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Daniel Barlet

Publications and source records attributed to Daniel Barlet.

At least 19 recordsLinked to original sources
arXiv2026

The $\mu$-Trace System

We study a simple 1-parameter perturbation of the regular holonomic Trace System satisfied by a complex power of the root of the universal polynomial of degree k as a holomorphic function of the coefficients. We prove that these systems have many analogous properties than the Trace System studied in [4] and we prove that they are, in general, minimal extensions of a simple pole meromorphic connection on a rank $k$ trivial bundle on $\mathbb{C}^k$. We also examine the structure of these $D$-modules for the special values of the parameters. This explicites many examples of perverse sheaves associated to representations of the $\pi_1$ of the complement of the hyper-surface $\{\sigma_k\Delta(\sigma) = 0\}$ in the affine space with coordinates $\sigma_1,\ldots,\sigma_k$, where $\Delta(\sigma)$ is the discriminant of the universal monic polynomial of degree $k$, $P_\sigma(z) := z^k + \sum_{h=1}^k (-1)^h \sigma_h z^{k-h}$.

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arXiv2025

Generalized Brieskorn Modules III. The algebra $\tilde{\mathcal{A}}\_{conv.}$

In this paper we introduce and study the ''convergent'' algebra (containing ''a'' and ''b'' and acting on holomorphic germs in ''a'') which naturally acts on the ''generalized Brieskorn modules'' associated to the Gauss-Manin connections of the germs at each point of the singular set of a holomorphic function on a complex manifold. We generalize to this convergent setting the results previously obtained (see 8], [9], [15] and [16]) in the formal case, and we show that, in suitable global situations (for instance when f is projective) we obtain also generalized (geometric) Brieskorn modules. So the question of the relationship between the left module structure on this algebra (which defines several interesting filtrations) and the mixte Hodge structure is raised

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arXiv2025

Generalized Brieskorn Modules II: Higher Bernstein Polynomials and Multiple Poles

Our main result is to show that the existence of a root in. --$\alpha$--Nfor the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphicform in the (convergent) Brieskorn (a,b)-module associated to f, under the hypothesis that f has an isolated singularity at the origin relative to the eigenvalue exp(2i$\pi$$\alpha$) of the monodromy, produces poles of order at least p for themeromorphic extension of the (conjugate) analytic functional given by polar partsat points--$\alpha$--N for N well chosen integer. This result is new, even forp= 1. As a corollary, this implies that, in the case of an isolated singularity for f,the existence of a root in. --$\alpha$--N for the p-th Bernstein polynomial of the (a,b)-module generated by a holomorphic form implies the existence of at leastp roots (counting multiplicities) for the usual reduced Bernstein polynomial of thegerm of f at the origin.In the case of an isolated singularity for f, we obtain that for each $\alpha$ thebiggest root --$\alpha$--m. of the reduced Bernstein polynomial of f in --$\alpha$--N producesa pole at--$\alpha$--m for the meromorphic extension of the associated distribution

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arXiv2024

Stable properties under weakly geometrically flat maps

In this note we show that a weakly geometrically flat map $\pi$ : M $\rightarrow$ N between pure dimensional complex spaces has the local lifting property for cycles. From this result we also deduce that, under these hypotheses, several properties of M are transferred to N.

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arXiv2024

Some comments on my study of period-integrals

This text is a presentation (without proofs) of some of my recent results on the singular terms of asymptotic expansions of period-integrals using (a,b)-modules. I try to explain why this simple algebraic structure is interesting and useful.

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arXiv2023

Complement to Higher Bernstein Polynomials and Multiple Poles of 1 $\Gamma$($\lambda$) X |f | 2$\lambda$ f --h $\rho$$\omega$ $\land$ $\omega$'

We give, using higher Bernstein polynomials defined in our paper [2], a stronger version of our previous result in [1] whose converse is proved in [2] and we give some complements to the results in [2] which help to compute these higher order Bernstein polynomials. Then we show some non trivial examples where we determine the root of the second Bernstein polynomial which is not a double root of the full Bernstein polynomial and where the main theorem of [2] applies and localizes where a double pole exists for the meromorphic extension of the (conjugate) analytic functional given by polar parts of $\omega$ ' $\rightarrow$ |f | 2$\lambda$ f --h $\rho$$\omega$ $\land$ $\omega$' when h $\in$ N is large enough.

