Geography problem of spin symplectic 4-manifolds

The geography problem asks whether every admissible lattice point $(\chi,c)$ can be realized by a simply connected spin symplectic $4$-manifold with $(\chi_h,c_1^2)=(\chi,c)$. The scan identifies no proposer or original date, and records only partial realization results.

Known results

• Park (2001): all but finitely many allowed points with $0\le c\le 8.76\chi$ are realized by simply connected spin symplectic $4$-manifolds.
• Park and Szabó (cited in 2010): broad realization results hold in the negative-signature region $0\le c_1^2<8\chi_h$.
• Akhmedov, Park, and Kürüz (2010): constructed families with $8.92<c_1^2/\chi_h<9$, with the ratio tending to $9$.
• A 2019 sequel produced infinitely many new smooth structures but did not settle the full problem.

August 2026 near-boundary constructions

A report dated August 26, 2026 describes constructions from complex surfaces of general type populating regions near the conjectural symplectic Bogomolov--Miyaoka--Yau boundary $c_1^2=9\chi_h$. This is a claimed advance toward the boundary, not a proof or disproof of the inequality or a complete solution of the geography problem.

Current status (as of August 2026): Partial realization results and a claimed, unverified near-boundary construction are known, while the full geography problem and the symplectic Bogomolov--Miyaoka--Yau inequality remain open.