Monodromy of hypergeometric functions and non-lattice integral monodromy
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.
scientific article; zbMATH DE number 3996010