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arXiv2023

Generalized Brieskorn Modules I: Convergent (a,b)-modules

This paper is the first one of two papers whose goal is to give a converse to the main result of my previous paper [6], so to prove the existence of multiple poles for the distribution |f|2$\lambda$ with an hypothesis on a Higher Bernstein Polynomial of the (a,b)-module generated by the germ $\omega$$\in$$\Omega$n+1 0 of a given holomorphic volum form. Note that, even for the existence of a simple pole this converse is already new. One difficulty to prove such a result comes from the use of the formal completion in f of the Brieskorn module of the holomorphic germ f\,: (Cn+1 ,0) $\rightarrow$(C,0) which does not give access to the cohomology of the Milnor's fiber of f, which by definition, is outside {f = 0}. This leads to introduce generalized Brieskorn modules (convergent geometric (a,b)-modules) which allow this passage. The first aim of this part I is to give a solid basis of the theory of convergent (a,b)-modules. In order to take in account Jordan blocs of the monodromy in our results we introduce the semi-simple filtration of a generalized Brieskorn module (convergent (a,b)-module) and we shall use it to define in part II the higher order Bernstein polynomials in this context. They correspond to a decomposition of the ``standard'' Bernstein polynomial of a generalized Brieskorn module, taking in account the nilpotent order of the monodromy. In this part I we obtain also a full description of generalized Brieskorn-modules in terms of (convergent) asymptotics expansions of Nilsson class which will be used as a starting point in part II. We conclude this part I by making explicite the relationship between the semi-simple filtration of a generalized Brieskorn module E and the nilpotent filtration of the monodromy on its saturation E___ .

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arXiv2023

Cycles of finite type

The aim of this book is to show that the use of f-analytic families of finite type cycles (cycles having finitely many irreducible components, but not compact in general) in a given complex space may be useful in complex geometry, despite the fact that the corresponding functor is not, in general, representable, in contrast to the compact case. This study leads to the notion of strongly quasi-proper map which is characterized by the existence of a geometric f-flattening which is a generalization of the Geometric Flattening Theorem for proper holomorphic maps. As applications we prove an existence theorem for meromorphic quotients of reduced complex spaces and a generalization of the classic Stein factorization.

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arXiv2022

On Principal Value and Standard Extension of Distributions

For a holomorphic function f on a complex manifold M we explain in this article that the distribution associated to |f | 2$\alpha$ (Log|f | 2) q f --N by taking the corresponding limit on the sets {|f | $\ge$ $\epsilon$} when $\epsilon$ goes to 0, coincides for ($\alpha$) non negative and q, N $\in$ N, with the value at $\lambda$ = $\alpha$ of the meromorphic extension of the distribution |f | 2$\lambda$ (Log|f | 2) q f --N. This implies that any distribution in the D Mmodule generated by such a distribution has the Standard Extension Property. This implies a non torsion result for the D M-module generated by such a distribution. As an application of this result we determine generators for the conjugate modules of the regular holonomic D-modules associated to z($\sigma$) $\lambda$ , the power $\lambda$, where $\lambda$ is any complex number, of the (multivalued) root of the universal equation of degree k, z k + k j=1 (--1) h $\sigma$ h z k--h = 0 whose structure is studied in [4].

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arXiv2021

Why (a,b)-modules, frescos and themes ?

This text is an advocacy for the use of (a,b)-modules (formal Brieskorn modules) in the study of the singularity of a holomorphic function on a complex manifold. It gives a short an elementary overview on this simple algebraic tool.

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arXiv2021

Algebraic differential equations of periods integrals

We explain that in the study of the asymptotic expansion at the origin of a period integral like $\gamma$z $\omega$/df or of a hermitian period like f =s $\rho$.$\omega$/df $\land$ $\omega$ /df the computation of the Bernstein polynomial of the "fresco" (filtered differential equation) associated to the pair of germs (f, $\omega$) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f $\in$ C[x 0 ,. .. , x n ] has n + 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when $\omega$ is a monomial holomorphic volume form. Several concrete examples are given.

