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Hirzebruch’s conjecture on cusp singularities. (English) Zbl 0810.14002
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 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 M.-N. Ishida [Math. Ann. 294, No. 1, 81-97 (1992; Zbl 0759.14003)], it is equivalent to the formula of this paper.

MSC:
14B05 Singularities in algebraic geometry
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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