Existence and characterization of a deeper obstruction to affineness beyond symplectic homology growth

Ascertain the existence and characterize the nature of a symplectic‑topological obstruction preventing a Liouville manifold from being biholomorphic to an affine variety that is not detected by the growth rate of symplectic homology, and describe this obstruction within the context of Stein/Liouville manifolds admitting normal crossing divisor compactifications.

Background

The paper constructs a Stein 4‑manifold obtained as the complement of a normal crossing divisor in the Kodaira–Thurston manifold that is not affine, even though standard techniques using symplectic homology growth (Seidel, McLean) do not obstruct affineness in this case.

This example suggests that existing growth‑rate criteria are insufficient and motivates the search for a new, deeper obstruction to affineness in the symplectic/Stein setting, beyond current homological growth methods.