Hirzebruch's conjecture on cusp singularities
This paper relates the signature defect \(\sigma (p)\) of an \(n\)- dimensional cusp singularity \((V,p)\) to \(\chi_ \infty (p)\), the contribution of the cusp to the arithmetic genus: \(\sigma (p) = 2^ n \chi_ \infty (p)\). The author gives a formula for the signature \(\text{Sign} (U, \partial U)\) of a neighbourhood \(U\) of the exceptional set of any resolution \(U \to V\) of a normal isolated singularity \(V\), in terms of primitive cohomology. For cusp singularities all relevant quantities can be expressed by data obtained from the toroidal description of the cusp. The proof of the main result involves elaborate calculations with binomial coefficients. Hirzebruch's original conjecture related the signature defect of a Hilbert modular cusp to special values of an \(L\)-function. By results of \textit{I. Satake} and the author [cf. Automorphic forms and geometry of arithmetic varieties, Adv. Stud. Pure Math. 15, 1-27 (1989; Zbl 0712.14009)] and \textit{M.-N. Ishida} [Math. Ann. 294, No. 1, 81-97 (1992; Zbl 0759.14003)], it is equivalent to the formula of this paper.
signature defect of a Hilbert modular cusp
scientific article; zbMATH DE number 537287
scientific article