Optimal Noether-type inequalities in higher dimensions

Determine whether optimal Noether-type inequalities exist for higher-dimensional non-general-type varieties of Kodaira dimension greater than one.

Background

The paper establishes sharp Noether-type inequalities for smooth projective threefolds of intermediate Kodaira dimensions: for Kodaira dimension two, the Iitaka volume satisfies Ivol⁡(V)≥34pg(V)−54\operatorname{Ivol}(V)\geq \frac34p_g(V)-\frac54, while for Kodaira dimension one it satisfies Ivol⁡(V)≥pg(V)−1\operatorname{Ivol}(V)\geq p_g(V)-1. The authors then ask whether analogous optimal inequalities are available beyond the threefold setting.

The question concerns higher-dimensional varieties that are not of general type and have Kodaira dimension κ>1\kappa>1. It is left unresolved and extends the paper’s investigation of quantitative relationships between Iitaka volume and geometric genus.

References

Naturally, out of curiosity about the statement of Theorem~\ref{thm:main1}, we would like to pose the following question: Do optimal Noether-type inequalities exist for higher dimensional non-general type varieties of Kodaira dimension $\kappa>1$?

— Sharp Noether-type inequalities for $3$-folds of intermediate Kodaira dimensions  (2610.02689 - Chen et al., 2 Oct 2026) in Open Question environment immediately following the final proposition in Section 4, “Examples”; conclusion of the paper