We study Nakai-Moishezon type question and Donaldson’s “tamed to compatible” question for almost complex structures on rational four manifolds. By extending Taubes’ subvarieties-currentform technique to J-nef genus 0 classes, we give affirmative answers of these two questions for all tamed almost complex structures on S^2 bundles over S^2 as well as for many geometrically interesting tamed almost complex structures on other rational four manifolds, including the del Pezzo ones.

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle$π$in the middle points of a certain pair of its skew edges.

We study Legendrian embeddings of a compact Legendrian submanifold L sitting in a closed contact manifold (M,ξ) whose contact structure is supported by a (contact) open book OB on M. We prove that if OB has Weinstein pages, then there exist a contact structure ξ′ on M, isotopic to ξ and supported by OB, and a contactomorphism f:(M,ξ)→(M,ξ′) such that the image f(L) of any such submanifold can be Legendrian isotoped so that it becomes disjoint from the closure of a page of OB.

No abstract uploaded!

No abstract uploaded!