Affine Hypersurfaces Research Articles

Torsion of differentials of hypersurfaces with isolated singularities

Let R = K[ X 1, X 2,…, X N ], where K is an algebraically closed field of characteristic 0 and consider the reduced, affine hypersurface algebra with an isolated singularity A = R (F) , where F ϵ K[ X 1, X 2,…, X N ]. For such algebras A the torsion (sub) modules of (Kaehler) differentials T(Ω A K N − 1) and Ω A K N are finite dimensional. Unlike in the case of a quasi-homogeneous hypersurface T(Ω A K N − 1) is not always cyclic even if some permutation of ∂F ∂X 1 ,…, ∂F ∂X N is an R-sequence. The main result of this paper proves that for reduced hypersurfaces with only isolated singularities dim KT(Ω A K N − 1) = dim K Ω A K N . We give an example of a reduced plane curve with a single isolated singularity at the origin such that the partial derivatives of F do not form an R-sequence.

Equivalence theorems for complex affine hypersurfaces.

We study equivalence problem for complex hypersurfaces endowed with different types of transversal vector fields. We mainly concentrate on the case where the induced shape operator is anti-complex.

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: Mn → An+1, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.

A general calculation of the number of vanishing cycles

We give an algebraic formula for calculating the change in the Euler characteristic of the Milnor fibre of a function f, as one “grows” from a hypersurface section out to the full Milnor fibre of f. This is a general calculation in that we allow the domain of f to be singular, the hypersurface section may have an isolated singularity, and we allow coefficients in an arbitrary bounded, constructible complex of sheaves. Our formula generalizes both a result of Lê for affine hypersurfaces and a result of Looijenga for isolated complete intersections.

On the Role of the Burstin-Mayer Metric for Surfaces in R

For submanifolds of the affine space Rn, it is very important to derive a riemannian or pseudo-riemannian metric on the manifold just from affine data of the configuration. It is in this way that the equi-affine hypersurface theory is initiated by the so called Blaschke-Berwald metric (for the most recent state of affine hypersurface theory see the book of Li-Simon-Zhao [1993] and the vast literature given there). The same is true for the centro-affine geometry of codimension-two submanifolds (cf. Walter [1988], [1991 a]). Another instance where such a metric has been constructed from affine data are the (two-dimensional) surfaces of R4 (Burstin-Mayer [1927]). Recently, the geometry of these surfaces has been taken up by Nomizu-Vrancken [1993] with respect to the construction of a new transversal plane bundle. In the present note, we deal with the existence and, in particular, non-existence of elliptic points of the Burstin-Mayer metric from a local and global viewpoint.

Calabi-type composition of affine spheres

We give several ways of constructing new nondegenerate affine hypersurfaces in R n n + 1 starting from some given proper or improper affine spheres. The resulting hypersurfaces can be forced to have nice properties, if the original affine spheres are chosen appropriately. In this way we obtain new examples of nondegenerate locally homogeneous affine hypersurfaces and of proper and improper affine spheres. We also discuss the properties of the affine metric of the hypersurface, such as the curvature or completeness. Finally we discuss the relation with projective transformations.

On locally symmetric affine hypersurfaces

1. Introduction. In the present paper, we investigate 3-dimensional. locally strongly convex affine hyperspheres M in N, ~. We will also assume that the Levi Civita connection ~' of the affine metric h is locally symmetric. This problem is related to the following two questions proposed at the conference on affine differential geometry at Oberwolfach ([10]) in 1986:

Affine hypersurfaces with parallel cubic form

As is well known, there exists a canonical transversal vector field on a non-degenerate affine hypersurface M. This vector field is called the affine normal. The second fundamental form associated to this affine normal is called the affine metric. If M is locally strongly convex, then this affine metric is a Riemannian metric. And also, using the affine normal and the Gauss formula one can introduce an affine connection ∇ on M which is called the induced affine connection. Thus there are in general two different connections on M: one is the induced connection ∇ and the other is the Levi Civita connection of the affine metric h. The difference tensor K is defined by K(X, Y) = KXY — ∇XY — XY. The cubic form C is defined by C = ∇h and is related to the difference tensor by.

