On the string-theoretic Euler number of a class of absolutely isolated singularities

Abstract

An explicit computation of the so-called string-theoretic E-function Estr (X

u, v) of a normal complex variety X with at most log-terminal singularities can be achieved by constructing one snc-desingularization of X, accompanied with the intersection graph of the exceptional prime divisors, and with the precise knowledge of their structure. In the present paper, it is shown that this is feasible for the case in which X is the underlying space of a class of absolutely isolated singularities (including both usual An-singularities and Fermat singularities of arbitrary dimension). As byproduct of the exact evaluation of estr (X) = limu,v→ 1 Estr (X

u, v), for this class of singularities, one gets counterexamples to a conjecture of Batyrev concerning the boundedness of the string-theoretic index. Finally, the string-theoretic Euler number is also computed for global complete intersections in ℙNℂ with prescribed singularities of the above type.