Multiplier ideals of normal surface singularities

Abstract

We study the multiplier ideals and the corresponding jumping numbers and multiplicities $\{m(c){\}}_{c\in \mathbb{R}}$ in the following context: $(X,o)$ is a complex analytic normal surface singularity, $\mathfrak{a}\subset {\mathcal{O}}_{X,o}$ is an ${\mathfrak{m}}_{X,o}$--primary ideal, $\ddot{I}•:\tilde{X}\to X$ is a log resolution of $\mathfrak{a}$ such that $\mathfrak{a}{\mathcal{O}}_{\tilde{X}}={\mathcal{O}}_{\tilde{X}}(-F)$, for some nonzero effective divisor $F$ supported on $\ddot{I}{•}^{-1}(0)$. We show that $\{m(c){\}}_{c>0}$ is combinatorially computable from $F$ and the resolution graph $\hat{I}“$ of $\ddot{I}•$, and we provide several formulae. We also extend Budur's result (valid for $(X,o)=({\mathbb{C}}^2,0)$), which makes an identification of ${\sum }_{c\in [0,1]}m(c)t^c$ with a certain Hodge spectrum. In our general case we use Hodge spectrum with coefficients in a mixed Hodge module. We show that $\{m(c){\}}_{c\le 0}$ 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 ${\sum }_{c}m(c)t^c$ from the Poincaré series of $(X,o)$ and when $(X,o)$ is a splice quotient then we recover ${\sum }_{c}m(c)t^c$ from the multivariable topological Poincaré (zeta) function of $\hat{I}“$.

23 pages, some minor corrections, list of references extended