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Daniel Barlet

Publications and source records attributed to Daniel Barlet.

At least 37 records · Page 2Linked to original sources
arXiv2017

The sheaf $α$ $\bullet$ X

We introduce in a reduced complex space, a "new coherent sub-sheaf" of the sheaf $ω\_{X}^{\bullet}$ which has the "universal pull-back property" for any holomorphic map, and which is in general bigger than the usual sheaf of holomorphic differential forms $Ω\_{X}^{\bullet}/torsion$. We show that the meromorphic differential forms which are sections of this sheaf satisfy integral dependence equations over the symmetric algebra of the sheaf $Ω\_{X}^{\bullet}/torsion$. This sheaf $α\_{X}^{\bullet}$ is also closely related to the normalized Nash transform. We also show that these $q-$meromorphic differential forms are locally square-integrable on any $q-$dimensional cycle in $X$ and that the corresponding functions obtained by integration on an analytic family of $q-$cycles are locally bounded and locally continuous on the complement of closed analytic subset.

math.AG↗
arXiv2017

Gauduchon's form and compactness of the space of divisors

We show that in a holomorphic family of compact complex connected manifolds parametrized by an irreducible complex space $S$, assuming that on a dense Zariski open set $S^{*}$ in $S$ the fibres satisfy the $\partial\bar\partial-$lemma, the algebraic dimension of each fibre in this family is at least equal to the minimal algebraic dimension of the fibres in $S^{*}$. For instance, if each fibre in $S^{*}$ are Moishezon, then all fibres are Moishezon.

math.CV↗
arXiv2016

Note on the semi-continuity of the algebraic dimension

In this short Note we show that the direct image sheaf R 1 $π$ * (O X) associated to an analytic family of compact complex manifolds $π$ : X $\rightarrow$ S parametrized by a reduced complex space S is a locally free (coherent) sheaf of O S --modules. This result allows to improve a semi-continuity type result for the algebraic dimension of compact complex manifolds in an analytic family given in [B.15]. AMS Classification 2010. 32G05-32A20-32J10.

math.CV↗
arXiv2015

A note on some fiber-integrals

We remark that the study of a fiber-integral of the type F (s) := f =s ($ω$/df) $\land$ ($ω$/df) either in the local case where $ρ$ $\not\equiv$ 1 around 0 is C $\infty$ and compactly supported near the origin which is a singular point of {f = 0} in C n+1 , or in a global setting where f : X $\rightarrow$ D is a proper holomorphic function on a complex manifold X, smooth outside {f = 0} with $ρ$ $\not\equiv$ 1 near {f = 0}, for given holomorphic (n+1)--forms $ω$ and $ω$' , that a better control on the asymptotic expansion of F when s $\rightarrow$ 0, is obtained by using the Bernstein polynomial of the "frescos" associated to f and $ω$ and to f and $ω$' (a fresco is a "small" Brieskorn module corresponding to the differential equation deduced from the Gauss-Manin system of f at 0) than to use the Bernstein polynomial of the full Gauss-Manin system of f at the origin. We illustrate this in the local case in some rather simple (non quasi-homogeneous) polynomials, where the Bernstein polynomial of such a fresco is explicitly evaluate. AMS Classification. 32 S 25, 32 S 40. Key words. Fiber-integrals @ Formal Brieskorn modules @ Geometric (a,b)-modules @ Frescos @ Gauss-Manin system.

math.CV↗
arXiv2015

Holomorphic families of $[λ]-$primitive themes

This article is the continuation of [B. 13-b] where we show how the isomorphism class of a $[λ]-$primitive theme with a given Bernstein polynomial may be characterized by a (small) finite number of complex parameters. We construct here a corresponding locally versal holomorphic deformation of $ [λ]-$primitive themes for each given Bernstein polynomial. Then we prove the universality of the corresponding "canonical family" in many cases. We also give some examples where no local universal family exists.

math.CV↗
arXiv2015

Meromorphic quotients for some holomorphic G-actions

Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ \ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good for the $B-$action, the existence of a strongly quasi-proper meromorphic quotient for the $G-$action on $X$.

