Sasaki, Takeshi; Yoshida, Masaaki The uniformizing differential equation of the complex hyperbolic structure on the moduli space of marked cubic surfaces. (English) Zbl 0976.33013 Proc. Japan Acad., Ser. A 75, No. 7, 129-133 (1999). D. Allcock, J. A. Carlsson and D. Toledo [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 1, 49-54 (1998)] proved that the moduli space of cubic surfaces carries a complex hyperbolic structure. The space is realized as a quotient space of the \(4\)-dimensional ball. It is classically known that it admits a bi-regular action of the Weyl group of type \(E_6\) and can be identified with a Zariski open subset of \({\mathbb C}^4\). Hence, regarding the complex ball lying in the projective space \({\mathbb P}^4\), we have a map from \({\mathbb C}^4\) to \({\mathbb P}^4\) with certain singularities that should have the invariance under the Weyl group and such a map is defined by a differential system of rank 5. This paper gives an explicit form of this system by using the theory of Schwarzian derivatives of several variables. Reviewer: T.Sasaki (Kobe) Cited in 2 ReviewsCited in 3 Documents MSC: 33C70 Other hypergeometric functions and integrals in several variables Keywords:uniformizing differential equation; hyperbolic structure; moduli space of cubic surfaces; hypergeometric differential equation × Cite Format Result Cite Review PDF Full Text: DOI References: [1] D. Allcock, J. Carlson and D. Toledo: A complex hyperbolic structure for moduli of cubic surfaces. C. R. Acad. Sci., 326 , 49-54 (1998). · Zbl 0959.32035 · doi:10.1016/S0764-4442(97)82711-5 [2] J. Carlson and D. Toledo: Discriminant compliments and kernels of monodromy representations, alg-geom/9708002, version 3, 11 May 1998. · Zbl 0978.14007 · doi:10.1215/S0012-7094-99-09723-5 [3] I. Naruki: Cross ratio variety as a moduli space of cubic surfaces. Proc. London Math. Soc., 45 , 1-30 (1982). · Zbl 0508.14005 · doi:10.1112/plms/s3-45.1.1 [4] T. Sasaki and M. Yoshida: Linear differential equations in two variables of rank 4, I, II. Math. Ann., 282 , 69-93, 95-111 (1988). · Zbl 0627.35014 · doi:10.1007/BF01457013 [5] J. Sekiguchi and M. Yoshida: \(W(E_6)\)-orbits of the configurations space of 6 lines on the real projective space. Kyushu J. Math., 51 , 1-58 (1997). · Zbl 0932.14027 · doi:10.2206/kyushujm.51.297 [6] M. Yoshida: The real loci of the configuration space of six points on the projective line and a Picard modular 3-fold. Kumamoto J. Math., 11 , 43-67 (1998). · Zbl 0920.52003 [7] M. Yoshida: Fuchsian Differential Equations. Vieweg Verlag, Wiesbaden, pp. 1-215 (1987). · Zbl 0618.35001 [8] M. Yoshida: Hypergeometric Functions, My Love. Vieweg Verlag, Wiesbaden, pp. 1-292 (1997). · Zbl 0889.33008 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.