Maciej Denkowski
Title: On the complex Lojasiewicz inequality with parameter
Abstract:
A parameter-version of the Lojasiewicz regular separation was first obtained by Lojasiewicz
and Wachta for compact subanalytic sets. The main feature of their result is that the regular
separation condition is satisfied with a uniform exponent. From this one may infer that the
Lojasiewicz inequality is satisfied with a uniform exponent in a continuous subanalytic family
of functions, too. The constant appearing in the inequality usually depends on the parameter.
In the talk we shall concentrate on the complex counterpart of this result. In particular, we will
show how a general continuity property (in the sense of currents) of the cycles of zeroes for
just a continuous family of holomorphic (or c-holomorphic) functions yields a parameter-version
of the Lojasiewicz inequality with an effective uniform exponent.