We study the algebraic dimension $a(X)$ of a compact hyperkähler manifold of dimension $2n$. We show that $a(X)$ is at most $n$ unless $X$ is projective. If a compact Kähler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0, $n$ and $2n$ are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If $n = 2$, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results for ”generalised hyperkähler” manifolds and studies nef lines bundles.

We construct a Kruskal-Szekeres-type analytic extension of the Emparan-Reall black ring, and investigate its geometry. We prove that the extension is maximal, globally hyperbolic, and unique within a natural class of extensions. The key to those results is the proof that causal geodesics are either complete, or approach a singular boundary in finite affine time. Alternative maximal analytic extensions are also constructed.

Based on large $N$ Chern-Simons/topological string duality, in a series of papers, Labastida, Mariño, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.