_{c \cap S}(F)_{= 1} \longrightarrow H^n_c(F)_{=1}$. On en déduit une forme hermitienne canonique
qui est non générée. Ceci généralise le cas d'une singularité isolé pour la valeur propre 1 (voir [B.90] et [B.97] ). Le phénomène de "suremmêlement" de strates pour la valeur 1 donne l'existence de pôles triples aux entiers négatifs (assez grands en valeur absolue) pour le prolongement méromorphe de la distribution $ \int_X \arrowvert f \arrowvert ^{2 \lambda} \square$ en présence de monodromies locales semi-simples pour $f$ en chaque point du lieu singulier de $\{f = 0\}$.
Mots Clés: Hypersurface ; Singularité non isolée ; Cycles Évanescents ; Emmêlement de Strates
BARLET, Daniel
Module de Brieskorn et forme hermitienne canonique pour une singularité isolée d'hypersurface
Abstract: This note intend to give a synthetic survey about the canonical hermitian form of a germ of holomorphic function $f$ with an isolated singularity at the origin in $\mathbb C ^{n+1}$. The link between this canonical hermitian form, the hermitian Poincaré duality on the Milnor ' fiber of $f$ and the variation map, proved by F. Loeser in [Lo.86], is presented in the (a,b)-module setting with purely local arguments.
Mots Clés: (a,b)-Module ; Brieskorn module ; Variation ; Canonical Hermitian Form ; Isolated Singularity
BARLET, Daniel
Sur certaines singularités non isolées d'hypersurfaces I
Abstract: The aim of this first part is to introduce, for an 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 reduced) and for the suspension of such non torsion cases with an isolated singularity.
2003
BARLET, Daniel; MAGNUSSON, Jon (à paraitre au Asian Journal of Math. 2004)
Integration of meromorphic cohomology classes and applications
Résumé: The main purpose of this article is to increase the efficiency of the tools introduced in [B.Mg. 98] and [B.Mg. 99], namely integration of meromorphic cohomology classes, and to generalize the results of [B.Mg. 99]. They describe how positively conditions on the normal bundle of a compact complex submanifold $Y$ of codimension $n+1$ in a complex manifold $Z$ can be transformed into positivity conditions for a Cartier divisor in a space parametrizing $n$-cycles in $Z$. As an application of our results we prove that the following problem has a positive answer in many cases : Let $Z$ be a compact connected complex manifold of dimension $n+p$. Let $Y \subset Z$ a submanifold of $Z$ of dimension $p - 1$ whose normal bundle $N_{Y \mid Z}$ is (Griffiths) positive. We assume that there exists a covering analyic family $(X_s)_{s \in S}$ of compact $n$-cycles in $Z$ parametrized by a compact normal complex space $S$. In the algebraic dimension of $Z \geq p$ ?
2002
BARLET, Daniel; KADDAR, Mohamed (Int. Journ. of Math. vol.14,4 (2003))
Incidence divisor (anglais corrigé, décembre 2002)
BARLET, Daniel
Singularités réelles isolées et développements asymptotiques d'intégrales oscillantes
Abstract: Let $(X_{\mathbb{R}},0)$ be a germ of real analytic subset in $(\{\mathbb{R}^N,0) of pure dimension $n+1$ with an isolated singularity at 0. Let $(f_{\mathbb{R}},0) : (X_{\mathbb{R}},0) \longrightarrow (\mathbb{}R,0) a real analytic germ with an isolated singularity at 0, such that its complexification $f_{\mathbb{C}}$ vanishes on the singular set $S$ of $X_{\mathbb{C}}$. We also assume that $X_{\mathbb{R}} -\{0\}$ is orientable. To each $A \in H^0(X_{\mathbb{R}} - \{0\},\mathbb{C})$ we associate a $n$-cycle $\Gamma(A)$ ("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 $\int_Ae^{irf(x)}\varphi(x)$ when $\tau \righttarrow \pm \infty$ can be read from the spectral decomposition of $\Gamma(A)$ relative to the monodromy of $f_{\mathbb{C}}$ at 0.
BARLET, Daniel
Intéraction de strates consécutives II (to appear in Publi. of RIMS (Kyoto)
Résumé: Cet article poursuit et complète l'étude commencée dans [B.91] du phénomène d'intéraction de strates consécutives pour les cycles evanescents d'un germe de fonction holomorphe à l'origine de $\Bbb C^{n+1}$. On y considère le cas où la valeur propre $e^{-2i \pi u} \neq 1$ de la monodromie n'apparaît que le long d'une courbe $S$ du milieu singulier de $(f = 0)$ et on y étudie le phénomène d'emmêlement pour toute classe $e$ donnée dans le sous-espace spectral pour la valeur propre $e^{-2i \pi u}$ de la monodromie agissant sur le n-ième groupe de cohomologie de la fibre de Milnor de $f$ en 0. Seul le cas des blocs de Jordan de taille $\geq k_0$ où $k_0$ est l'ordre de nilpotence de la monodromie agissant sur $H^{n-1} (\psi_f(u))$ était considéré dans [B.91] (ici $\psi_f$ désigne le complexe des cycles évanescents de $f$ et $psi_f(u)$ le sous-complexe spectral pour la valeur propre $e^{-2i \pi u}$ de la monodromie). On y montre que ce phénomène d'emmêlement, quand il se produit est toujours détectable par des pôles "inattendus" en $-u+\Bbb Z$ liés à la classe $e$ dans le prolongement méromorphe de $\mid f \mid^{2 \lambda}$ et on donne des critères en termes topologiques pour calculer effectivement ces pôles "exédentaires".