Volume 8, Issue 3

In this article, we investigate projective domains with a strictly convex point in the boundary and their automorphisms. We prove that ellipsoids can be characterized as follows: A domain Ω is an ellipsoid if and only if ∂Ω is locally strongly convex at some boundary point where an Aut(Ω)-orbit accumulates. We also show that every quasi-homogeneous projective domain in an affine space which is locally strictly convex at a boundary point, is the universal covering of a closed projective manifold.

We study the geometry of n -Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n -dimensional totally geodesic submanifolds. We introduce the n -Grassmannian structure as a distinguished distribution on the Grassmann bundle, and then compute the n -Grassmannian invariants, recovering for n = 1 the projective invariants of Thomas.

A 6-parametric family of 6-dimensional quasi-Kähler manifolds with Norden metric is constructed on a Lie group. This family is characterized geometrically.

Let J be a unitary almost complex structure on a Riemannian manifold ( M, g ). If x is a unit tangent vector, let π := Span{ x, Jx } be the associated complex line in the tangent bundle of M . The complex Jacobi operator and the complex curvature operators are defined, respectively, by and . We show that if ( M, g ) is Hermitian or if ( M, g ) is nearly Kähler, then either the complex Jacobi operator or the complex curvature operator completely determine the full curvature operator; this generalizes a well known result in the real setting to the complex setting. We also show that this result fails for general almost Hermitian manifolds.

Let P be a finite polar space of rank r ≥ 2 with q + 1 ≥ 3 points on each line. In [J. Bamberg, S. Kelly, M. Law, T. Penttila, Tight sets and m -ovoids of finite polar spaces. J. Combin. Theory Ser. A 114 (2007), 1293–1314. MR2353124] and [J. Bamberg, M. Law, T. Penttilla, Tight sets and m -ovoids of generalised quadrangles. Combinatorica , to appear.] it was shown that every m -ovoid of P intersects every i -tight set of P in precisely mi points. In the present paper, we characterize m -ovoids of P as those sets of points which have constant intersection size with each member of a “nice family” of i -tight sets of P and conversely, we characterize i -tight sets of P as those sets of points which have constant intersection size with each member of a “nice family” of m -ovoids. Some interesting corollaries of these characterization theorems are given.

In this paper we give a sharp lower bound on the dimension of Grassmann secant varieties of a given variety and we classify varieties for which the bound is attained.

We classify projective manifolds X of dimension n ≥ 3 with nef anticanonical class and index n – 1 such that .

The aim of this paper is to describe algebraic K3 surfaces with an even set of rational curves or of nodes. Their minimal possible Picard number is nine. We completely classify these K3 surfaces and after a careful analysis of the divisors contained in the Picard lattice we study their projective models, giving necessary and sufficient conditions to have an even set. Moreover we investigate their relation with K3 surfaces with a Nikulin involution.

Let X be a general divisor in |3 M + nL | on the rational scroll Proj where d i and n are integers, M is the tautological line bundle, L is a fibre of the natural projection to ℙ 1 , and d 1 d 4 = 0. We prove that X is rational d 1 = 0 and n = 1.

Let H 0 denote the kernel of the endomorphism, defined by , of the real algebraic group given by the Weil restriction of . Let X be a nondegenerate anisotropic conic in . The principal -bundle over the complexification defined by the ample generator of Pic(), gives a principal H 0 -bundle over X through a reduction of structure group. Given any principal G -bundle E G over X , where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that E G is isomorphic to the principal G -bundle obtained by extending the structure group of using . The tautological line bundle over the real projective line and the principal -bundle together give a principal -bundle F on , Given any principal G -bundle E G over , where G is any connected reductive linear algebraic group defined over ℝ, we prove that there is a homomorphism such that E G is isomorphic to the principal G -bundle obtained by extending the structure group of F using .