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Monodromy of hypergeometric functions and non-lattice integral monodromy. (English) Zbl 0615.22008
The object of study is the monodromy of integrals of the type
\[ \int^{\infty}_{1}u^{-\mu_0} (u-1)^{-\mu_1}\prod^{d+1}_{i=2}(u-x_i)^{-\mu_i} \,du \]
on the space \(Q=\{(x_i)\mid x_i\neq 0,1,\infty\) and \(x_i\ne x_j\) for \(i\ne j\}\). This generalizes the classical work of Schwarz in the case \(d=1\) and of Picard for \(d=2\). Under a certain integrality condition (INT) on the numbers \((1-\mu_i-\mu_j)^{-1}\), the authors prove that the monodromy group \(\Gamma\) is a lattice in the projective unitary group \(\mathrm{PU}(1,d)\). They also give criteria for \(\Gamma\) to be arithmetic.
This paper is very rich and instructive. Aside from some results on algebraic and Lie groups (not indispensable for understanding), this paper is essentially self-contained. The authors redefine cohomologically the integrals above, construct a compactification \(Q_{st}\) of \(Q\) and a completion \(\tilde Q_{st}\) over \(Q_{st}\) of the minimal covering space \(\tilde Q\) of \(Q\) on which those integrals are single-valued. The main point is the study of the mapping properties of a certain function \(\tilde w_{\mu}: \tilde Q_{st}\to B = \) the complex unit ball with a hermitian metric, which identifies the fibers of the projection \(\tilde Q_{st}\to Q_{st}\) with the orbits of a group of isometries of \(B\). The authors list all integrals satisfying condition (INT) and determine in each case whether the corresponding \(\Gamma\) is arithmetic and whether \(\mathrm{PU}(1,d)/\Gamma\) is compact. For \(d>5\) condition (INT) is never satisfied.

MSC:
22E40 Discrete subgroups of Lie groups
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
33C60 Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions)
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
11F06 Structure of modular groups and generalizations; arithmetic groups
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