[For the entire collection see Zbl 0745.00034.]
The article under review is a report on results obtained by the author in Commun. Math. Phys. 131, No. 1, 179-218 (1990; Zbl 0704.32007); erratum: ibid. 140, No. 2, 415-416 (1991; Zbl 0739.32025). Starting from the data: \(X\) a smooth complex projective curve of genus \(g\), \(A=\{Q_ 1,Q_ 2,\ldots,Q_ n\}\) a set of points on the curve and \(B\) a closed disc containing \(A\), he defines complex divisors on the curve by allowing the divisors to have complex coefficients at the points in \(A\). The degree is required to be an integer. The author introduces principal divisors as divisors for certain multivalued meromorphic functions and the associated class group. The latter coincides with the usual class group of integral divisors. The Weil-Deligne pairing and the Arakelov-Deligne metric are extended to this setting if the degrees of the complex divisors equal zero. In string theory the string partition function can be given by integrating the Polyakov measure over the moduli space \({\mathcal M}_ g\) of smooth projective curves. By the Belavin-Knizhnik theorem this measure can be expressed in terms of the Mumford form. The author uses the theory of complex divisors to derive an analogous result for the tachyon scattering amplitude which can be calculated by integration over \({\mathcal M}_{g,n}\), the moduli space of smooth projective curves with \(n\) marked points. The marked points (the points in \(A)\) correspond to the scattering points, the complex coefficients of the divisors to momentum and the degree zero condition to the conservation of momentum.