Complex powers of irreducible algebroid curves. (English) Zbl 0605.14029

Igusa, Jun-ichi

Geometry today, Int. Conf., Rome 1984, Prog. Math. 60, 207-230 (1985).

[For the entire collection see Zbl 0563.00006.]
The purpose of this paper is the study of the poles of the so called complex power of a K-analytic function f defining an algebroid curve C in a neighbourhood of 0. Here K is a local field of characteristic 0. The author determines all the poles and gives explicit formulas for the residues thus extending results of L. Strauss [Trans. Am. Math. Soc. 278, 481-493 (1983; Zbl 0524.14024)] and himself [Complex Anal. Algebr. Geom., Collect. Pap. dedic. K. Kodaira, 357-368 (1977; Zbl 0355.14012)].
A main technical ingredient is a precise knowledge of the desingularisation \(\tilde C\) of C and the intersection behaviour of the exceptional curves on \(\tilde C\).

Reviewer: F.Herrlich

MSC:

14H25	Arithmetic ground fields for curves
14E15	Global theory and resolution of singularities (algebro-geometric aspects)
14H20	Singularities of curves, local rings
11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
14G20	Local ground fields in algebraic geometry

Keywords:

Schwartz-Bruhat space; complex power of a K-analytic function; desingularisation; exceptional curves