Volume 14, Issue 4

We prove that if a hypersurface M with constant scalar curvature in S 6 (1) has the sectional curvatures ≤ 1, then the type number of M is not greater than 1.

Ehrhart’s conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kähler- Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree by proving the corresponding convex-geometric statement. We also discuss a generalized version of Ehrhart’s conjecture involving an invariant corresponding to the so-called greatest lower bound on the Ricci curvature.

Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold with nondegenerate induced metric which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation formula of the volume of L with respect to Hamiltonian variations and apply this formula to several cases. We observe that a minimal Lagrangian submanifold L in a Ricci-flat pseudo- or para-Kähler manifold is H-stable, i.e. its second variation is definite and L is in particular a local extremizer of the volume with respect to Hamiltonian variations. We also give a stability criterion for spacelike minimal Lagrangian submanifolds in para-Kähler manifolds, similar to Oh’s stability criterion for minimal Lagrangian manifolds in Kähler-Einstein manifolds (cf. [20]). Finally, we determine the H-stability of a series of examples of H-minimal Lagrangian submanifolds: the product S 1 (r 1 ) x ··· x S 1 (r n ) of n circles of arbitrary radii in complex space Cn is H-unstable with respect to any indefinite flat Hermitian metric, while the product ℍ 1 (r 1 ) x ···x ℍ 1 (r n ) of n hyperbolas in para-complex vector space D n is H-stable for n = 1; 2 and H-unstable for n ≥ 3. Recently, minimal Lagrangian surfaces in the space of geodesics of space forms have been characterized ([4], [11]); on the other hand, a class of H-minimal Lagrangian surfaces in the tangent bundle of a Riemannian, oriented surface has been identified in [6]. We discuss the H-stability of all these examples.

Hyper-Kähler manifolds of type A∞ are noncompact complete Ricci-flat Kähler manifolds of complex dimension 2. We study the holomorphic symplectic structures preserved by the natural ℂ x -actions on these manifolds, and we give sufficient and necessary conditions for the existence of ℂ x -equivariant biholomorphisms between two hyper-Kähler manifolds of type A∞ preserving their holomorphic symplectic structures. As a consequence, we can show the existence of a complex manifold of dimension 2 on which there is a continuous family of complete Ricci-flat Kähler metrics with distinct volume growth.

We show a tight relation between bifurcation values of a fixed polynomial f ∈ R[X; Y] and the family of polynomials bounded on a set { x ∈ R 2 | f(x) ≤ c}. If g ∈ R[X; Y] is bounded on {x ∈ R 2 | f(x) ≤ c}, then it is bounded on the bigger set {x ∈ R 2 | f(x) ≤ c̃} with c < c̃ provided that the interval [c; c̃] does not contain any complex bifurcation value at infinity of f.

A cone curve is a reduced sextic space curve which lies on a quadric cone and does not pass through the vertex. We classify families of bitangent planes of cone curves. The methods we apply can be used for any space curve with ADE singularities, though in this paper we concentrate on cone curves. An embedded complex projective surface which is adjoint to a degree one weak Del Pezzo surface contains families of minimal degree rational curves, which cannot be defined by the fibers of a map. Such families are called minimal non-fibration families. Families of bitangent planes of cone curves correspond to minimal non-fibration families. The main motivation of this paper is to classify minimal non-fibration families. We present algorithms which compute all bitangent families of a given cone curve and their geometric genus. We consider cone curves to be equivalent if they have the same singularity configuration. For each equivalence class of cone curves we determine the possible number of bitangent families and the number of rational bitangent families. Finally we compute an example of a minimal non-fibration family on an embedded weak degree one Del Pezzo surface.

In this paper, vectorial type deformations of a 7-dimensional manifold with structure group G 2 are considered. The fundamental 3-form of such a manifold is deformed by a vector field, and the new cross product and the new Levi-Civita covariant derivation of the new metric are obtained in terms of the old ones. An example of a 7-dimensional manifold with structure group G 2 which is in the widest class W in the classification of Fernández and Gray is given; this manifold has a nontrivial Killing vector field corresponding to the new metric.

We compute the Betti numbers and describe the cohomology algebras of the ordered and unordered configuration spaces of three points in complex projective spaces, including the infinite dimensional case. We also compute these invariants for the configuration spaces of three collinear and non-collinear points.

Let (X,-K X ) be a Fano fourfold and let S⊂X be a smooth surface. We show that (X,-K X ) is slope semistable with respect to S if c 1 (N S|X ) is a nontrivial and nef class.

We study the geometry of quartic surfaces in P 3 that contain a line of the second kind over algebraically closed fields of characteristic different from 2; 3. In particular, we correct Segre’s claims made for the complex case in 1943.