On the string-theoretic Euler number of a class of absolutely isolated singularities (Q5945012)

Let \(X\) be a normal complex variety with at most log-terminal singularities and \(\pi:\overline X\to X\) be a desingularities such that the exceptional locus \(\displaystyle\bigcup^m_{i=1} D_i\) \((D_i\) smooth with normal crossings) has discrepancy \(K_{\overline X}-\pi^* (K_X)=\displaystyle\sum^m_{i=1} a_iD_i\). Let \(I=\{1,\dots, m\}\) and define for \(J\subset I\), \[ D_J:= \begin{cases} \overline X&\text{for }J= \emptyset \\ \bigcap_{j\in J}D_j \end{cases},\qquad D_J^0=D_J \setminus \bigcup_{ j\in I\setminus J} D_j. \] Then the string-theoretic \(E\)-function \(E_{\text{str}} (X,u,v)\) is defined by \[ E_{\text{str}} (X,u,v)=\sum_{J\subseteq I} E(D^0_J,u,v) \prod_{j\in J}{uv-1 \over(uv)^{a_j+1} -1}. \] Here \(E(X,u,v) =\sum e^{pq}(X)u^p v^q\in\mathbb{Z} [u,v]\) is the \(E\)-polynomial with \[ e^{pq}(X)= \sum(-1)^i h^{pq} (H^i_e (X,\mathbb{C})),\qquad h^{pq}(H^i_e (X,\mathbb{C}))\text{ the Hodge numbers.} \] The string-theoretic Euler number is defined by \[ e_{\text{str}} (X)= \lim_{u,v\to 1}E_{ \text{str}}(X,u,v) \] and the string theoretic index by \[ \text{ind}_{\text{str}} (X)=\min\left\{ \ell\in\mathbb{Z}_+\left|e_{\text{str}} (X)\in {1\over\ell} \mathbb{Z}\right\}\right. . \] Let \(X\) be an \(A^{(r)}_{n,\ell}\)-singularity, that is, defined by \(x_1^{n+1} +x_2^\ell +\cdots+x^\ell_{r+1} =0\); then an explicit computation of \(E_{\text{str}} (X,u,v)\), \(e_{\text{str}}(X)\) is given. This is applied to give a counterexample to a conjecture of \textit{V. V. Batyrev} [in: Integrable systems and algebraic geometry Proc. 41st Taniguchi Symp., Kobe 1997, Kyota 1997,World Scientific, 1-32 (1998; Zbl 0963.14015)] concerning the boundedness of the string-theoretic index.