## 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.

### MSC:

 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
Full Text:

### References:

 [1] 1. E. Artal-Bartolo, P. Cassou-Nogues, and A. Dimca, Sur la topologie des polynômes complexes. Progr. Math. 162 (1998). · Zbl 0964.32028 [2] 2. V. I. Arnold, A. N. Varchenko, and S. M. Goussein-Zade, Singularities of differentiable mappings. Mir, Moscow (1986). [3] 3. G. Bailly-Maitre. La connexion de Gauss–Manin des polynômes hyperelliptiques. Bull. Sci. Math. 124 (2000), 621–657. · Zbl 0997.32023 [4] 4. S. A. Broughton, Milnor number and the topology of polynomial hypersurfaces. Inv. Math. 92 (1988), 217–241. · Zbl 0658.32005 [5] 5. A. Dimca, A D-module look at a polynomial of two variables. Preprint (1998). · Zbl 0935.32024 [6] 6. —-, Sheaves in topology. Springer-Verlag, Berlin–Heidelberg (2004). · Zbl 1043.14003 [7] 7. —-, Monodromy and Hodge theory of regular functions. In: New Developments in Singularity Theory, Kluwer Academic Publ. (2001), p. 257–278. · Zbl 0993.32017 [8] 8. A. Dimca and M. Saito, Algebraic Gauss–Manin systems and Brieskorn modules. Amer. J. Math. 123 (2001), 163–184. · Zbl 0990.14004 [9] 9. —-, Monodromy at infinity and the weights of cohomology. Compos. Math. 138 (2003), 55–71. · Zbl 1039.32037 [10] 10. A. Douai, Très bonnes bases du réseau de Brieskorn d’un polynôme modéré. Bull. Soc. Math. France 127 (1999), 255–287. · Zbl 0935.32025 [11] 11. —-, Sur le système de Gauss–Manin d’un polynôme modéré. Bull. Sci. Math. 5 (2001), 395–405. · Zbl 0984.32017 [12] 12. A. Douai and C. Sabbah, Gauss–Manin systems, Brieskorn lattices and Frobenius structures I. Ann. Inst. Fourier 53 (2003), No. 4, 1055–1116. · Zbl 1079.32016 [13] 13. D. Eisenbud, Commutative algebra with a view toward algebraic geometry. Grad. Texts Math. 150, Springer-Verlag (1996). · Zbl 0819.13001 [14] 14. L. Gavrilov, Petrov modules and zero of Abelian integrals. Bull. Sci. Math. 122 (1998), 571–584. · Zbl 0964.32022 [15] 15. C. Hertling, Frobenius manifolds and moduli spaces for singularities. Cambridge Tracts in Math. 151 (2002). [16] 16. C. Hertling and Y. Manin, Unfoldings of meromorphic connections and a construction of Frobenius manifolds. In: Frobenius Manifolds, C. Hertling and M. Marcolli (Eds.), Aspects Math. E 36. · Zbl 1101.32013 [17] 17. A. G. Kouchnirenko, Polyèdres de Newton et nombres deMilnor. Invent. Math. 32 (1976), 1–31. · Zbl 0328.32007 [18] 18. V. P. Kostov, Gauss–Manin systems of polynomials of two variables can be made Fuchsian. Proc. Int. Conf. Geometry, Integrability and Quantization, St. Constantine and Elena, Bulgaria, September 1–10 (1999), p. 105–126. · Zbl 0979.34066 [19] 19. D. Novikov and S. Yakovenko, Redundant Picard–Fuchs system for Abelian integrals. J. Differential Equations 177 (2001), 267–306. · Zbl 1011.37042 [20] 20. F. Pham, Singularités des systèmes différentiels de Gauss–Manin. Progr. Math. 2, Birkhäuser, Boston (1980). [21] 21. C. Sabbah, Hypergeometric periods for a tame polynomial. C. R. Acad. Sci. Paris 328 (1999), 603–608. · Zbl 0967.32028 [22] 22. —-, Déformations isomonodromiques et variétés de Frobenius. Savoirs Actuels, CNRS Editions, Paris (2002). [23] 23. M. Schulze, Good bases for tame polynomials. e-print math.AG/0306120. · Zbl 1128.32017 [24] 24. S. Yakovenko, Bounded decomposition in the Brieskorn lattice and Pfaffian Picard–Fuchs systems for Abelian integrals. e-print math.DS/0201114. · Zbl 1021.34026
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.