The theme of a vanishing period
Daniel Barlet (IECN, IUF)

TL;DR
This paper introduces the concept of a 'theme' associated with multivalued formal functions, characterizes their isomorphism classes via finite complex parameters, and constructs analytic invariants in degenerating families of complex manifolds.
Contribution
It defines the 'theme' as a minimal filtered differential equation, characterizes their classes with fixed Bernstein polynomial, and applies this to construct invariants in degenerating complex manifolds.
Findings
Isomorphism classes characterized by finite complex parameters.
Construction of analytic invariants for degenerating families.
Application to vanishing periods in complex geometry.
Abstract
Let \ \ and consider a multivalued formal function of the type where \ \ for \ . The {\bf theme} associated to such a \ \ is the "minimal filtered differential equation" with generator \ , 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 \ , 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 \ \ be a proper…
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions · Geometry and complex manifolds
