| Abstract: |
The link \( \Sigma \) of a normal surface singularity \( (X,0) \) is a
compact \( 3 \)-manifold constructed by plumbing using the graph \(
\Gamma \) of a resolution of \(X \). Our 1990 JAMS paper defined a
topological invariant \( I(X) \) from \( \Gamma \) with: (a) \(
I(X)\in \mathbb{Q}_{\geq 0} \) ; (b) \( I(X)=0 \) iff \( X \) is
log-canonical; (c) \( I(X) \) is ``characteristic'', multiplying by
degree in unramified covers; (d) \( X \) Gorenstein (e.g.,
hypersurface), \( I(X)\neq 0 \) implies \( I(X)\geq 1/42 \), and \(
\{I(X)|X \text{Gorenstein}\} \) satisfies the ACC. We introduce
definitions now for the more general log (or orbifold) situation of a
pair \( (X,\sum c_iC_i) \), where the \( C_i \) are curves on \( X \)
(resp. knots in \( \Sigma \) ), and \( c_i\in [0,1] \) (usually \(
c_i=1/n_i \) or \( 1-1/n_i \) ). Like \( I(X) \), it is based on the
Zariski decomposition of some \( \mathbb{Q} \)-line bundles on a resolution of
\( X \).
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