Isolated singularities defined by weighted homogeneous polynomials
TL;DR: In this paper, the authors give simple explicit formulas for the multiplicity p and the characteristic polynomial A(t) associated with a weighted homogeneous polynomial in several complex variables, having an isolated critical point at the origin.
About: This article is published in Topology.The article was published on 1970-11-01 and is currently open access. It has received 249 citations till now. The article focuses on the topics: Homogeneous polynomial & Isolated singularity.
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TL;DR: In this article, a survey of critical points of smooth functions and their bifurcations is presented, and the connections with the theories of groups generated by reflections, automorphic forms, and degenerations of elliptic curves.
Abstract: This paper contains a survey of research on critical points of smooth functions and their bifurcations. We indicate applications to the theory of Lagrangian singularities (caustics), Legendre singularities (wave fronts) and the asymptotic behaviour of oscillatory integrals (the stationary phase method). We describe the connections with the theories of groups generated by reflections, automorphic forms, and degenerations of elliptic curves. We give proofs of the theorems on the classification of critical points with at most one modulus, and also a list of all singularities with at most two moduli. The proofs of the classification theorems are based on a geometric technique associated with Newton polygons, on the study of the roots of certain Lie algebras resembling the Enriques-Demazure technique of fans, and on spectral sequences that are constructed with respect to quasihomogeneous filtrations of the Koszul complex defined by the partial derivatives of a function.
291 citations
219 citations
TL;DR: In this article, the authors studied η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds and used a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of ηEinstein structures on many different compact manifolds including exotic spheres.
Abstract: A compact quasi-regular Sasakian manifold M is foliated by one-dimensional leaves and the transverse space of this characteristic foliation is necessarily a compact Kahler orbifold . In the case when the transverse space is also Einstein the corresponding Sasakian manifold M is said to be Sasakian η-Einstein. In this article we study η-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem in the context of Sasakian geometry to prove the existence of η-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl and Lorenzian Sasakian-Einstein structures.
159 citations
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Book•
01 Jan 1968
TL;DR: The Singular Points of Complex Hypersurfaces (AM-61) as mentioned in this paper is a seminal work in the area of complex hypersurfaces, and is based on as mentioned in this paper.
Abstract: The description for this book, Singular Points of Complex Hypersurfaces. (AM-61), will be forthcoming.
2,676 citations
320 citations
219 citations
TL;DR: In this article, the authors implique l'accord avec les conditions generales d'utilisation (http://www.numdam.org/legal.php).
Abstract: L’acces aux archives de la revue « Bulletin de la S. M. F. » (http://smf. emath.fr/Publications/Bulletin/Presentation.html), implique l’accord avec les conditions generales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systematique est constitutive d’une infraction penale. Toute copie ou impression de ce fichier doit contenir la presente mention de copyright.
124 citations