Viro Method for the Construction of Real Complete Intersections

The Viro method is a powerful construction method of real nonsingular algebraic hypersurfaces with prescribed topology. It is based on polyhedral subdivisions of Newton polytopes. A combinatorial version of the Viro method is called combinatorial patchworking and arises when the considered subdivisions are triangulations. B. Sturmfels has generalized the combinatorial patchworking to the case of real complete intersections. We extend his result by generalizing the Viro method to the case of real complete intersections.     

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ _ℂ ⁿ . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ _ℂ ³ for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.     

The signature of generalized flag manifolds and applications

Let G be a linear algebraic group over C and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in CP^n are prime if n satisfies certain conditions.     

Recognizing unstable equidimensional maps, and the number of stable projections of algebraic hypersurfaces

We study the recognition of -classes of multi-germs in families of corank-1 maps from n-space into n-space. From these recognition conditions we deduce certain geometric properties of bifurcation sets of such families of maps. As applications we give a formula for the number of -codimension-1 classes of corank-1 multi-germs from ℂ ⁿ to ℂ ⁿ and an upper bound for the number of stable projections of algebraic hypersurfaces in ℝ ^{n +1} into hyperplanes. Received: 23 July 1998     

On Isotopies and Homologies of Subvarieties of Toric Varieties

In ℂⁿ we consider an algebraic surface Y and a finite collection of hypersurfaces S_i. Froissart’s theorem states that if Y and S_i are in general position in the projective compactification of ℂⁿ together with the hyperplane at infinity then for the homologies of Y \∪ S_i we have a special decomposition in terms of the homology of Y and all possible intersections of S_i in Y. We prove the validity of this homological decomposition on assuming a weaker condition: there exists a smooth toric compactification of ℂn in which Y and S_i are in general position with all divisors at infinity. One of the key steps of the proof is the construction of an isotopy in Y leaving invariant all hypersurfaces Y ∩ S_k with the exception of one Y ∩ S_i, which is shifted away from a given compact set. Moreover, we consider a purely toric version of the decomposition theorem, taking instead of an affine surface Y the complement of a surface in a compact toric variety to a collection of hypersurfaces in it.     

Quadratic convexity

We say that a setA ⊂ ℂ ⁿ is quadratically convex if its complement is a union of quadratic hypersurfaces. Some geometric properties of quadratically convex sets are investigated; in particular, they are related to lineally convex sets in a space of higher dimension. We say thatA is strongly quadratically convex if a certain generalization of the Fantappiè transform is surjective, which in effect means that we have a representation for any function holomorphic onA as a superposition of reciprocals of quadratic expressions. The main theorem in this paper gives a sufficient condition for a compact set to be strongly quadratically convex. Using integral representation formulas for holomorphic functions, an explicit inversion formula for the transform is obtained.     

Critical Points of Real Polynomials and Topology of Real Algebraic T-Surfaces

The paper is devoted to a special class of real polynomials, so-called T-polynomials, which arise in the combinatorial version of the Viro theorem. We study the relation between the numbers of real critical points of a given index of a T-polynomial and the combinatorics of lattice triangulations of Newton polytopes. We obtain upper bounds for the numbers of extrema and saddles of generic T-polynomials of a given degree in three variables, and derive from them upper bounds for Betti numbers of real algebraic surfaces in defined by T-polynomials. The latter upper bounds are stronger than the known upper bounds for arbitrary real algebraic surfaces in . Another result is the existence of an asymptotically maximal family of real polynomials of degree min three variables with 31m ³/36 + O(m ²) saddle points.     

Curvature surfaces of Hopf hypersurfaces in complex space forms

We study the principal curvatures of a Hopf hypersurfaceM in ℂP ⁿ or ℂH ⁿ . The respective eigenspaces of the shape operator often turn out to induce totally real foliations ofM, whose leaves are spherical in the ambient space. Finally we classify the Hopf hypersurfaces with three distinct principal curvatures satisfying a certain non-degeneracy condition.     

On dimensions of the groups of biholomorphic automorphisms of real-analytic hypersurfaces

In this paper we study the dependence of the local geometry of real-analytic hypersuffaces in ℂ ⁿ on the dimension of the group of biholomorphic automorphisms of this surface. We also classify the hypersurfaces in terms of this group. We present some examples showing that the classes of the given construction are not empty. We find a new formulation of the Freeman theorem on the so-called straightening of a real-analytic CR-submanifold in ℂ ⁿ with degenerate Levi form of constant rank. Translated fromMatematicheskie Zametki, Vol. 61, No. 3, pp. 349–358, March, 1997. Translated by E. G. Anisova     

Neighborly cubical spheres and a cubical lower bound conjecture

Using mirrors and cyclic polytopes, we construct cubicald-spheres which are the analogs of cyclic polytopes in the sense that they have the ⌉d−1/2⌈-skeleta of cubes. The existence of these neighborly cubical spheres leads to a special case of an upper bound conjecture for cubical spheres, suggested by Kalai. We extend the same construction to show that the closed convex hull off-vectors of cubical spheres contains a cone described by Adin, as an analog to the generalized lower bound theorem for simplicial polytopes. Supported in part, respectively, by an NSF Postdoctoral Fellowship; NSF Grants DMS 9207700 and 9500581; and an NSF Planning Grant.     

