On tight projective designs

It is shown that among all tight designs in , where is or , or (quaternions), only 5-designs in (Lyubich, Shatalora Geom Dedicata 86: 169–178, 2001) have irrational angle set. This is the only case of equal ranks of the first and the last irreducible idempotent in the corresponding Bose-Mesner algebra.      

Analytic properties in the spectrum of certain Banach algebras

We show a sufficient condition for a domain in to be a H ^∞-domain of holomorphy. Furthermore if a domain has the Gleason property at a point and the projection of the n − 1th order generalized Shilov boundary does not coincide with Ω then is schlicht. We also give two examples of pseudoconvex domains in which the spectrum is non-schlicht and satisfy several other interesting properties.      

Foliations and polynomial diffeomorphisms of $${\mathbb{R}^{3}}$$

Let be a C ² map and let Spec(Y) denote the set of eigenvalues of the derivative DY _p , when p varies in . We begin proving that if, for some ϵ > 0, then the foliation with made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek’s Jacobian Conjecture for polynomial maps of The first author was supported by CNPq-Brazil Grant 306992/2003-5. The first and second author were supported by FAPESP-Brazil Grant 03/03107-9.     

Lagrangian minimal surfaces in Lorentzian complex plane

Flat Lagrangian minimal surfaces in the Lorentzian complex plane are classified by B. Y. Chen and L. Vrancken in [8]. On the other hand, Vrancken proves in [11] that Lagrangian minimal surfaces of constant curvature in are flat surfaces. In this article, we classify all Lagrangian minimal surfaces in which are free from flat points. Received: 16 December 2007     

A conic bundle degenerating on the Kummer surface

Let C be a genus 2 curve and the moduli space of semi-stable rank 2 vector bundles on C with trivial determinant. In Bolognesi (Adv Geom 7(1):113–144, 2007) we described the parameter space of non stable extension classes of the canonical sheaf ω of C by ω⁻¹. In this paper, we study the classifying rational map that sends an extension class to the corresponding rank two vector bundle. Moreover, we prove that, if we blow up along a certain cubic surface S and at the point p corresponding to the bundle , then the induced morphism defines a conic bundle that degenerates on the blow up (at p) of the Kummer surface naturally contained in . Furthermore we construct the -bundle that contains the conic bundle and we discuss the stability and deformations of one of its components.     

On the Integral Representation of Spherical k-Regular Functions in Clifford Analysis

We study the null solutions of iterated applications of the spherical (Atiyah-Singer) Dirac operator on locally defined polynomial forms on the unit sphere of ; functions valued in the universal Clifford algebra , here called spherical k-regular functions. We construct the kernel functions, get the integral representation formula and Cauchy integral formula of spherical k-regular functions, and as applications, the weak solutions of higher order inhomogeneous spherical (Atiyah-Singer) Dirac equations . We obtain, in particular, the weak solution of an inhomogeneous spherical Poisson equation Δ _s g = f. This work was partially supported by NNSF of China (No.10471107) and RFDP of Higher Education (No.20060486001).     

Bauer–Furuta invariants under $${\mathbb{Z}_2}$$ -actions

S. Bauer and M. Furuta defined a stable cohomotopy refinement of the Seiberg–Witten invariants. In this paper, we prove a vanishing theorem of Bauer–Furuta invariants for 4-manifolds with smooth -actions. As an application, we give a constraint on smooth -actions on homotopy K3#K3, and construct a nonsmoothable locally linear -action on K3#K3. We also construct a nonsmoothable locally linear -action on K3.      

Quasiperiodic Group Factorizations

If is any ring or semi-ring (e.g., ) and G is a finite abelian group, two elements a, b of the group (semi-)ring are said to form a factorization of G if ab = rΣ _g∈G g for some . A factorization is called quasiperiodic if there is some element g ∈ G of order m > 1 such that either a or b – say b – can be written as a sum b ₀ + ... + b _m−1 of m elements of such that ab _h = g ^h ab ₀ for h = 0, ... , m − 1. Hajós [5] conjectured that all factorizations are quasiperiodic when and r = 1 but Sands [15] found a counterexample for the group . Here we show however that all factorizations of abelian groups are quasiperiodic when and that all factorizations of cyclic groups or of groups of the type are quasiperiodic when . We also give some new examples of non-quasiperiodic factorizations with for the smaller groups and . Received: May 12, 2006. Revised: October 3, 2007.     

The $${\overline{\partial}}$$-equation on homogeneous varieties with an isolated singularity

Let X be a regular irreducible variety in , Y the associated homogeneous variety in , and N the restriction of the universal bundle of to X. In the present paper, we compute the obstructions to solving the -equation in the L ^p -sense on Y for 1 ≤  p ≤  ∞ in terms of cohomology groups . That allows to identify obstructions explicitly if X is specified more precisely, for example if it is equivalent to or an elliptic curve.      

