We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\d... more 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.
Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X... more Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X_{\mathbb C} \subset {\mathbb C}^{N}$ is normal with an isolated singularity at $0$. Let $f_{\mathbb R} : X_{\mathbb R} \rightarrow {\mathbb R}$ a real analytic function such that its complexification $f_{\mathbb C} : X_{\mathbb C} \rightarrow {\mathbb C}$ has an isolated singularity at $0$ in $X_{\mathbb
This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 c... more This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 case. Of course this is a more involved situation because the existence of the smooth stratum for the hypersurface {f = 0} forces to consider three strata for the nearby cycles. And we already know that the smooth stratum is
Existence of oblique polar lines for the meromorphic extension of the current valued function $\i... more 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
The main purpose of this article is to increase the efficiency of the tools introduced in [B.Mg. ... more 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 positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n + 1 in a complex manifold Z
Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated... more 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.
In complex geometry, the use of n-convexity and the use of ampleness of the normal bundle of a d-... more In complex geometry, the use of n-convexity and the use of ampleness of the normal bundle of a d-codimensional submanifold are quite difficult for n > 0 and d > 1. The aim of this paper is to explain how some constructions on the cycle space (the Chow variety in the quasiprojective setting) allows one to pass from the n-convexity of Z to the 0-convexity of Cn(Z) and from a (n + 1)-codimensional submanifold of Z having an ample normal bundle to a Cartier divisor of Cn(Z) having the same property. We illustrate the use of these tools with some applications.
Publications of the Research Institute for Mathematical Sciences, 1999
Nous donnons une condition necessaire et suffisante topologique sur A E H°({f / 0},<C), pour un g... more Nous donnons une condition necessaire et suffisante topologique sur A E H°({f / 0},<C), pour un germe analytique reel / : (IR" +1 ,0) -> (IR,0), dont la complexifiee presente une singularite isolee relativement a la valeur propre 1 de la monodromie, pour que le prolongement analytique de J^ / A D presente un pole multiple aux entiers negatifs assez "grands". On montre en particulier que si un tel pole multiple existe, il apparait deja pour 1 = -(n + 1) avec 1'ordre maximal que nous calculons topologiquement.
We show in this note that for a germ $g$ of holomorphic function with an isolated singularity at ... more 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 | g |^{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 $b^{-1}a$. 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 pole...
We compute the Gauss-Manin differential equation for any period of a polynomial in \ $\C[x_{0},\d... more 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.
Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X... more Let $X_{\mathbb R} \subset {\mathbb R}^{N}$ a real analytic set such that its complexification $X_{\mathbb C} \subset {\mathbb C}^{N}$ is normal with an isolated singularity at $0$. Let $f_{\mathbb R} : X_{\mathbb R} \rightarrow {\mathbb R}$ a real analytic function such that its complexification $f_{\mathbb C} : X_{\mathbb C} \rightarrow {\mathbb C}$ has an isolated singularity at $0$ in $X_{\mathbb
This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 c... more This text is a study of the missing case in our article [B.91], that is to say the eigenvalue 1 case. Of course this is a more involved situation because the existence of the smooth stratum for the hypersurface {f = 0} forces to consider three strata for the nearby cycles. And we already know that the smooth stratum is
Existence of oblique polar lines for the meromorphic extension of the current valued function $\i... more 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
The main purpose of this article is to increase the efficiency of the tools introduced in [B.Mg. ... more 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 positivity conditions on the normal bundle of a compact complex submanifold Y of codimension n + 1 in a complex manifold Z
Let (X_R, 0) be a germ of real analytic subset in (R^N, 0) of pure dimension n+1 with an isolated... more 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.
In complex geometry, the use of n-convexity and the use of ampleness of the normal bundle of a d-... more In complex geometry, the use of n-convexity and the use of ampleness of the normal bundle of a d-codimensional submanifold are quite difficult for n > 0 and d > 1. The aim of this paper is to explain how some constructions on the cycle space (the Chow variety in the quasiprojective setting) allows one to pass from the n-convexity of Z to the 0-convexity of Cn(Z) and from a (n + 1)-codimensional submanifold of Z having an ample normal bundle to a Cartier divisor of Cn(Z) having the same property. We illustrate the use of these tools with some applications.
Publications of the Research Institute for Mathematical Sciences, 1999
Nous donnons une condition necessaire et suffisante topologique sur A E H°({f / 0},<C), pour un g... more Nous donnons une condition necessaire et suffisante topologique sur A E H°({f / 0},<C), pour un germe analytique reel / : (IR" +1 ,0) -> (IR,0), dont la complexifiee presente une singularite isolee relativement a la valeur propre 1 de la monodromie, pour que le prolongement analytique de J^ / A D presente un pole multiple aux entiers negatifs assez "grands". On montre en particulier que si un tel pole multiple existe, il apparait deja pour 1 = -(n + 1) avec 1'ordre maximal que nous calculons topologiquement.
We show in this note that for a germ $g$ of holomorphic function with an isolated singularity at ... more 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 | g |^{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 $b^{-1}a$. 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 pole...
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