Suppose $c_1,\ldots,c_{n+k}$ are real numbers, $\{a_1,\ldots,a_{n+k}\}\!\subset\!\mathbb{R}^n$ is a set of points not all lying in the same affine hyperplane, $y\!\in\!\mathbb{R}^n$, $a_j\cdot y$ denotes the standard real inner product of $a_j$ and $y$, and we set $g(y)\!:=\!\sum^{n+k}_{j=1} c_j e^{a_j\cdot y}$. We prove that, for generic $c_j$, the number of connected components of the real ze...

Let M be a complex manifold, X ⊂ M a singular hypersurface. We study residues of top-dimensional forms with poles along X. Applying resolution of singularities we obtain residue classes either in L-cohomology of X or in the intersection cohomology. The conditions allowing to construct these residue classes coincide. They can be formulated in terms of weight filtration.

It was shown by J.N. Mather and S.S.-T. Yau that an isolated complex hypersurface singularity is completely determined by its moduli algebra. In this article it is shown, for the simple elliptic singularities, how to construct continuous invariants from the moduli algebras and, hence, associate invariants to the singularities themselves.

In the Hyperbolic space Hn (n ≥ 3) there are uncountably many topological types of convex hypersurfaces. When is a locally convex hypersurface in Hn globally convex, that is, when does it bound a convex set? We prove that any locally convex proper embedding of an (n− 1)-dimensional connected manifold is the boundary of a convex set whenever the complement of (n−1)-flats of the resulting hypersu...

We show, using a Knapp-type homogeneity argument, that the (L, L) restriction theorem implies a growth condition on the hypersurface in question. We further use this result to show that the optimal (L, L) restriction theorem implies the sharp isotropic decay rate for the Fourier transform of the Lebesgue measure carried by compact convex finite hypersurfaces. Section 0: Introduction Let S be a ...

A contact hypersurface in a Kähler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kähler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces with constant mean curvature in the complex quadric Q = SOn+2/SOnSO2 and its no...

The present paper contains some results about the algebraic sets of planar normal sections associated to the focal manifolds of homogeneous isoparametric hypersurfaces in spheres. With the usual identification of the tangent spaces to the focal manifold with subspaces of the tangent spaces to the isoparametric hypersurface, it is proven that the algebraic set of planar normal sections of the fo...

Tropicalization takes a curve in a hypersurface in 3-space to a balanced rational weighted graph in a tropical surface. In this paper, we study the ‘lifting’ problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a hypersurface? We develop specific combinatorial obstructions to a graph lifting by studying the factorizations of polynomials with particular N...

We study lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtai...