Compact type-(γ) embeddings in complex surfaces

Construct an embedding of a connected compact complex curve into a compact complex surface that is of Ueda type (γ), meaning that the embedding is formally vertically linearizable but not vertically linearizable.

Background

The paper describes Ueda type (γ) embeddings as embeddings of connected compact complex curves with degree-zero normal bundle that are formally vertically linearizable—equivalently, all Ueda classes vanish—but are not vertically linearizable. For compact surfaces, vertical linearizability is equivalent to Hermitian semipositivity of the divisor line bundle.

The authors note that examples of type (γ) embeddings are known for elliptic curves in noncompact surfaces, while their Theorem 1.6 proves that no such embedding exists when the curve is a smooth elliptic curve in a compact complex surface. Thus the explicitly stated construction problem concerns the remaining general compact setting, not the elliptic-curve case ruled out by the paper.

References

Constructing an example of a type $(\gamma)$ embedding into a compact complex surface is a well-known open problem p.1464, p.242, p.24.

— Compact Kähler surfaces with semipositive anticanonical bundle  (2609.26716 - Chen et al., 22 Sep 2026) in Section 1, subsection “Ueda Theory”

The compactness of $S$ is used only in the implication (1)$\Rightarrow$(3). It is conjectured in that this should hold even when $S$ is noncompact.

— Compact Kähler surfaces with semipositive anticanonical bundle  (2609.26716 - Chen et al., 22 Sep 2026) in Proposition 2.1 and the paragraph immediately following its proof, Section 2 “Vertical linearization and Ueda Theory”