Lattice-free polytopes and their diameter

A convex polytope in real Euclidean space islattice-free if it intersects some lattice in space exactly in its vertex set. Lattice-free polytopes form a large and computationally hard class, and arise in many combinatorial and algorithmic contexts. In this article, affine and combinatorial properties of such polytopes are studied. First, bounds on some invariants, such as the diameter and layer-number, are given. It is shown that the diameter of ad-dimensional lattice-free polytope isO(d ³). A bound ofO(nd+d ³) on the diameter of ad-polytope withn facets is deduced for a large class of integer polytopes. Second, Delaunay polytopes and [0, 1]-polytopes, which form major subclasses of lattice-free polytopes, are considered. It is shown that, up to affine equivalence, for anyd≥3 there are infinitely manyd-dimensional lattice-free polytopes but only finitely many Delaunay and [0, 1]-polytopes. Combinatorial-types of lattice-free polytopes are discussed, and the inclusion relations among the subclasses above are examined. It is shown that the classes of combinatorial-types of Delaunay polytopes and [0,1]-polytopes are mutually incomparable starting in dimension six, and that both are strictly contained in the class of combinatorial-types of all lattice-free polytopes. This research was supported by DIMACS—the Center for Discrete Mathematics and Theoretical Computer Science at Rutgers University.     

Cotorsion pairs associated with Auslander categories

Let N ≥ n + 1, and denote by K the convex hull of N independent standard gaussian random vectors in ℝⁿ. We prove that with high probability, the isotropic constant of K is bounded by a universal constant. Thus we verify the hyperplane conjecture for the class of gaussian random polytopes. Supported by the Clay Mathematics Institute and by NSF grant #DMS-0456590     

Integer points on curves and surfaces

Various upper bounds are given for the number of integer points on plane curves, on surfaces and hypersurfaces. We begin with a certain class of convex curves, we treat rather general surfaces in ³ which include algebraic surfaces with the exception of cylinders, and we go on to hypersurfaces in ⁿ with nonvanishing Gaussian curvature.Written with partial supports from NSF grant No. MCS-8211461.     

On the behavior of strictly plurisubharmonic functions near real hypersurfaces

We describe the behavior of certain strictly plurisubharmonic functions near some real hypersurfaces in ℂ ⁿ , n≥3. Given a hypersurface we study continuous plurisubharmonic functions which are zero on the hypersurface and have Monge–Ampère mass greater than one in a one-sided neighborhood of the hypersurface. If we can find complex curves which have sufficiently high contact order with the hypersurface then the plurisubharmonic functions we study cannot be globally Lipschitz in the one-sided neighborhood.     

The Levi Monge-Ampère equation: Smooth regularity of strictly Levi convex solutions

We prove smoothness of strictly Levi convex solutions to the Levi equation in several complex variables. This equation is fully non linear and naturally arises in the study of real hypersurfaces in ℂⁿ⁺¹, for n ≥ 2. For a particular choice of the right-hand side, our equation has the meaning of total Levi curvature of a real hypersurface ℂⁿ⁺¹ and it is the analogous of the equation with prescribed Gauss curvature for the complex structure. However, it is degenerate elliptic also if restricted to strictly Levi convex functions. This basic failure does not allow us to use elliptic techniques such in the classical real and complex Monge-Ampère equations. By taking into account the natural geometry of the problem we prove that first order intrinsic derivatives of strictly Levi convex solutions satisfy a good equation. The smoothness of solutions is then achieved by mean of a bootstrap argument in tangent directions to the hypersurface.     

Real Hypersurfaces with Isometric Reeb Flow in Complex Two-Plane Grassmannians

 We classify all real hypersurfaces with isometric Reeb flow in the complex Grassmann manifold G ₂ (ℂ ^m+2 ) of all 2-dimensional linear subspaces in ℂ ^m+2 , m ≥ 3.     

Extreme points for convex mappings ofB n

We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ ⁿ forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials of degree 2 in ℂ ⁿ⁻¹, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the closed convex hull of the set of extreme points here exhibited.     

An Inscribing Model for Random Polytopes

For convex bodies K with boundary in ℝ ^d , we explore random polytopes with vertices chosen along the boundary of K. In particular, we determine asymptotic properties of the volume of these random polytopes. We provide results concerning the variance and higher moments of this functional, as well as an analogous central limit theorem. The research of V.H. Vu was done under the support of A. Sloan Fellowship and an NSF Career Grant. The research of L. Wu is done while the author was at University of California San Diego.