On the curvature of singular complex hypersurfaces
نویسنده
چکیده
We study the behavior of the Gauss-Bonnet integrand on the level sets of a holomorphic function in a neighborhood of an isolated critical point. This is a survey of some older results of Griffiths, Langevin, Lê and Teissier. It is a blend of classical integral geometry and complex Morse theory (a.k.a Picard-Lefschetz theory). Motivation Consider the family of plane complex curves Ct = { (x, y)) ∈ C; xy = t, |x| + |y| ≤ 1 } , |t| 1. Ct is non-singular for t 6= 0 , while for t = 0 the complex curve C0 consists of the two plane disks Dx = { (x, 0); |x| = 1 } , Dy = { (0, y); |y| ≤ 1 } . Denote by gt the metric on Ct induces by the Euclidean metric on C . The boundary of Ct is ∂Ct := Ct ∩ S1(0), where Sr(p) denotes the sphere of radius r centered at p ∈ C. Observe that ∂Ct consists of two boundary components corresponding to the two solutions of the equation (see Figure 1) √ ρ2 + |t|2 ρ2 = 1 ⇐⇒ ρ + |t| = ρ, ρ > 0. For t 6= 0 the Riemann surface Ct is homotopy equivalent with the vanishing circle (see Figure 1) δt = { (x, y) ∈ Ct; |x| = |y| = √ |t| } . Thus χ(Ct) = χ(δt) = 0. Clearly χ(C0) = χ(pt) = 1 so that lim t→0 χ(Ct) 6= χ(C0). Denote by Kt the sectional curvature of the Riemann surface (Ct, gt) and by κt the geodesic curvature of ∂Ct ↪→ Ct (see [9, vol 3, Chap. 4] or [10, §4.1] for a definition of ∗Talk at the Felix Klein Seminar, Notre Dame, Fall 2003.
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