Volume 24, Issue 4

In this paper, we introduce the notion of local quasi-isometry for metric germs and prove that two definable germs are quasi-isometric if and only if their tangent cones are bi-Lipschitz homeomorphic. Since bi-Lipschitz equivalence is a particular case of local quasi-isometric equivalence, we obtain Sampaio’s tangent cone theorem as a corollary. As an application, we provide a different proof of the theorem by Fernandes-Sampaio, which states that the tangent cone of a Lipschitz normally embedded germ is also Lipschitz normally embedded.

We introduce the notion of moving absolute geometry of a geometry with triality and show that, in the classical case where the triality is of type ( I σ ) and the absolute geometry is a generalized hexagon, the moving 5 3 6 absolute geometry also gives interesting flag-transitive geometries with Buekenhout diagram for the groups G 2 ( k ) and 3 D 4 ( k ), for any prime power k ≥ 2. We also classify the absolute geometries for geometries with trialities but no dualities coming from maps of Class III with automorphism group L 2 ( q 3 ), where q ≥ 2 is prime power. We then investigate the moving absolute geometries for these geometries, illustrating their interest in this case.

An isomorphism between two hermitian unitals is provided, and used to treat isomorphisms of classical groups that are related to the isomorphism between certain simple real Lie algebras of types A and D (and rank 3).

Two subsets A , B of the plane are betweenness isomorphic if there is a bijection f : A → B such that, for every x , y , z ∈ A , the point f ( z ) lies on the line segment connecting f ( x ) and f ( y ) if and only if z lies on the line segment connecting x and y . In general, it is quite difficult to tell whether two given subsets of the plane are betweenness isomorphic. We concentrate on the case when each of the sets A , B is of the form C ∪ D where C is a circle and D is a finite set. We fully characterize the betweenness isomorphism classes in the family of all circles with three collinear points inside. In particular, we show that there are only countably many isomorphism classes, for which we provide an algebraic description.

We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them was locally homogeneous. In contrast, our new examples — all of them geodesically incomplete — realize all odd dimensions starting from 5 and are of rank 2, as well as locally homogeneous.

Given a closed, orientable Lagrangian submanifold L in a symplectic manifold ( X , ω ), we show that if L is relatively exact then any Hamiltonian diffeomorphism preserving L setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on L . Curiously, the proof uses only mod 2 coefficients in its singular and Floer cohomology rings.

A p -set of equiangular lines in ℂ 3 is a set of p lines spanning ℂ 3 each pair of which has the same nonzero angle arccos c , where 0 < c < 1. It is known that via a real matrix representation, a pair of lines in ℝ 3 with angle arccos c yields a pair of isoclinic planes in ℝ 6 with angle arccos c . In this article we characterize all p -tuples of equi-isoclinic planes in ℝ 6 which come via our real matrix representation from p -tuples of equiangular lines in ℂ 3 . More precisely, we describe quadruples, quintuples up to nine-tuples of equiangular lines in ℂ 3 .

In this paper, we establish several results related to vanishing theorems for Mather–Jacobian multiplier ideals on a Gorenstein projective variety, including an injectivity theorem, a Nadel-type vanishing theorem, a Griffiths-type vanishing theorem for vector bundles, and a vanishing theorem for the asymptotic Mather–Jacobian multiplier ideals.

The notion of a linear deformation of a codimension one foliation into contact structures was introduced in [5]. This concept is a special type of deformation of confoliations. In this paper, we study linear deformations of pairs of codimension one foliations into contact pairs. Applications of our main result are also provided.

Recently, the first and third author proved a correspondence theorem which recovers the Levine- Welschinger invariants of toric del Pezzo surfaces as a count of tropical curves weighted with arithmetic multiplicities. In this paper, we study properties of the arithmetic count ofplane tropical curves satisfying point conditions. We prove that this count is independent of the configuration of point conditions. Moreover, a Caporaso- Harris formula for the arithmetic count of plane tropical curves is obtained by moving one point to the very left. Repeating this process until all point conditions are stretched, we obtain an enriched count of floor diagrams which coincides with the tropical count. Finally, we prove polynomiality properties for the arithmetic counts using floor diagrams.

We study effective global generation of adjoint line bundles on smooth projective varieties. To measure the effectivity we introduce the concept of the convex Fujita number of a smooth projective variety and compute its value for a class of varieties with prescribed dimension d ≥ 2 and an arbitrary projective group as fundamental group.