The theme of a vanishing period

Abstract

Let \ $\lambda \in \mathbb{Q}^{*+}$ \ and consider a multivalued formal function of the type $$ \phi(s) : = \sum_{j=0}^k \ c_j(s).s^{\lambda + m_j}.(Log\, s)^j $$ where \ $c_j \in \C[[s]], m_j \in \mathbb{N}$ \ for \ $j \in [0,k-1]$. The {\bf theme} associated to such a \ $\phi$ \ is the "minimal filtered differential equation" with generator \ $\phi$, in a sens which is made precise in this article. We study such objects and show that their isomorphism classes may be characterized by a finite set of complex numbers, when we assume the Bernstein polynomial fixed. For a given \ $\lambda$, to fix the Bernstein polynomial is equivalent to fix a finite set of integers associated to the logarithm of the monodromy in the geometric stuation described above. Our purpose is to construct some analytic invariants, for instance in the following situation : Let \ $f : X \to D$ \ be a proper holomorphic function defined on a complex manifold \ $X$ \ with value in a disc \ $D$. We assume that the only critical value is \ $0 \in D$ \ and we consider this situation as a degenerating family of compact complex manifolds to a singular compact complex space \ $f^{-1}(0)$. To a smooth \ $(p+1)-$form \ $\omega$ \ on \ $X$ \ such that \ $d\omega = 0 = df \wedge \omega$ \ and to a vanishing \ $p-$cycle \ $\gamma$ \ choosen in the generic fiber \ $f^{-1}(s_0), s_0 \in D \setminus \{0\}$, we associated a vanishing period \ $\phi(s) : = \int_{\gamma_s} \ \omega\big/df $ \ which is, when \ $\gamma$ \ is choosen in the spectral subspace of \ $H_p(f^{-1}(s_0), \C)$ \ for the eigenvalue \ $e^{2i\pi.\lambda}$ \ of the monodromy of \ $f$, of the form above. Here \ $(\gamma_s)_{s \in D^*}$ is the horizontal multivalued family of \ $p-$cycles in the fibers of \ $f$ \ obtained from the choice of \ $\gamma$. The result obtained allows, for instance, to associate "natural" holomorphic functions of the parameter space when we have a family of such degenerations depending holomorphically on a parameter.