Hyperbolic surfaces in P3{\bf P}^3: examples

This is a recent conference report on the Kobayashi Problem on hyperbolicity of generic projective hypersurfaces. As an appendix, a (non-updated) author's survey article of 1992 on the same subject, published in an edition with a limited distribution, is added.Comment: 25 pages; an updated version of a conference repor

Selected problems

This is a renovated list of open problems, to appear in: "Affine Algebraic Geometry" conference Proceedings volume in Contemporary Mathematics series of the Amer. Math. Soc. Ed. by Jaime Gutierrez, Vladimir Shpilrain, and Jie-Tai Yu

Discrete convolution operators in positive characteristic: a variation on the Floquet-Bloch Theory

The classical Floquet theory deals with Floquet-Bloch solutions of periodic PDEs (see e.g., P. Kuchment. Floquet Theory for Partial Differential Equations. Basel: Birkhauser, 1993). Peter Kuchment developed as well a discrete version of this theory for difference vector equations on lattices, including the Floquet theory on infinite periodic graphs. Here we propose a variation on this theory for matrix convolution operators acting on vector functions on lattices with values in a field of positive characteristic

On exotic algebraic structures on affine spaces

By an exotic algebraic structure on the affine space ${\bf C}^n$ we mean a smooth affine algebraic variety which is diffeomorphic to ${\bf R}^{2n}$ but not isomorphic to ${\bf C}^n$. This is a survey of the recent developement on the subject, which emphasizes its analytic aspects and points out some open problems.Comment: 30 pages, author-supplied DVI file available at http://www.math.duke.edu/preprints/95-00.dvi , LaTe

Periodic binary harmonic functions

A function on a (generally infinite) graph \G with values in a field $K$ of characteristic 2 will be called {\it harmonic} if its value at every vertex of \G is the sum of its values over all adjacent vertices. We consider binary pluri-periodic harmonic functions f: \Z^s\to\F_2=\GF(2) on integer lattices, and address the problem of describing the set of possible multi-periods $\bar n=(n_1,...,n_s)\in\N^s$ of such functions. Actually this problem arises in the theory of cellular automata. It occurs to be equivalent to determining, for a certain affine algebraic hypersurface $V_s$ in \A_{\bar\F_2}^s, the torsion multi-orders of the points on $V_s$ in the multiplicative group (\bar\F_2^\times)^s. In particular $V_2$ is an elliptic cubic curve. In this special case we provide a more thorough treatment. A major part of the paper is devoted to a survey of the subject.Comment: 36 pages, 3 figure

Lectures on Exotic Algebraic Structures on Affine Spaces

These notes are based on the lecture courses given at the Ruhr-Universit{\"a}t-Bochum (03--08.02.1997) and at the Universit{\'e} Paul Sabatier (Toulouse, 08-12.01.1996).Comment: Second revised version, extended. 54 Pages, LaTe

Hyperbolic hypersurfaces in P^n of Fermat-Waring type

In this note we show that there are algebraic families of hyperbolic, Fermat-Waring type hypersurfaces in P^n of degree 4(n-1)^2, for all dimensions n>1. Moreover, there are hyperbolic Fermat-Waring hypersurfaces in P^n of degree 4n^2-2n+1 possessing complete hyperbolic, hyperbolically embedded complements.Comment: 5 page

Liouville and Carath\'eodory coverings in Riemannian and complex geometry

A Riemannian manifold resp. a complex space $X$ is called Liouville if it carries no nonconstant bounded harmonic resp. holomorphic functions. It is called Carath\'eodory, or Carath\'eodory hyperbolic, if bounded harmonic resp. holomorphic functions separate the points of $X$. The problems which we discuss in this paper arise from the following question: When a Galois covering $X$ with Galois group $G$ over a Liouville base $Y$ is Liouville or, at least, is not Carath\'eodory hyperbolic?Comment: 20 pages, AMSTeX. A revised version. The proof of Theorem 3.1 has been completed, and some other minor correction has been don

On the uniqueness of C∗{\bf C}^*-actions on affine surfaces

We prove that a normal affine surface $V$ over $\bf C$ admits an effective action of a maximal torus ${\bf T}={\bf C}^{*n}$ ($n\le 2$) such that any other effective ${\bf C}^*$-action is conjugate to a subtorus of $\bf T$ in Aut $(V)$, in the following particular cases: (a) the Makar-Limanov invariant ML$(V)$ is nontrivial, (b) $V$ is a toric surface, (c) $V={\bf P}^1\times {\bf P}^1\backslash \Delta$, where $\Delta$ is the diagonal, and (d) $V={\bf P}^2\backslash Q$, where $Q$ is a nonsingular quadric. In case (a) this generalizes a result of Bertin for smooth surfaces, whereas (b) was previously known for the case of the affine plane (Gutwirth) and (d) is a result of Danilov-Gizatullin and Doebeli.Comment: 11/06/2004 2 version 14/06/200

Plane Curves with Hyperbolic and C-hyperbolic Complements

The general problem which initiated this work is: What are the quasiprojective varieties which can be uniformized by means of bounded domains in \cz^n ? Such a variety should be, in particular, C--hyperbolic, i.e. it should have a Carath\'{e}odory hyperbolic covering. We study here the plane projective curves whose complements are C--hyperbolic. For instance, we show that most of the curves whose duals are nodal or, more generally, immersed curves, belong to this class. We also give explicit examples of irreducible such curves of any even degree d greater or equal 6.Comment: Final version, published in 1996, subsuming version 1 of this preprint and another work by the same authors on "Examples of Plane Curves with Hyperbolic and C-hyperbolic Complements