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arXiv2021

On partial differential operators which annihilate the roots of the universal equation of degree k

The aim of this paper is to study in details the regular holonomic $D-$module introduced in \cite{[B.19]} whose local solutions outside the polar hyper-surface $\{\Delta(\sigma).\sigma_k = 0 \}$ are given by the local system generated by the local branches of the multivalued function which is the root of the universal degree $k$ equation $z^k + \sum_{h=1}^k (-1)^h.\sigma_h.z^{k-h} = 0 $. Note that it is surprising that this regular holonomic $D-$module is given by the quotient of $D$ by a left ideal which has very simple explicit generators despite the fact it necessary encodes the analogous systems for any root of the universal degree $l$ equation for each $l \leq k$. Our main result is to relate this $D-$module with the minimal extension of the irreducible local system associated to the difference of two branches of the multivalued function defined above. Then we obtain again a very simple explicit description of this minimal extension in term of the generators of its left ideal in the Weyl algebra. As an application we show how these results allow to compute the Taylor expansion of the root near $-1$ of the equation $z^k + \sum_{h=-1}^k (-1)^h.\sigma_h.z^{k-h} - (-1)^k = 0 $.

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arXiv2020

New singularity invariants : the sheaf $\beta_X^\bullet$

The graded coherent sheaf $\alpha_X^\bullet$ constructed in [B.18] for any reduced pure dimensional complex space $X$ is stable by exterior product but not by the de Rham differential. We construct here a new graded coherent sheaf $\alpha_X^\bullet$ containing $\alpha_X^\bullet$ and stable both by exterior product and by the de Rham differential. We show that it has again the ``pull-back property'' for holomorphic maps $f : X \to Y$ between irreducible complex spaces such that $f(X)$ is not contained in the singular set of $Y$. Moreover, this graded coherent sheaf $\alpha_X^\bullet$ comes with a natural coherent exhaustive filtration and this filtration is also compatible with the pull-back by such holomorphic maps. These sheaves define new invariants on singular complex spaces. We show on some simple examples that these invariants are new.

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arXiv2019

On symmetric partial differential operators

Let s 1 ,. .. , s k be the elementary symmetric functions of the complex variables x 1 ,. .. , x k. We say that F $\in$ C[s 1 ,. .. , s k ] is a trace function if their exists f $\in$ C[z] such that F (s 1 ,. .. , s k ] = k j=1 f (x j) for all s $\in$ C k. We give an explicit finite family of second order differential operators in the Weyl algebra W 2 := C[s 1 ,. .. , s k ] $\partial$ $\partial$s 1 ,. .. , $\partial$ $\partial$s k which generates the left ideal in W 2 of partial differential operators killing all trace functions. The proof uses a theorem for symmetric differential operators analogous to the usual symmetric functions theorem and the corresponding map for symbols. As a corollary, we obtain for each integer k a holonomic system which is a quotient of W 2 by an explicit left ideal whose local solutions are linear combinations of the branches of the multivalued root of the universal equation of degree k: z k + k h=1 (--1) h .s h .z k--h = 0.

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arXiv2019

On Lisbon integrals

We introduce new complex analytic integral transforms, the Lisbon Integrals, which naturally arise in the study of the affine space $\mathbb{C}^k$ of unitary polynomials $P_s(z)$ where $s\in\mathbb{C}^k$ and $z\in \mathbb{C}$, $s_i$ identified to the $i-$th symmetric function of the roots of $P_s(z)$. We completely determine the $\mathcal{D}$-modules (or systems of partial differential equations) the Lisbon Integrals satisfy and prove that they are their unique global solutions. If we specify a holomorphic function $f$ in the $z$-variable, our construction induces an integral transform which associates a regular holonomic module quotient of the sub-holonomic module we computed. We illustrate this correspondence in the case of a $1$-parameter family of exponentials $f_t(z) = exp(t z)$ with $t$ a complex parameter.

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arXiv2017

On the nearly smooth complex spaces

We introduce a class of normal complex spaces having only mild sin-gularities (close to quotient singularities) for which we generalize the notion of a (analytic) fundamental class for an analytic cycle and also the notion of a relative fundamental class for an analytic family of cycles. We also generalize to these spaces the geometric intersection theory for analytic cycles with rational positive coefficients and show that it behaves well with respect to analytic families of cycles. We prove that this intersection theory has most of the usual properties of the standard geometric intersection theory on complex manifolds, but with the exception that the intersection cycle of two cycles with positive integral coefficients that intersect properly may have rational coefficients. AMS classification. 32 C 20-32 C 25-32 C 36.

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