Asymptotics of a vanishing period

Let begin by the definition of a vanishing period. Let $X$ be a complex connected manifold of dimension $n + 1$ and $f : X \to D$ be a holomorphic function with value in a disc $D\subset\mathbb{C}$ with center $0$. Assume\footnote{if the hypersurface $\{f=0\}$ in $X$ is reduced, this is always satisfied, up to localize around $0$ in $D$.} that the critical set $S := \{df = 0\}$ is a closed subset in $\{f = 0\}$ with no interior point in $\{f = 0\}$. Let $\omega$ be a $(p + 1)-\mathscr{C}^{\infty}-$differential form on $X$ such that $d\omega = 0 = df \wedge \omega$, and let $\gamma\in H_{p} (X_{s_0},\mathbb{C})$ be a vanishing $p-$cycle on the generic fiber $X_{s_0}$ of $f$. Then we call \textbf{a vanishing period} for $f$ the multivalued\footnote{so well defined and holomorphic on the universal cover $\exp: H\to D^{\ast}$} holomorphic function on $D^\ast$, defined as \begin{equation*} F_{\gamma}(s):=\int_{\gamma_s}\frac{\omega}{df} \end{equation*} where $(\gamma_s)_{s\in H}, \gamma_s\in H_p(X_s,\mathbb{C})$ is the horizontal family of $p-$cycles in the fibers of $f$ whose value\footnote{in fact at some base point in $H$ choosen over $s_0$.} at $s_0$ is $\gamma$. It is known that such a function $F_{\gamma}$ is solution of regular singular meromorphic linear differential system around $s = 0$ which is the Gauss-Manin connexion of $f$. In fact, for such a differential form, there exists a minimal filtered regular singular differential equation which annihilates the corresponding function $F_{\gamma}$ for any choice of the vanishing cycle $\gamma$. Let me explain what we mean by ``filtered regular singular differential equation". Let $\tilde{A} := \Sigma^{\infty}_{\nu=0} P_{\nu}(a). b^{\nu}$, where $P_{\nu}$ are polynomials, the $\mathbb{C}-$algebra whose product is defined by the following two conditions : \begin{enumerate} \item The commutation relation $a.b - b.a = b^2$. \item The left and right multiplication by $a$ are continuous for the $b-$adic topology in $\tilde{\mathcal{A}}$. \end{enumerate} Then there exists an integer $k\geq 1$ and a $P\in\tilde{\mathcal{A}}$ which is a monic polynomial of degree $k$ in a with coefficients in $\mathbb{C}[[b]]$, such its initial form in $(a, b)$ is homogeneous of degree $k$ and such that $P.F_{\gamma} = 0$ for any choice of a vanishing cycle $\gamma$. Here we let $\tilde{\mathcal{A}}$ acts on (multivalued) holomorphic functions on $D^{\ast}$ via $a := \times s$ and $b := \int^s_0$. This corresponds to a special class of left $\tilde{\mathcal{A}}$ modules $E :=\tilde{\mathcal{A}} / \tilde{\mathcal{A}}.P$ which are generated by one element on $\tilde{\mathcal{A}}$, are free and finite type over the sub-algebra $\mathbb{C}[[b]]$, (rank $k$ for $E$ associated to $P$), with an extra condition (called ``geometric") which encodes at the same time the regularity of the Gauss-Manin connexion, the monodromy theorem and the positivity theorem of B. Malgrange. Such an object will be called a \textbf{fresco}. It is defined from the choice of $\omega$. Now each choice of a vanishing cycle $\gamma$ will produce, via the expansion of the function $F_{\gamma}$, an $\tilde{\mathcal{A}}-$linear map $\varphi_{\gamma}: E_{\omega}\to\Xi$ where \begin{equation*} \Xi:=\sum_{\lambda\in\mathbb{Q}^{+}, j\in\mathbb{N}} \mathbb{C}[[b]].s^{\lambda-1}.(\log s)^{j}\equiv\sum_{\lambda\in\mathbb{Q}^{+},j\in\mathbb{N}}\mathbb{C}[[a]].s^{\lambda-1}.(\log s)^j \end{equation*} is the $\tilde{\mathcal{A}}-$module of (formal) multiform asymptotic expansions, where $a$ and $b$ acts again by multiplication by $s$ and integration without constant. The image in $\Xi$ by a $\tilde{\mathcal{A}}-$linear map of a fresco is again a fresco, but it belongs to a special subclass of fresco we call \textbf{theme}. Choosing the cycle $\gamma$ in the generalized eigenspace of the monodromy for the eigenvalue $\exp(-2i\pi.\lambda)$ we obtain a $[\lambda]$-primitive theme. This corresponds to Jordan blocs of the monodromy associated to $\omega$ and $\gamma$. The structure of themes is rather simple and, as they corresponds to logarithmic terms of the asymptotic expansions of the vanishing periods, it is rather important to know what are the possible quotient themes for a given fresco. In these talks we shall first recall the classical theory of Brieskorn modules for an holomorphic function in $\mathbb{C}^{n+1}$ with an isolated singularity and we shall explain how this generalizes for a function with arbitrary singularities with sheaves of $\tilde{\mathcal{A}}-$modules. Then we shall give some general results on the structure of fresco and explain how to compute the Bernstein polynomial of a fresco. In the last part we shall describe some general theorems on themes and sketch the theory of holomorphic families of $[\lambda]-$primitive themes and show how to build a versal family of $[\lambda]-$primitive themes with given Bernstein polynomial. \vspace{1cm} My recent works on this subject. \begin{itemize} \item D. Barlet, Sur certaines singularit\'es non isol\'ees d’hypersurfaces II J. Alg. Geom. 17 (2008), p.199-254. \item D. Barlet, P\'eriodes \'evanescentes et (a,b)-modules monog\`enes, Boll. U.M.I. (9) II (2009), p.651-697. \item D. Barlet, Le th\`eme d’une p\'eriode \'evanescente, preprint Inst. E. Cartan (Nancy) 2009 n$^{\textrm{o}}$33, 57 pages. \item D. Barlet, Changement de variable pour un th\`eme, preprint Inst. E. Cartan (Nancy) 2010 n$^\textrm{o}$09, 19 pages. \end{itemize}