Contemporary Mathematics Volume 368, 2005 Cusp-type singularities of real analytic curves in the complex plane Patrick Ahern and Xianghong Gong 1. Introduction A parameterized real analytic curve C in C is given by (1.1) C:t----+x=a(t), y=b(t), tER where a, bare real-valued real analytic functions defined near the origin. We always assume that 0 E C, and that all changes of coordinates are local and fix the origin. We say that two parameterized curves C and C are equivalent if there exist a bi- holomorphic map hand a real analytic diffeomorphism¢ so that h(C(t)) = C( ¢(t) ). We also say that C, C are formally equivalent, if h is a formal biholomorphic map and ¢ is a formal real power series with ¢'(0) -=f. 0. Without parametrization, we shall see that (1.1) defines an irreducible real analytic set in C, or an analytic arc having the origin as the end point. An obvious invariant is the smallest integer n so that C(t) = 'Y(t)n for some real analytic 'Y(t) with 'Y'(O) -=f. 0. Here the classification for 'Y is restricted to biholomorphic maps commuting with vn: z----+ e2rri/nz. We shall see, in next section, that classifications for the parameterized curves C that are not arcs, for the unparametrized curves C, and for the liftings 'Y are determined by each other, when the curves are not arcs. Some simple modifications will suffice to reduce the classification of arcs to that of non arcs. We first formulate some results for smooth real analytic curves in C, under a change of holomorphic coordinates commuting with v = Vn· Let 'Y be a smooth real analytic curve, and r = r'Y be the anti-holomorphic reflection ofT Associated with v the indicator of 'Y is x = vrv- 1 r. We start with PROPOSITION 1.1. Let 'Y be a germ of smooth real analytic curve at 0 E C. Then the indicator x'Y = Vn1""(v 1 r'Y is linear if and only if x'Y is linearizable by some local biholomorphic mapping commuting with Vn. Also X 'Y is linear if and only if 'Y is locally equivalent to y = 0 by some biholomorphic mapping commuting with Vn. 2000 Mathematics Subject Classification. Primary 32B10, 37G05. Research of the second author is supported in part by an NSF grant. © 2005 American Mathematical Society
Previous Page Next Page