Kobayashi's ampleness conjecture for compact Kobayashi-hyperbolic manifolds
Let be a compact Kähler or projective manifold that is Kobayashi hyperbolic.
Kobayashi's conjecture. The canonical bundle is ample.
This conjecture asks whether Kobayashi hyperbolicity forces positivity of the canonical bundle. The paper notes that it has a positive answer for compact Kähler hyperbolic manifolds, but the general case remains open.
References
Primary source
Abdelouahab Khelifati, “Holomorphic Deformations of Compact Kähler Hyperbolic Manifolds”, arXiv:2508.07096 (2025).
Additional references
21 papers in this index state this conjecture (2001–2025). The statement above is taken from the most recent of them; the others are arXiv:2406.19003, arXiv:2307.04288, arXiv:2306.15036, arXiv:2107.08972, arXiv:2011.11379, arXiv:1909.11417, arXiv:1909.11495, arXiv:1908.04723, arXiv:1806.02364, arXiv:1804.01719, arXiv:1609.06918, arXiv:1601.07514, and 8 more.
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