Commensurabilities Among Lattices in Pu

The first part of this monograph is devoted to a characterization of hypergeometric-like functions, that is,twistsof hypergeometric functions inn-variables. These are treated as an (n+1) dimensional vector space of multivalued locally holomorphic functions defined on the space ofn+3 tuples of distinct points on the projective linePmodulo, the diagonal section of AutoP=m. Forn=1, the characterization may be regarded as a generalization of Riemann's classical theorem characterizing hypergeometric functions by their exponents at three singular points. This characterization permits the authors to compare monodromy groups corresponding to different parameters and to prove commensurability modulo inner automorphisms ofPU(1,n). The book includes an investigation of elliptic and parabolic monodromy groups, as well as hyperbolic monodromy groups. The former play a role in the proof that a surprising number of lattices inPU(1,2) constructed as the fundamental groups of compact complex surfaces with constant holomorphic curvature are in fact conjugate to projective monodromy groups of hypergeometric functions. The characterization of hypergeometric-like functions by their exponents at the divisors "at infinity" permits one to prove generalizations inn-variables of the Kummer identities forn-1 involving quadratic and cubic changes of the variable.Deligne, Pierre is the author of 'Commensurabilities Among Lattices in Pu' with ISBN 9780691000961 and ISBN 0691000964.