Hirzebruch's conjecture on cusp singularities
type: Item
instance of: scholarly article
title: Hirzebruch's conjecture on cusp singularities
zbMATH Open document ID: 0810.14002
DOI: 10.1007/BF01445095
author: Shoetsu Ogata
published in: Mathematische Annalen
publication date: 1995-04-17T00:00:00Z
full work available at URL: https://eudml.org/doc/165075
review text: 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.
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 32S50
Mathematics Subject Classification ID: 14G10
zbMATH DE Number: 537287
zbMATH Keywords: \(L\)-function
zbMATH Keywords: cusp singularity
zbMATH Keywords: signature defect of a Hilbert modular cusp
Wikidata QID: Q122982623
reviewed by: Jan Stevens
MaRDI profile type: Publication
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OpenAlex ID: W1968099476
description: scientific article; zbMATH DE number 537287