Volume 23, Issue 2

For each contact diffeomorphism ϕ : ( Q , ξ ) → ( Q , ξ ) of ( Q , ξ ), we equip its mapping torus M ϕ with a locally conformal symplectic form of Banyaga’s type, which we call the lcs mapping torus of the contact diffeomorphism ϕ . In the present paper, we consider the product Q × S 1 = M id (corresponding to ϕ = id) and develop basic analysis of the associated J -holomorphic curve equation, which has the form ∂ ˉ π w = 0 , w ∗ λ ∘ j = f ∗ d θ $$\bar{\partial}^{\pi} w=0, \quad w^{*} \lambda \circ j=f^{*} d \theta$$ for the map u = ( w , f ) : Σ˙→Q×S1$\dot{\Sigma} \rightarrow Q \times S^{1}$for a λ -compatible almost complex structure J and a punctured Riemann surface (Σ˙,j).$(\dot{\Sigma}, j).$In particular, w is a contact instanton in the sense of [31], [32].We develop a scheme of treating the non-vanishing charge by introducing the notion of charge class in H1(Σ˙,Z)$H^{1}(\dot{\Sigma}, \mathbb{Z})$and develop the geometric framework for the study of pseudoholomorphic curves, a correct choice of energy and the definition of moduli spaces towards the construction of a compactification of the moduli space on the lcs-fication of ( Q , λ ) (more generally on arbitrary locally conformal symplectic manifolds).

An h -tiling on a finite simplicial complex is a partition of its geometric realization by maximal simplices deprived of several codimension one faces together with possibly their remaining face of highest codimension. In this last case, the tiles are said to be critical. An h -tiling thus induces a partitioning of its face poset by closed or semi-open intervals. We prove the existence of h -tilings on every finite simplicial complex after finitely many stellar subdivisions at maximal simplices. These tilings are moreover shellable. We also prove that the number of tiles of each type used by a tiling, encoded by its h -vector, is determined by the number of critical tiles of each index it uses, encoded by its critical vector. In the case of closed triangulated manifolds, these vectors satisfy some palindromic property. We finally study the behavior of tilings under any stellar subdivision.

We examine to what extent a projection theorem is possible in the non-commutative (free) setting. We first review and extend some results that count against a full free projection theorem. We then obtain a weak version of the projection theorem: projections along linear and separated variables yield semialgebraically parametrised free semi-algebraic sets.

We consider Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced in [5; 4], and show that they have Jordan-Hölder filtrations whose quotients are stable, locally free and H-nflat. This is applied to show that curve semistable Higgs bundles on simply connected Calabi–Yau varieties have vanishing discriminant.

The convex-geometric Minkowski tensors contain information about shape and orientation of the underlying convex body. They therefore yield valuable summary statistics for stationary marked point processes with marks in the family of convex bodies, or, slightly more specialised, for stationary particle processes. We show here that if the distribution of the typical particle is invariant under rotations about a fixed k -plane, then the average volume tensors of the typical particle can be derived from k + 1-dimensional sections. This finding extends the well-known three-dimensional special case to higher dimensions. A corresponding result for the surface tensors is also proven. In the last part of the paper we show how Minkowski tensors can be used to define three ellipsoidal set-valued summary statistics, discuss their estimation and illustrate their construction and use in a simulation example. Two of these, the so-called Miles ellipsoid and the inertia ellipsoid, are based on mean volume tensors of ranks up to 2. The third, based on the mean surface tensor of rank 2, will be called the Blaschke ellipsoid and is only defined when the typical particle has a rotationally symmetric distribution about an axis, as we then can use uniqueness and reconstruction results for centred ellipsoids of revolution from their rank-2 surface tensor. The latter are also established here.

We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k -connected closed n -manifold ( n ≥ 7, k < ( n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊 2 n − k . We then prove that every closed manifold M 2 n +1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊 2⌊3 n /2⌋+3 . We also prove that every closed manifold M 2 n +1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ 2 n +3 . Finally, we give various examples of contact open book embeddings of contact (2 n + 1)-manifolds in the trivial supporting open book of the standard contact structure on 𝕊 4 n +1 .

We prove a classification result for smooth complex Fano fourfolds of Picard number 3 having a prime divisor of Picard number 1, after a characterisation result in arbitrary dimension by Casagrande and Druel [5]. These varieties are obtained by blowing-up a ℙ 1 -bundle over a smooth Fano variety of Picard number 1 along a codimension 2 subvariety. We study in detail the case of dimension 4, and show that they form 28 families. We compute the main numerical invariants, determine the base locus of the anticanonical system, and study their deformations to give an upper bound to the dimension of the base of the Kuranishi family of a general member.

A polar space 𝒮 is called symplectic if it admits a projective embedding ε : 𝒮 → PG( V ) such that the image ε (𝒮) of 𝒮 by ε is defined by an alternating form of V . In this paper we characterize symplectic polar spaces in terms of their incidence properties, with no mention of peculiar properties of their embeddings. This is relevant especially when 𝒮 admits different (non-isomorphic) embeddings, as it is the case when 𝒮 is defined over a field of characteristic 2.

We compute the Chow groups of smooth Gushel–Mukai varieties of dimension 5.