×

Picard-Vessiot groups of Lauricella’s hypergeometric systems \(E_C\) and Calabi-Yau varieties arising integral representations. (English) Zbl 1456.14014

The authors study the Zariski closure of the monodromy group \(Mon\) of Lauricella’s hypergeometric function \(F_C(a,b,c;x)=\sum_{m_1,\ldots ,m_n=0}^{\infty}\frac{(a)_{m_1+\cdots +m_n}(b)_{m_1+\cdots +m_n}}{(c_1)_{m_1}\cdots (c_n)_{m_n}m_1!\cdots m_n!}x_1^{m_1}\cdots x_n^{m_n}\), where \(a,b\in \mathbb{C}\), \(c_i\in \mathbb{C}\setminus \{ 0,-1,-2,\ldots \}\), \((c_i)_{m_i}=\Gamma (c_i+m_i)/\Gamma (c_i)\), and Calabi-Yau varieties arising from its integral representation. When the identity component of \(Mon\) acts irreducibly, then \(\overline{Mon}\cap SL_{2^n}(\mathbb{C})\) is one of the classical groups \(SL_{2^n}(\mathbb{C})\), \(SO_{2^n}(\mathbb{C})\) or \(Sp_{2^n}(\mathbb{C})\).

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J28 \(K3\) surfaces and Enriques surfaces
33C65 Appell, Horn and Lauricella functions
32S40 Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects)
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] K.Aomoto and M.Kita, Theory of hypergeometric functions, Springer Monographs in Mathematics (Springer, Tokyo, 2011). (Translated by K. Iohara.) · Zbl 1229.33001
[2] W. N.Bailey, ‘A reducible case of the fourth type of Appell’s hypergeometric functions of two variables’, Q. J. Math.4 (1933) 305-308. · Zbl 0008.11402
[3] F.Beukers and G.Heckman, ‘Monodromy for the hypergeometric function \({}_n F_{n - 1}\)’, Invent. Math.95 (1989) 325-354. · Zbl 0663.30044
[4] S.Cynk and D.vanStraten, ‘Infinitesimal deformations of double covers of smooth algebraic varieties’, Math. Nachr.279 (2006) 716-726. · Zbl 1101.14006
[5] S.Cynk and T.Szemberg, ‘Double covers and Calabi-Yau varieties’, Banach Center Publ.44 (1998) 337-344. · Zbl 0915.14025
[6] P.Deligne and G. D.Mostow, ‘Monodromy of hypergeometric functions and non‐lattice integral monodromy’, Publ. Math. Inst. Hautes Études Sci.63 (1986) 5-88. · Zbl 0615.22008
[7] H.Esnault and E.Viehweg, Lectures on vanishing theorems, DMV Seminar 20 (Birkhäuser, Basel, 1992). · Zbl 0779.14003
[8] W.Fulton and J.Harris, Representation theory, Springer Graduate Texts in Mathematics 129 (Springer, Berlin, 1991). · Zbl 0744.22001
[9] Y.Goto, ‘Twisted cycles and twisted period relations for Lauricella’s hypergeometric function \(F_C\)’, Internat. J. Math.24 (2013) 1350094. · Zbl 1285.33011
[10] Y.Goto, ‘The monodromy representation of Lauricella’s hypergeometric function \(F_C\)’, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)XVI (2016) 1409-1445. · Zbl 1362.33017
[11] Y.Goto, J.Kaneko and K.Matsumoto, ‘Pfaffian of Appell’s hypergeometric system \(F_4\) in terms of the intersection form of twisted cohomology groups’, Publ. Res. Inst. Math. Sci.52 (2016) 223-247. · Zbl 1347.33032
[12] Y.Goto and M.Matsumoto, ‘The monodromy representation and twisted period relations for Appell’s hypergeometric function \(F_4\)’, Nagoya Math. J.217 (2015) 61-94. · Zbl 1327.32001
[13] Y.Goto and M.Matsumoto, ‘Irreducibility of the monodromy representation of Lauricella’s \(F_C\)’, Hokkaido Math. J.48 (2019) 489-512. · Zbl 1429.33025
[14] C.Hall, ‘Big symplectic or orthogonal monodromy modulo \(\ell \)’, Duke Math. J.141 (2008) 179-203. · Zbl 1205.11062
[15] R.Hattori and N.Takayama, ‘The singular locus of Lauricella’s \(F_C\)’, J. Math. Soc. Japan66 (2014), 981-995. · Zbl 1312.33043
[16] J. E.Humphreys, Linear algebraic groups, Springer Graduate Texts in Mathematics 21 (Springer, New York-Heidelberg, 1975). · Zbl 0325.20039
[17] M.Kato, ‘Appell’s \(F_4\) with finite irreducible monodromy group’, Kyushu J. Math.51 (1997) 125-147. · Zbl 0890.33007
[18] M.Kita, ‘On hypergeometric functions in several variables 1. New integral representations of Euler type’, Jpn. J. Math.18 (1992) 25-74. · Zbl 0767.33009
[19] C.Meyer, Modular Calabi‐Yau threefolds, Fields Institute Monographs 22 (American Mathematical Society, Providence, RI, 2005). · Zbl 1096.14032
[20] K.Mimachi and M.Noumi, ‘Solutions in terms of integrals of multivalued functions for the classical hypergeometric equations and the hypergeometric system on the configuration space’, Kyushu J. Math.70 (2016) 315-342. · Zbl 1354.33008
[21] D.Morrison, ‘On K3 surfaces with large Picard number’, Invent. Math.35 (1984) 105-121. · Zbl 0509.14034
[22] M.van derPut and M. F.Singer, Galois theory of linear differential equations, Grundlehren der mathematischen Wissenschaften 328 (Springer, Berlin-Heidelberg, 2003). · Zbl 1036.12008
[23] T.Sasaki, ‘Picard‐Vessiot group of Appell’s system of hypergeometric differential equations and infiniteness of monodromy group’, Kumamoto J. Sci. (Math.)14 (1980/81) 85-100. · Zbl 0439.33002
[24] T.Sasaki and M.Yoshida, ‘Linear differential equations in two variables of rank four I’, Math. Ann.282 (1988) 69-93. · Zbl 0627.35014
[25] M.Schütt and T.Shioda, ‘Elliptic surfaces’, Algebraic geometry in East Asia ‐ Seoul 2008, Advanced Studies in Pure Mathematics 60 (eds J.Keun (ed.), S.Kondō (ed.) and K.Oguisu (ed.); World Scientific, Singapore, 2010) 51-160. · Zbl 1216.14036
[26] M.Yoshida, Fuchsian differential equations, Aspects of Mathematics E11 (Friedrich Vieweg & Sohn, Braunschweig, 1987). · Zbl 0618.35001
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.