Splitting Polytopes

A split of a polytope $P$ is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of $P$ admits a unique decomposition as a linear combination of weight functions corresponding to the splits of $P$ (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope $P$, the split complex of $P$. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].Comment: 25 pages, 7 figures; minor corrections and change

Eulerian digraphs and toric Calabi-Yau varieties

We investigate the structure of a simple class of affine toric Calabi-Yau varieties that are defined from quiver representations based on finite eulerian directed graphs (digraphs). The vanishing first Chern class of these varieties just follows from the characterisation of eulerian digraphs as being connected with all vertices balanced. Some structure theory is used to show how any eulerian digraph can be generated by iterating combinations of just a few canonical graph-theoretic moves. We describe the effect of each of these moves on the lattice polytopes which encode the toric Calabi-Yau varieties and illustrate the construction in several examples. We comment on physical applications of the construction in the context of moduli spaces for superconformal gauged linear sigma models.Comment: 27 pages, 8 figure

Every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces

We prove that every symplectic toric orbifold is a centered reduction of a Cartesian product of weighted projective spaces. A theorem of Abreu and Macarini shows that if the level set of the reduction passes through a non-displaceable set then the image of this set in the reduced space is also non-displaceable. Using this result we show that every symplectic toric orbifold contains a non-displaceable fiber and we identify this fiber.Comment: 20 pages, 11 figures; Final version. Accepted at IMRN. Comments from the referees included. Section about Gromov width added. Moreover we fixed some small mistakes that unfortunately made it to the published version (moment polytope for the weighted projective space was not fully correct; at some point a not connected subgroup was called a torus

Small Covers, infra-solvmanifolds and curvature

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real moment-angle manifolds with certain conditions on the Ricci or sectional curvature. We will see that these curvature conditions put very strong restrictions on the topology of the corresponding small covers and real moment-angle manifolds and the combinatorial structure of the underlying simple polytopes.Comment: 22 pages, no figur