Picard-Fuchs equations and Gauss-Manin systems with a view towards the Riemann-Hilbert problem. (English) Zbl 1096.32015

The paper is devoted to the study of the Picard-Fuchs equations associated with a tame polynomial function \(f:\mathbb C^n \to \mathbb C\). These Picard-Fuchs equations describe the differential systems satisfied by period matrices, i.e. matrices whose elements are Abelian integrals, that is the integrals of polynomial differential \((n-1)\) forms over a continuous family of cycles on the level sets \(f=t\). Since the integral vanishes on exact forms, Abelian integrals depend on the classes of the polynomial forms involved in the \(\mathbb C[t]\)-module \(P_f\) called Petrov module, this is the top cohomology group of a relative de Rham complex. The author looks for a basis of the Petrov module, in which Picard-Fuchs equation equations become as simple as posible.
The principal result of the first part of the paper is the following theorem:
Theorem. Let \(f\) be cohomologically tame. Then \(P_f\) is free of rank \(\mu\) over \(\mathbb C[t]\). Moreover, there exists a basis \((\eta_1\ldots,\eta_\mu)\) of \(P_f\) over \(\mathbb C[t]\) such that \[ (t\text{Id}-A_0)\frac{d}{dt}I(t)=A_1I(t), \] where Id is the identity matrix, \[ I(t)=\bigg(\int_{\delta(t)} \eta_1 ,\ldots, \int_{\delta(t)} \eta_\mu \bigg)^T, \] and \(A_0\) and \(A_1\) are constant matrices. The matrix \(A_1\) is diagonalizable, its eigenvalues are strictly positive rational numbers, \((d\eta_1\ldots,d\eta_\mu)\) project onto a basis of \[ \Omega^n(U)/df\wedge \Omega^{n-1}(U) \] and \(A_0\) is the matrix of the multiplication by \([f]\) in this basis.
The aim of the paper is to show how the Picard-Fuchs equations can be studied using the theory of algebraic Gauss-Manin systems and to connect the Petrov module with Brieskorn lattice. The author works inside the Gauss-Manin system \(M\) of \(f\) which is a regular, holonomic \(\mathbb C[t](\partial_t)\)-module. Let \(c_1,\ldots,c_l\) be the singular points of \(M\), \(\Gamma={c_1,\ldots,c_l}\) and \(p(t)=\Pi_{i=1}^l(t-c_i)\). The localized module \(M^{\text{loc}}= \mathbb C[t,1/p(t)]\otimes_{\mathbb C[t]}M\) is then free over the ring of rational functions with poles contained in \(\Gamma\). \(M^{\text{loc}}\) is a meromorphic bundle on \(P^1\) with poles on \(\Gamma\cup\infty\), equipped with a connection \(\partial_t\). If \(\Omega\) is a chosen basis of \(M^{\text{loc}}\), we have \[ \partial_t\Omega=A(t)\Omega \] where \(A(t)\) is a matrix whose coefficients belong to \(\mathbb C[t,1/p(t)]\). A natural \(\mathbb C[t]\)-submodule lies in \(M\), the Brieskorn module \(M_0\), generating \(M\) over \(\mathbb C[t](\partial_t)\). If \(f\) has only isolated critical points \[ M_0=\frac{\Omega^n}{df\wedge d\Omega^{n-2}}. \] If \(f\) moreover satisfies some tameness assumptions at infinity, \(M_0\) is a free \(\mathbb C[t]\)-module of rank \(\mu\), the global Milnor number of \(f\): \(M_0\) is then a lattice in \(M\). The following proposition holds: Proposition. Let \(f\) be cohomologically tame. There exists a basis of \(M_0\) such that \[ \partial_t\Omega=\sum_{i=1}^r\frac{B_i(t)}{t-c_i}\Omega \] for some constant matrices \(B_i, i=1,\ldots,r\) if and only if \([f]\) is diagonalizable. (Here \([f]\) is the endomorphism induced by the multiplication by \(f\) on the vector space \(\Omega^n(U)/df\wedge \Omega^{n-1}(U)\) which is of finite dimension if \(f\) has only isolated critical points). This is the result of Sect. 4. In sections 5 and 6 the Riemann-Hilbert problem (in family) for the Gauss-Manin system is discussed. Some examples are given.


32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
32G20 Period matrices, variation of Hodge structure; degenerations
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