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Lie brackets and singular type of real hypersurfaces

This note is the sequel of Barraud and Mazzilli Math Z 248:757–772, 2004, where the regular type of a real hypersurface H in (eventually endowed with an almost complex structure) was characterized in terms of Lie brackets of complex tangent vector fields on H. This note extends these results to the singular type.     

Closed geodesics on compact nilmanifolds with Chevalley rational structure

We study the distribution of closed geodesics on nilmanifolds Γ \ N arising from a 2-step nilpotent Lie algebra constructed from an irreducible representation of a compact semisimple Lie algebra on a real finite dimensional vector space U. We determine sufficient conditions on the semisimple Lie algebra for Γ \ N to have the density of closed geodesics property where Γ is a lattice arising from a Chevalley rational structure on .     

Anisotropic version of a theorem of H. Hopf

Given a positive function F on S ² which satisfies a convexity condition, we define a function for surfaces in which is a generalization of the usual mean curvature function. We prove that an immersed topological sphere in with = constant is the Wulff shape, up to translations and homotheties.      

Smooth structures,normalized Ricci flows,and finite cyclic groups

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors [Ishida, The normalized Ricci flow on four-manifolds and exotic smooth structures; Şuvaina, Einstein metrics and smooth structures on non-simply connected 4-manifolds] we prove that for any finite cyclic group , where d > 1, there exist infinitely many compact topological 4-manifolds, with fundamental group , which admit at least one smooth structure for which non-singular solutions of the normalized Ricci flow exist, but also admit infinitely many distinct smooth structures for which no non-singular solution of the normalized Ricci flow exists. We show that there are no non-singular -equivariant, d > 1, solutions to the normalized Ricci flow on appropriate connected sums of and .     

Syzygies of grids of projective lines

In this paper we study the syzygy modules of a grid or a fat grid of . We compute the minimal free resolution for the ideal of a complete grid in , and we conjecture this resolution in . Moreover we compute the minimal free resolution for the ideal of an incomplete grid of . We also conjecture the minimal free resolution for the ideal of a fat complete grid in .      

OPC Lattices and Congruence Heredity

We prove that if is a finite algebra which satisfies a nontrivial idempotent Mal’cev condition, and if Con contains a copy of an order polynomially complete lattice other than , , or Con, then Con is not hereditary. Received March 7, 2006; accepted in final form December 5, 2006.     

Lifshits Tails for Random Schrödinger Operators with Nonsign Definite Potentials

This paper is devoted to the study of Lifshits tails for random Schr?dinger operator acting on of the form , where H ₀ is a -periodic Schr?dinger operator, λ is a positive coupling constant, are i.i.d and bounded random variables and V is the single site potential with changing sign. We prove that, in the weak disorder regime, at an open band edge, a true Lifshits tail for the random Schr?dinger operator occurs under a certain set of conditions on H ₀ and on V. Submitted: April 17, 2007. Accepted: December 13, 2007.     

Density property for hypersurfaces $$UV = P({\bar X})$$

We study hypersurfaces of given by equations of form where the zero locus of a polynomial p is smooth reduced. The main result says that the Lie algebra generated by algebraic completely integrable vector fields on such a hypersurface coincides with the Lie algebra of all algebraic vector fields. Consequences of this result for some conjectures of affine algebraic geometry and for the Oka-Grauert-Gromov principle are discussed.      

Projective curves of degree=codimension+2

In this article we study nondegenerate projective curves of degree d which are not arithmetically Cohen-Macaulay. Note that for a rational normal curve and a point . Our main result is about the relation between the geometric properties of X and the position of P with respect to . We show that the graded Betti numbers of X are uniquely determined by the rank of P with respect to . In particular, X satisfies property N _2,p if and only if . Therefore property N _2,p of X is controlled by and conversely can be read off from the minimal free resolution of X. This result provides a non-linearly normal example for which the converse to Theorem 1.1 in (Eisenbud et al., Compositio Math 141:1460–1478, 2005) holds. Also our result implies that for nondegenerate projective curves of degree d which are not arithmetically Cohen–Macaulay, there are exactly distinct Betti tables.     

On commuting polynomial automorphisms of {\mathbb{C}}^{k} , k ≥ 3

We characterize the polynomial automorphisms of ${\mathbb{C}}^3We characterize the polynomial automorphisms of , which commute with a regular automorphism. We use their meromorphic extension to and consider their dynamics on the hyperplane at infinity. We conjecture the additional hypothesis under which the same characterization is true in all dimensions. We give a partial answer to a question of S. Smale that in our context can be formulated as follows: can any polynomial automorphism of be the uniform limit on compact sets of polynomial automorphisms with trivial centralizer (i.e. )? Partially supported by Progetto MURST di Rilevante Interesse Nazionale Proprietà geometriche delle varietà reali e complesse. Supported by Istituto Nazionale Alta Matematica, “F. Severi”, Roma and G.N.S.A.G.A., Roma.