Volume 8, Issue 4

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [G. Calvaruso, Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57 (2007), 1279–1291. MR2287304 (2008b:53066) Zbl 1112.53051] and [G. Calvaruso, R. A. Marinosci, Homogeneous geodesics of three-dimensional unimodular Lorentzian Lie groups. Mediterr. J. Math. 3 (2006), 467–481. MR2274738 Zbl pre05151042], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

For plurisubharmonic solutions of the complex homogeneous Monge–Ampère equation whose level sets are hypersurfaces of finite type, in dimension 2, it is shown that the Monge–Ampère foliation is defined even at points of higher degeneracy. The result is applied to provide a positive answer to a question of Burns on homogeneous polynomials whose logarithms satisfy the complex Monge–Ampère equation and to generalize the work of P. M. Wong on the classification of complete weighted circular domains.

We investigate the horofunction boundary of the Hilbert geometry defined on an arbitrary finite-dimensional bounded convex domain D . We determine its set of Busemann points, which are those points that are the limits of “almost-geodesics”. In addition, we show that any sequence of points converging to a point in the horofunction boundary also converges in the usual sense to a point in the Euclidean boundary of D . We prove that all horofunctions are Busemann points if and only if the set of extreme sets of the polar of D is closed in the Painlevé–Kuratowski topology.

We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underlie modern techniques.

Let X be a smooth real curve of genus g ≥ 1 having some real point. Define M( X ) as being the smallest integer m such that each line bundle L on X of even degree at least 2 m having restrictions of even degree to each connected component of X (ℝ) contains a totally non-real divisor inside | L | (hence a divisor containing no real point of X ). In this paper we prove that M( X ) = g .

We consider ramified (Galois) covers of the upper half plane in the category of Klein surfaces. We study the connection between the group theoretical ramification data of the cover and its geometrical properties, such as the number of the connected components of the boundary and orientability of the surface.

Let ( X, L ) be a polarized manifold of dimension n defined over the field of complex numbers. Assume that L is spanned. Then, in this paper, we will classify ( X, L ) by the second sectional Betti number b 2 ( X, L ) and the second sectional Hodge number ( X, L ) of type (1, 1), which were defined by the author in a previous paper. Moreover, for the case where L is very ample, we will give a classification of ( X, L ) with b 4 ( X, L ) = h 4 ( X , ℂ).

Let B be an o -symmetric convex body in ℝ d , and M d be the normed space with unit ball B . The M d -thickness Δ B ( K ) of a convex body K ⊆ ℝ d is the smallest possible M d -distance between two distinct parallel supporting hyperplanes of K . Furthermore, K is said to be M d -reduced if Δ B ( K ′) < Δ B ( K ) for every convex body K ′ with K ′ ⊆ K and K ′ ≠ K . In our main theorems we describe M d -reduced polytopes as polytopes whose face lattices possess certain antipodality properties. As one of the consequences, we obtain that if the boundary of B is regular, then a d -polytope with m facets and n vertices is not M d -reduced provided m = d + 2 or n = d + 2 or n > m . The latter statement yields a new partial answer to Lassak's question on the existence of Euclidean reduced d -polytopes for d ≥ 3.