Which Powers of Holomorphic Functions Are Integrable?

Question 1. Let f(z1, . . . , zn) be a holomorphic function on an open set U ⊂ C. For which t ∈ R is |f |t locally integrable? The positive values of t pose no problems, for these |f |t is even continuous. If f is nowhere zero on U then again |f |t is continuous for any t ∈ R. Thus the question is only interesting near the zeros of f and for negative values of t. More generally, if h is an invertible function then |f |t locally integrable iff |fh|t is locally integrable. Thus the answer to the question depends only on the hypersurface (f = 0) but not on the actual equation. (A hypersurface (f = 0) is not just the set where f vanishes. One must also remember the vanishing multiplicity for each irreducible component.) It is traditional to change the question a little and work with s = −t/2 instead. Thus we fix a point p ∈ U and study the values s such that |f |−s is L in a neighborhood of p. It is not hard to see that there is a largest value s0 (depending on f and p) such that |f |−s is L in a neighborhood of p for s < s0 but not L for s > s0. Our aim is to study this “critical value” s0. Definition 2. Let f be a holomorphic function in a neighborhood of a point p ∈ C. The log canonical threshold or complex singularity exponent of f at p is the number cp(f) such that • |f |−s is L in a neighborhood of p for s < cp(f), and • |f |−s is not L in any neighborhood of p for s > cp(f). It is convenient to set cp(0) = 0.