Volume 23, Issue 3

We study the topology of moduli spaces of weighted stable tropical curves Δ g , w with fixed genus and unit volume. The space Δ g , w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space M g , w of weighted stable curves. When the genus is positive, we show that Δ g , w is simply connected for any choice of weight vector w . We also give a formula for the Euler characteristic of Δ g , w in terms of the combinatorics of the weight vector.

We use the method of eigenfamilies to construct explicit complex-valued proper p -harmonic functions on the compact real Grassmannians. We also find proper p -harmonic functions on the real flag manifolds which do not descend onto any of the real Grassmannians.

A Kähler metric is called central if the determinant of its Ricci endomorphism is constant; see [12]. For the case in which this constant is zero, we study on 4-manifolds the existence of complete metrics of this type which have cohomogeneity one for three unimodular 3-dimensional Lie groups: SU(2), the group E (2) of Euclidean plane motions, and a quotient by a discrete subgroup of the Heisenberg group nil 3 . We obtain a complete classification for SU(2), and some existence results for the other two groups, in terms of specific solutions of an associated ODE system.

The tropical fundamental class of a rational balanced polyhedral fan induces cap products between tropical cohomology and tropical Borel–Moore homology. When all these cap products are isomorphisms, the fan is said to be a tropical Poincaré duality space . If all the stars of faces also are such spaces, such as for fans of matroids, the fan is called a local tropical Poincaré duality space . In this article, we first give some necessary conditions for fans to be tropical Poincaré duality spaces and a classification in dimension one. Next, we prove that tropical Poincaré duality for the stars of all faces of dimension greater than zero and a vanishing condition implies tropical Poincaré duality of the fan. This leads to necessary and sufficient conditions for a fan to be a local tropical Poincaré duality space. Finally, we use such fans to show that certain abstract balanced polyhedral spaces satisfy tropical Poincaré duality.

We introduce the notion of a wall-connected twin building and show that the local-to-global principle holds for these twin buildings. As each twin building satisfying Condition (co) (introduced in [7]) is wall-connected, we obtain a strengthening of the main result of [7] that covers also the thick irreducible affine twin buildings of rank at least 3.

We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G -bundles on a smooth projective curve for a reductive algebraic group G . For example, our result applies to the stack of semistable G -bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G -bundles. Similar arguments apply to Gieseker semistable G -bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G -bundles admitting a quasiprojective good moduli space can be written naturally as a G -linearized global quotient Y / G , so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G -Higgs bundles on a family of curves is smooth over any base in characteristic 0.

We study abelian covers of rational surfaces branched over line arrangements. We use these covers to address the geography problem for closed simply connected nonspin irreducible symplectic 4-manifolds with positive signature.

In this work we describe horofunction compactifications of metric spaces and finite-dimensional real vector spaces through asymmetric metrics and asymmetric polyhedral norms by means of nonstandard methods, that is, by ultrapowers of the spaces at hand. The polyhedral compactifications of the vector spaces carry the structure of stratified spaces with the strata indexed by dual faces of the polyhedral unit ball. Explicit neighborhood bases and descriptions of the horofunctions are provided.