MULTIPLIER IDEALS OF NORMAL SURFACE SINGULARITIES

Koltai, László and László, Tamás and Némethi, András (2025) MULTIPLIER IDEALS OF NORMAL SURFACE SINGULARITIES. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. ISSN 0002-9947 (In Press)

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Abstract

We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c)\}_{c\in \R}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, ${\mathfrak a}\subset \cO_{X,o}$ is an ${\mathfrak m}_{X,o}$--primary ideal, and $\phi:\tX\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}\cO_{\tX}=\cO_{\tX}(-F)$, for some nonzero effective divisor $F$ supported on $\phi^{-1}(0)$. A priori, $\{m(c)\}_{c\in \R}$ depends on the Hilbert function associated with the resolution and $F$. We prove that $\{m(c)\}_{c>0}$ is combinatorially computable from $F$ and the dual resolution graph $\Gamma$ of $\phi$, and we provide several formulae. We also make an identification of $\sum_{c\in[0,1)}m(c)t^c$ with a certain Hodge spectrum. This extends a result, valid for $(X,o)$ smooth, proved in different versions by Varchenko, Vaqui\'e, Loeser--Vaqui\'e and Budur. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c)\}_{c\leq 0}$ usually depends on the analytic type of $(X,o)$. However, for some distinguished analytic types we determine it concretely. E.g., when $(X,o)$ is weighted homogeneous (and $F$ is associated with the central vertex), we recover $\sum_cm(c)t^c$ from the Poincar\'e series of $(X,o)$. Also, when $(X,o)$ is a splice quotient then we recover $\sum_cm(c)t^c$ from the multivariable topological Poincar\'e (zeta) function of $\Gamma$.

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