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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Euler characteristics associated to exceptional divisors

Author: Willem Veys
Journal: Trans. Amer. Math. Soc. 347 (1995), 3287-3300
MSC: Primary 11S40; Secondary 14E15, 32S45
MathSciNet review: 1308026
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Abstract: Let $ k$ be an algebraically closed field and $ f \in k[{x_1}, \ldots ,{x_{n + 1}}]$. Fix an embedded resolution $ h:X \to {\mathbb{A}^{n + 1}}\quad {\text{of}}\quad {f^{ - 1}}\{ 0\} $ and denote by $ {E_i}$, $ i \in S$, the irreducible components of $ {h^{ - 1}}({f^{ - 1}}\{ 0\} )$ with multiplicity $ {N_i}$ in the divisor of $ f{\text{o}}h$. Put also $ {\mathop E\limits^{\text{o}} _i}: = {E_i}\backslash { \cup _{j \ne i}}{E_j}$, and denote by $ \chi ({E_i})$ its Euler characteristic.

Several conjectures concerning Igusa's local zeta function and the topological zeta function of $ f$ motivate the study of Euler characteristics associated to subsets $ { \cup _{i \in T}}{E_i}$ of $ { \cup _{i \in S}}{E_i}$, which are maximal connected with respect to the property that $ d\vert{N_i}$ for all $ i \in T$. Here $ d \in \mathbb{N},d > 1$. We prove that if $ h$ maps $ { \cup _{i \in T}}{E_i}$ to a point, then

$\displaystyle {( - 1)^n}\sum\limits_{i \in T} {\chi ({{\mathop E\limits^{\text{o}} }_i}) \geqslant 0} $

This generalizes a well-known result for curves. We also prove some vanishing results concerning the $ \chi ({\mathop E\limits^{\text{o}} _i})$ for such a maximal connected subset $ { \cup _{i \in T}}{E_i}$ and give an application on the above-mentioned zeta functions, yielding some confirmation of the holomorphy conjecture for those zeta functions.

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Article copyright: © Copyright 1995 American Mathematical Society