Let V be a degree d, reduced hypersurface in CP, n ≥ 1, and fix a generic hyperplane, H. Denote by U the (affine) hypersurface complement, CP−V ∪H, and let U be the infinite cyclic covering of U corresponding to the kernel of the linking number homomorphism. Using intersection homology theory, we give a new construction of the Alexander modules Hi(U ;Q) of the hypersurface complement and show t...

We study the topology of the boundary manifold of a regular neighborhood of a complex projective hypersurface. We show that, under certain Hodge theoretic conditions, the cohomology ring of the complement of the hypersurface functorially determines that of the boundary. When the hypersurface defines a hyperplane arrangement, the cohomology of the boundary is completely determined by the combina...

LetX be a compact, strictly convex C-hypersurface in the (n+1)-dimensional Euclidean space R. The Gauss map ofX maps the hypersurface one-to-one and onto the unit n-sphere S. One may parametrize X by the inverse of the Gauss map. Consequently, the Gauss curvature can be regarded as a function on S. The classical Minkowski problem asks conversely when a positive function K on S is the Gauss curv...

In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form [CM74]. The latter appears in our class as the one with the ”maximum normalization” in the lowest degree. However, there are other n...

We propose a new model for representing a hypersurface describing a scalar field in any dimension at different levels of detail. The model is based on a decomposition of the domain into a hierarchy of simplicial complexes. We present a data structure for encoding such model as well as an algorithm for building a multiresolution representation of a hypersurface based on a decreasing sequence of ...

Kähler’s paper Über die Verzweigung einer algebraischen Funktion zweier Veränderlichen in der Umgebung einer singulären Stelle” offered a more perceptual view of the link of a complex plane curve singularity than that provided shortly before by Brauner. Kähler’s innovation of using a “square sphere” became standard in the toolkit of later researchers on singularities. We describe his contributi...

Abstract A real hypersurface M in a complex projective space inherits an almost contact metric structure from the Kählerian of ambient space. This allows us to define, for any nonzero number k , so-called -th generalized Tanaka–Webster connection. With this connection and Levi-Civita one we can associate two tensors type (1,2) Jacobi operator $$R_{\xi }$$ <mml:math xmlns:mml="http://www.w3.org/...

(1) f = det(xM1 + yM2 + zM3), where M1,M2,M3 are Hermitian d × d matrices. The representation is definite if there is a point e ∈ R for which the matrix e1M1+e2M2+e3M3 is positive definite. This imposes an immediate condition on the projective curve VC(f). Because the eigenvalues of a Hermitian matrix are real, every real line passing through e meets this hypersurface in only real points. A pol...