On the Monodromy of Higher Logarithms

The (multivalued) higher logarithms are interpreted, by studying their monodromy, as giving well-defined maps from P¿ — (3 points} into certain complex nilmanifolds with C*-actions. The purpose of this note is to exhibit a family of unipotent representations of Z * Z arising naturally from the monodromy of the higher logarithms ln¿. (see [4]), and thereby interpret each ln¿. as yielding a well-defined map pk of P¿ — (0,1, 00} into a (fc + l)-dimensional complex nilmanifold Mk+i equipped with a C*-action. Moreover, a natural holomorphic connection vt is shown to exist on each fibration Mk+] -» Mk+i/C*, with respect to which pk is flat. The role of dilogarithm in the study of volumes of hyperbolic 3-manifolds [5], arithmetic [1,2], TT-theory and Kac-Moody Lie algebras [3] leads one to hope for such interesting links in the case of higher logarithms as well. Furthermore, the tower of nilmanifolds associated to P¿ — {3 points} via {ln¿} suggests, following a remark of P. Deligne, relations to Sullivan's theory of differential forms. We hope to pursue this at some future time. Finally, we have come to learn recently of an independent and very elegant construction of these nilmanifolds due to J. W. Milnor (unpublished). We would like to thank Spencer Bloch for his constant and friendly encouragement. For x in C, set ln0(x) = x/(l — x) and L(x) = i2iri)~xlog(x). Then ln¿ is defined recursively by M*) = f^k-\(t)àL(t). Jo It follows easily that each ln¿. satisfies the (fc + l)st order, homogeneous, algebraic differential equation <•> ¿(o^HP'"))-This equation has regular singular points at 1 and 00 if fc = 1, and at 0,1, and 00 iffc> 1. Received by the editors August 3, 1981. 1980 Mathematics Subject Classification. Primary 53C30, 22E40; Secondary 33A75. 1982 American Mathematical Society 0002-9939/82/0000-1081 /S02.00 596 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use MONODROMY OF HIGHER LOGARITHMS Now let, for each integer m > 2,