Volume 22, Issue 2

We analyze Higgs bundles ( V , ϕ ) on a class of elliptic surfaces π : X → B , whose underlying vector bundle V has vertical determinant and is fiberwise semistable. We prove that if the spectral curve of V is reduced, then the Higgs field ϕ is vertical , while if the bundle V is fiberwise regular with reduced (respectively, integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of B and the degree of the fundamental line bundle of π (respectively, if the fundamental line bundle is sufficiently ample), then ϕ is scalar . We apply these results to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on the class of elliptic surfaces considered, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to X .

We introduce the notion of rigidity for harmonic-Ricci solitons and provide some characterizations of rigidity, generalizing known results for Ricci solitons. In the complete case we restrict to steady and shrinking gradient solitons, while in the compact case we treat general solitons without further assumptions. We show that the rigidity can be traced back to the vanishing of certain modified curvature tensors that take into account the geometry of a Riemannian manifold equipped with a smooth map φ , called φ -curvature, which is a natural generalization in the setting of harmonic-Ricci solitons of the standard curvature tensor.

Given a nilpotent Lie algebra, we study the space of all diagonalizable derivations such that the corresponding one-dimensional solvable extension admits a left-invariant metric with negative Ricci curvature. Lauret and Will have conjectured that such a space coincides with an open and convex subset of derivations defined in terms of the moment map for the variety of nilpotent Lie algebras. We prove the validity of the conjecture in dimensions ≤ 5, as well as for Heisenberg Lie algebras and standard filiform Lie algebras.

For an embedded submanifold Σ ⊂ ℝ N , Belkin and Niyogi showed that one can approximate the Laplacian operator using heat kernels. Using a definition of coarse Ricci curvature derived by iterating Laplacians, we approximate the coarse Ricci curvature of submanifolds Σ in the same way. For this purpose, we derive asymptotics for the approximation of the Ricci curvature proposed in [2]. Specifically, we prove Proposition 3.2 in [2].

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace.

We present various results about the combinatorial properties of line arrangements in terms of the Chern numbers of the corresponding log surfaces. This resembles the study of the geography of surfaces of general type. We prove some new results about the distribution of Chern slopes, we show a connection between their accumulation points and the accumulation points of linear H -constants on the plane, and we conclude with two open problems in relation to geography over ℚ and over ℂ.

A cube tiling of ℝ d is a family of pairwise disjoint cubes [0, 1) d + T = {[0, 1) d + t : t ∈ T } such that ∪ t ∈ T ([0, 1) d + t ) = ℝ d . Two cubes [0, 1) d + t , [0, 1) d + s are called a twin pair if | t j − s j | = 1 for some j ∈ [ d ] = {1, ⋅, d } and t i = s i for every i ∈ [ d ]∖{ j }. In 1930, Keller conjectured that in every cube tiling of ℝ d there is a twin pair. For x ∈ ℝ d and i ∈ [ d ], let L ( T , x , i ) be the set of all i th coordinates t i of vectors t ∈ T such that ([0, 1) d + t )∩([0, 1] d + x )≠∅ and t i ≤ x i . Let r−(T)=minx∈Rdmax1≤i≤d|L(T,x,i)|$r^-(T)=\min_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$ and r+(T)=maxx∈Rdmax1≤i≤d|L(T,x,i)|$r^ + (T)=\max_{x\in \mathbb{R}^d} \max_{1\leq i\leq d}|L(T,x,i)|$. Before 2019 it was known that Keller’s conjecture is true for dimensions d ≤ 6 and false for all dimensions d = 8. Moreover, in dimension 7 it was known to be true if r − ( T ) ≤ 2 or r + ( T ) = 5. The present paper resolves the case r + ( T ) = 4. At the end of 2019, when the paper was still under review, Brakensiek et al. resolved the cases r + ( T ) ∈ {3, 4, 6}, proving thereby Keller’s conjecture in dimension 7.