Unirational Compact Algebraic Manifolds: Oka or aOka-1?

Determine whether every compact algebraic manifold that is unirational—equivalently, admits a surjective morphism C^n → X—is an Oka manifold and whether it satisfies the algebraic Oka‑1 (aOka‑1) property.

Background

Using results of Arzhantsev–Kaliman–Zaidenberg, the authors note that a compact algebraic manifold X is unirational if and only if there is a surjective morphism C^n → X. Such X is therefore densely dominable by C^n, implying X is Oka‑1 by Alarcón–Forstnerič’s theorem.

The stronger properties—being Oka or aOka‑1—remain unresolved for unirational compact algebraic manifolds, and settling this would clarify the precise placement of unirational manifolds within the hierarchy of Oka-type classes.