math.AG↗
arXiv2015

Strongly quasi-proper maps and the f-flattening theorem

We complete and precise the results of [B.13] and we prove a strong version of the semi-proper direct image theorem with values in the space C f n (M) of finite type closed n--cycles in a complex space M. We describe the strongly quasi-proper maps as the class of holomorphic surjective maps which admit a meromorphic family of fibers and we prove stability properties of this class. In the Appendix we give a direct and short proof of D. Mathieu's flattening theorem (see [M.00]) for a strongly quasi-proper map which is easier and more accessible.

math.CV↗
arXiv2015

Some examples due to H. Hironaka

The aim of this paper is to explain the construction by H. Hironaka [H.61] of a holomorphic (in fact "algebraic") family of compact complex manifolds parametrized by $\C$ such for all $s \in \C\setminus \{0\}$ the fiber is projective, but such that the fiber at the origin in non k{ä}hlerian, to mathematicians which are not algebraic geometers. We also explain why it is not possible to make in the same way such an example with fiber at $0$ a simpler example of non k{ä}hlerian Moishezon manifold which is also due to H. Hironaka (see section 5).

math.AG↗
arXiv2014

Algebraic differential equations associated to some polynomials

We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\dots, x_{n}]$ \ with \ $(n+2)$ \ monomials. We give two general factorizations theorem in the algebra \ $\C< z, (\frac{\partial}{\partial z})^{-1}>$ \ for such a differential equations.

math.AG↗
arXiv2013

Asymptotics of a vanishing period : characterization of semi-simplicity

In this paper we introduce the word {\em fresco} to denote a monogenic geometric (a,b)-module. This "basic object" (generalized Brieskorn module with one generator) corresponds to the formal germ of the minimal filtered (regular) differential equation. Such an equation is satisfied by a relative de Rham cohomology class at a critical value of a holomorphic function on a smooth complex manifold. In [B.09] the first structure theorems are proved. Then in [B.10] we introduced the notion of {\em theme} which corresponds in the \ $[λ]-$primitive case to frescos having a unique Jordan-H{ö}lder sequence (a unique Jordan block for the monodromy). Themes correspond to asymptotic expansion of a given vanishing period, so to an 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. \\ We show that for any fresco there exists an {\em unique} Jordan-H{ö}lder sequence, called the {\em principal J-H. sequence}, with corresponding quotients giving the opposite of the roots of the Bernstein polynomial in increasing order. We study the semi-simple part of a given fresco and we characterize the semi-simplicity of a fresco by the fact for any given order on the roots of its Bernstein polynomial we may find a J-H. sequence making them appear with this order. Then we construct a numerical invariant, called the \ $β-$invariant, and we show that it produces numerical criteria in order to give a necessary and sufficient condition on a fresco to be semi-simple. We show that these numerical invariants define a natural algebraic stratification on the set of isomorphism classes of fresco with given fundamental invariants (or equivalently with given roots of the Bernstein polynomial).

math.AG↗
arXiv2012

Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version)

In this paper we consider holomorphic families of frescos (i.e. filtered differential equations with a regular singularity) and we construct a locally versal holomorphic family for every fixed Bernstein polynomial. We construct also several holomorphic parameters (a holomorphic parameter is a function defined on a set of isomorphism classes of frescos) which are quasi-invariant by changes of variable. This is motivated by the fact that a fresco is associated to a relative de Rham cohomology class on a one parameter degeneration of compact complex manifolds, up to a change of variable in the parameter. Then the value of a quasi-invariant holomorphic parameter on such data produces a holomorphic (quasi-)invariant of such a situation.

math.AG↗
arXiv2012

Asymptotics of a vanishing period : General existence theorem and basic properties of frescos

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. The first part of this paper shows that, for any \ $[λ]-$primitive fresco there exists an unique Jordan-H{ö}lder sequence (called the principal J-H. sequence) with corresponding quotients giving the opposite of the roots of the Bernstein polynomial in a non decreasing order. Then we introduce and study the semi-simple part of a given fresco and we characterize the semi-simplicity of a fresco by the fact for any given order of the roots of its Bernstein polynomial we may find a J-H. sequence making them appear with this order. Then, using the parameter associated to a rank \ 2 \ \ $[λ]-$primitive theme, we introduce inductiveley a numerical invariant, that we call the \ $α-$invariant, which depends polynomially on the isomorphism class of a fresco (in a sens which has to be defined) and which allows to give an inductive way to produce a sub-quotient rank \ 2 \ theme of a given \ $[λ]-$primitive fresco assuming non semi-simplicity. In the last section 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. This is, of course, the motivation for the study of frescos.

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