In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the cohomology groups equal zero. Here, denotes the sheaf of holomorphic (p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative, Alternatively, if the first Chern class of F is positive,
equal zero. Here, 
denotes the 
Alternatively, if the first Chern class of F is positive,
See also
References
Original publications
- Akizuki, Yasuo; Nakano, Shigeo (1954). "Note on Kodaira-Spencer's proof of Lefschetz theorems". Proceedings of the Japan Academy. 30 (4): 266–272. doi:10.3792/pja/1195526105. ISSN 0021-4280.
- Nakano, Shigeo (1973). "Vanishing theorems for weakly 1-complete manifolds". Number theory, algebraic geometry and commutative algebra — in honor of Yasuo Akizuki. Kinokuniya. pp. 169–179.
- Nakano, Shigeo (1974). "Vanishing Theorems for Weakly 1-Complete Manifolds II". Publications of the Research Institute for Mathematical Sciences. 10 (1): 101–110. doi:10.2977/prims/1195192175.
Secondary sources
- Hitchin, N. J. (1981-07-01). "Kählerian Twistor Spaces" (PDF). Proceedings of the London Mathematical Society. s3-43 (1): 133–150. doi:10.1112/plms/s3-43.1.133. ISSN 1460-244X. S2CID 121623969.
- Raufi, Hossein (2012-12-18). "The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles". arXiv:1212.4417 .
- Kobayashi, Shoshichi (2014-07-14). Differential Geometry of Complex Vector Bundles. Princeton University Press. p. 68. ISBN 9781400858682.
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