Singular points of complex hypersurfaces.

*(English)*Zbl 0184.48405
Annals of Mathematics Studies. No. 61. Princeton, N.J.: Princeton University Press and the University of Tokyo Press. 122 p. (1968).

From the author’s introduction: Let \(f(z_1,\ldots,z_n)\) be a non-constant polynomial in \(n+1\) complex variables, and let \(V\) be the algebraic set consisting of all \((n+1)\) tuples \(z=(z_1,\ldots,z_{n+1})\) of complex numbers with \(f(z)=0\). (Such a set is called a complex hypersurface.) We want to study the topology of \(V\) in the neighborhood of some point \(z^0\). Intersect the hypersurface \(V\) with a small sphere \(S_{\varepsilon}\) centered at the given point \(z^0\). Then the topology of \(V\) within the disk bounded by \(S_{\varepsilon}\) is closely related to the topology of the set \(K= V\cap S_{\varepsilon}\subset S_{\varepsilon}\). The object of this paper is to introduce a fibration which is useful in describing the topology of such intersections \(K= V\cap S_{\varepsilon}\subset S_{\varepsilon}\).

Here are some of the main results, which will be proved in Sections 4 through 7.

Fibration Theorem. If \(z^0\) is any point of the complex hypersurface \(V= f^{-1}(0)\) and if \(S_{\varepsilon}\) is a sufficiently small sphere centered at \(z^0\), then the mapping \(\varphi(z)=f(z)/| f(z)|\) from \(S_{\varepsilon}-K\) to the unit circle is the projection map of a smooth fiber bundle. Each fiber \(F_\theta=\varphi^{-1}(e^{i\theta})\subset S_{\varepsilon}-K\) is a smooth parallelizable \(2n\)-dimensional manifold. If the polynomial \(f\) has no critical points near \(z^0\), except for \(z^0\) itself, then we can give a much more precise description.

Theorem. If \(z^0\) is an isolated critical point of \(f\), then each fiber \(F_\theta\) has the homotopy type of a bouquet \(S^n\vee\ldots\vee S^n\) of \(n\)-spheres, the number of spheres in this bouquet (i.e., the middle Betti number of \(F_\theta)\), being strictly positive.

Each fiber can be considered as the interior of a smooth compact manifold with boundary closure \((F_\theta)=F_\theta\cup K\), where the common boundary \(K\) is an \((n-2)\)-connected manifold. The Brieskorn examples of singular varieties which are topologically manifolds are described in §9, and the classical theory of singular points of complex curves is described in §10. The last section proves a generalization of the fibration theorem to certain systems of real polynomials.

Here are some of the main results, which will be proved in Sections 4 through 7.

Fibration Theorem. If \(z^0\) is any point of the complex hypersurface \(V= f^{-1}(0)\) and if \(S_{\varepsilon}\) is a sufficiently small sphere centered at \(z^0\), then the mapping \(\varphi(z)=f(z)/| f(z)|\) from \(S_{\varepsilon}-K\) to the unit circle is the projection map of a smooth fiber bundle. Each fiber \(F_\theta=\varphi^{-1}(e^{i\theta})\subset S_{\varepsilon}-K\) is a smooth parallelizable \(2n\)-dimensional manifold. If the polynomial \(f\) has no critical points near \(z^0\), except for \(z^0\) itself, then we can give a much more precise description.

Theorem. If \(z^0\) is an isolated critical point of \(f\), then each fiber \(F_\theta\) has the homotopy type of a bouquet \(S^n\vee\ldots\vee S^n\) of \(n\)-spheres, the number of spheres in this bouquet (i.e., the middle Betti number of \(F_\theta)\), being strictly positive.

Each fiber can be considered as the interior of a smooth compact manifold with boundary closure \((F_\theta)=F_\theta\cup K\), where the common boundary \(K\) is an \((n-2)\)-connected manifold. The Brieskorn examples of singular varieties which are topologically manifolds are described in §9, and the classical theory of singular points of complex curves is described in §10. The last section proves a generalization of the fibration theorem to certain systems of real polynomials.

Reviewer: V. Villani