On the Hodge structure of projective hypersurfaces in toric varieties

This paper generalizes classical results of Griffiths, Dolgachev and Steenbrink on the cohomology of hypersurfaces in weighted projective spaces. Given a $d$-dimensional projective simplicial toric variety $P$ and an ample hypersurface $X$ defined by an polynomial $f$ in the homogeneous coordinate ring $S$ of $P$ (as defined in an earlier paper of the first author), we show that the graded pieces of the Hodge filtration on $H^d(P - X)$ are naturally isomorphic to certain graded pieces of $S/J(f)$, where $J(f)$ is the Jacobian ideal of $f$. We then discuss how this relates to the primitive cohomology of $X$. Also, if $T$ is the torus contained in $X$, then the intersection of $X$ and $T$ is an affine hypersurface in $T$, and we show how recent results of the second author can be stated using various ideals in the ring $S$. To prove our results, we must give a careful description (in terms of $S$) of $d$-forms and $(d-1)$-forms on the toric variety $P$. For completeness, we also provide a proof of the Bott-Steenbrink-Danilov vanishing theorem for simplicial toric varieties. Other topics considered in the paper include quasi-smooth hypersurfaces and $V$-submanifolds, the structure of the complement of $U$ when $P$ is represented as the quotient of an open subset $U$ of affine space, a generalization of the Euler exact sequence on projective space, and the relation between graded pieces of $R/J(f)$ and the moduli of ample hypersurfaces in $P$.

Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces

Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces

The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces

The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces

Topological invariants of affine hypersurfaces: connectivity, ends, and signature

Topological invariants of affine hypersurfaces: connectivity, ends, and signature

Newton Polygons of Zeta Functions and L Functions

In this article we give a systematic treatment of Newton polygons of exponential sums. The Newton polygon is a nice way to describe p-adic values of the zeroes or poles of zeta functions and L functions. Our main objective is to show that the Adolphson-Sperber conjecture 12], which asserts that under a simple condition the generic Newton polygon of L functions coincides with its lower bound, is false in its full form, but true in a slightly weaker form. We also show that the full form is true in various important special cases. For example, we show that for a generic projective hypersurface of degree d, the Newton polygon of the interesting part of the zeta function coincides with its lower bound (the Hodge polygon). This gives a p-adic proof of a recent theorem of Illusie, conjectured by Dwork and Mazur. For more examples, let us consider the family of affine hypersurfaces of degree d or the family of affine hypersurfaces defined by polynomials f(xi, . . ., x7n) of degree di with respect to xi (1 < i < n), where the di are fixed positive integers. Then, for all large prime numbers p, the generic Newton polygon for the zeta functions of each of the two families of hypersurfaces coincides with its lower bound. We obtain our main results, namely several decomposition theorems, using certain maximizing functions from linear programming. Our work suggests a possible connection between Newton polygons and the resolution of singularities of toric varieties. Let p be a prime, q = pa, and let Fq be the finite field of q elements and Fqm its extension of degree m. Fix a nontrivial additive character qP of Fp. For any Laurent polynomial f(xi, . . ., xn) E Fq[xl, xj1,.. . ., x, xi1] we form

Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori

Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori

Some equivalence theorems in affine hypersurface theory

Equivalence problems in affine geometry of hypersurfaces with type number greater than one are considered.

Certain curvature characterizations of affine hypersurfaces

Certain curvature characterizations of affine hypersurfaces

On normal and conormal maps for affine hypersurfaces

On normal and conormal maps for affine hypersurfaces

3-dimensional affine hypersurfaces in ℝ4 with parallel cubic form

In this paper, we study 3-dimensional locally strongly convex affine hypersurfaces in ℝ4. Since the publication of Blaschke’s book [B] in the early twenties, it is well-known that on a nondegenerate affine hyper-surface M there exists a canonical transversal vector field called the affine normal. The second fundamental form associated to the affine normal is called the affine metric. In the special case that M is locally strongly convex, this affine metric is a Riemannian metric. Also, using the affine normal, by the Gauss formula one can introduce an affine connection on M, called the induced connection ∇. So on M, we can consider two connections, namely the induced affine connection ∇ and the Levi Civita connection of the affine metric h.

Harmonic Gauss maps and Codazzi tensors for affine hypersurfaces

Harmonic Gauss maps and Codazzi tensors for affine hypersurfaces

Affine hypersurfaces with parallel cubic forms

Affine hypersurfaces with parallel cubic forms