Simplicial-volume spectrum of closed Kähler surfaces

Determine the set SV_{\mathrm{Kah}(4)} of simplicial volumes of closed connected Kähler surfaces; in particular, decide whether it contains every nonnegative rational number and whether every closed oriented connected manifold has the same simplicial volume as some closed connected Kähler surface.

Background

The paper introduces SV_{\mathrm{Kah}(4)} as the set of simplicial volumes realized by closed connected Kähler surfaces. Although every nonnegative rational number is known to occur as the simplicial volume of a closed oriented connected 4-manifold, the cited construction does not produce Kähler surfaces. The Kähler realization spectrum, including the proposed universality for nonnegative rationals and realization of arbitrary manifold simplicial volumes, is therefore unresolved.

References

Determine \operatorname{SV}_{\mathrm{Kah}(4)}. In particular, does it contain every nonnegative rational number? More generally, given a closed oriented connected manifold M, does there exist a closed connected Kähler surface X such that

\lVert X\rVert=\lVert M\rVert?

— Simplicial Volume and Scalar Curvature on Closed Kähler Surfaces  (2608.17335 - Min et al., 18 Aug 2026) in Section 5, final Section 5 problem