Nakano vanishing theorem

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.^[1]^[2]^[3] 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 ${\textstyle H^{q}(M;\Omega ^{p}(F))}$ equal zero. Here, ${\textstyle \Omega ^{p}(F)}$ 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, $H^{q}(M;\Omega ^{p}(F))=0{\text{ when }}q+p<n.$ Alternatively, if the first Chern class of F is positive, $H^{q}(M;\Omega ^{p}(F))=0{\text{ when }}q+p>n.$

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 [math.CV].
^ Kobayashi, Shoshichi (2014-07-14). Differential Geometry of Complex Vector Bundles. Princeton University Press. p. 68. ISBN 9781400858682.

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[1] 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.

[2] Raufi, Hossein (2012-12-18). "The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles". arXiv:1212.4417 [math.CV].

[3] Kobayashi, Shoshichi (2014-07-14). Differential Geometry of Complex Vector Bundles. Princeton University Press. p. 68. ISBN 9781400858682.

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