The diversity of symplectic Calabi-Yau six-manifolds

Given an integer b and a finitely presented group G we produce a compact symplectic six-manifold with c_1 = 0, b_2 > b, b_3 > b and fundamental group G. In the simply-connected case we can also arrange for b_3 = 0; in particular these examples are not diffeomorphic to K\"ahler manifolds with c_1 = 0. The construction begins with a certain orientable four-dimensional hyperbolic orbifold assembled from right-angled 120-cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi-Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.Comment: 18 pages, 1 figure. v2 added proof that b_3 can also be taken arbitrarily larg

Intersections of Quadrics, Moment-angle Manifolds and Connected Sums

The topology of the intersection of two real homogeneous coaxial quadrics was studied by the second author who showed that its intersection with the unit sphere is in most cases diffeomorphic to a connected sum of sphere products. Combining that approach with a recent one (due to Antony Bahri, Martin Bendersky, Fred Cohen and the first author) we study here the intersections of k>2 quadrics and we identify very general families of such manifolds that are diffeomorphic to connected sums of sphere products. These include those moment-angle manifolds for which the result was conjectured by Frederic Bosio and Laurent Meersseman. As a byproduct, a simpler and neater proof of the result for the case k=2 is obtained. Two new sections contain results not included in the first version of this article: Section 2 describes the topological change on the manifolds after the operations of cutting off a vertex or an edge of the associated polytope, which can be combined in a special way with the previos results to produce new infinite families of manifolds that are connected sums of sphere products. In other cases we get slightly more complicated manifolds: with this we solve another question by Bosio-Meersseman about the manifold associated to the truncated cube. In Section 3 we use this to show that the known rules for the cohomology product of a moment-angle manifold have to be drastically modified in the general situation. We state the modified rule, but leave the details of this for another publication. Section 0 recalls known definitions and results and in section 2.1 some elementary topological constructions are defined and explored. In the Appendix we state and prove some results about specific differentiable manifolds, which are used in sections 1 and 2.Comment: We have included many clarifying suggestions and minor corrections from some colleagues who read the manuscript carefully. The only change in content from the previous version is the suppression a special case (item 3) of Theorem 1.3 because we have not been able to fill in the details of any of the known sketched proofs (including ours

Theoretical evidence for unexpected O-rich phases at corners of MgO surfaces

Realistic oxide materials are often semiconductors, in particular at elevated temperatures, and their surfaces contain undercoordiated atoms at structural defects such as steps and corners. Using hybrid density-functional theory and ab initio atomistic thermodynamics, we investigate the interplay of bond-making, bond-breaking, and charge-carrier trapping at the corner defects at the (100) surface of a p-doped MgO in thermodynamic equilibrium with an O2 atmosphere. We show that by manipulating the coordination of surface atoms one can drastically change and even reverse the order of stability of reduced versus oxidized surface sites.Comment: 5 papges, 4 figure

CP^n, or, entanglement illustrated

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP^3 and give a few new results concerning them.Comment: 28 pages, 9 figure