Oka manifolds are ubiquitous

Abstract

We prove that a holomorphic family of connected complex Lie groups has Oka total space if and only if its base is Oka. Extension presentations and generating sections give Oka results for stable and parabolic bundle moduli and for holomorphic mapping and embedding spaces. Every nonempty degree component of maps from a compact Oka curve to projective space is Oka; rational-function spaces are also holomorphically elliptic. For a vector bundle on a smooth projective curve, the Oka property of its nested Quot scheme is equivalent to that of the symmetric powers determined by the length increments. Complete Hamiltonian flows give holomorphic ellipticity of irreducible \(\mathrm{SL}_2\)- and \(\mathrm{GL}_2\)-character and representation loci, for closed surfaces of genus at least two and for surfaces with fixed regular semisimple boundary classes. The Riemann--Hilbert correspondence yields Oka moduli spaces of nonresonant parabolic connections for arbitrary parabolic weights, and holomorphic ellipticity when reducible monodromy is absent. A proper holomorphic fibration over a Riemann surface, with a section and connected complex tori as smooth general fibres, has Oka total space exactly when its base is Oka. For a Schoen fibre product \(X\) of rational elliptic surfaces with disjoint singular values, we construct an entire local biholomorphism \(\mathbb C^3\to X\) whose image contains a dense Zariski-open subset. Further applications treat Hilbert schemes, compactified relative Jacobians, and logarithmic complements.