Indecomposability
The bounded derived category $\mathbf{D}^{\mathrm{b}}(X)$ is indecomposable if it admits no non-trivial semiorthogonal decomposition. It is conjectured that $\mathbf{D}^{\mathrm{b}}(X)$ is indecomposable for every hyperelliptic variety (Belmans–Demleitner–Núñez).
This is established (indeed stably indecomposable) whenever
- $G$ is cyclic, or
- $q = \dim X - 1$ or $q = \dim X - 2$, or
- the Albanese fiber has trivial canonical bundle.
In particular all hyperelliptic threefolds are indecomposable. In dimension $4$ the criteria above settle most cases; the regular fourfolds with holonomy $\mathrm{D}_4 \times \mathrm{C}_2$ (where $q = 0$ and $\omega_X \not\cong \mathcal{O}_X$) escape all three and remain open. On object pages this is recorded as "indecomposable" or "open".
The relevant machinery is Pirozhkov's theory of stably indecomposable varieties and the criterion of Kawatani–Okawa.
References
- P. Belmans, A. Demleitner, P. Núñez, The Albanese morphism for hyperelliptic varieties, Indag. Math. (N.S.) 37 (2026) 1450–1475. MR5103244 doi
- D. Pirozhkov, Stably semiorthogonally indecomposable varieties, Épijournal Géom. Algébrique 7 (2023). MR4582884 doi
- F. Caucci, Paracanonical base locus, Albanese morphism, and semi-orthogonal indecomposability, Selecta Math. (N.S.) 30 (2024) no. 86. MR4805086 doi
- K. Kawatani, S. Okawa, Nonexistence of semiorthogonal decompositions and sections of the canonical bundle. arXiv:1508.00682