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 [math]\displaystyle{ H^q(M; \Omega^p(F)) }[/math] equal zero. Here, [math]\displaystyle{ \Omega^p(F) }[/math] 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,[math]\displaystyle{ H^q(M; \Omega^p(F)) = 0 \text{ when } q + p \lt n. }[/math] Alternatively, if the first Chern class of F is positive,[math]\displaystyle{ H^q(M; \Omega^p(F)) = 0 \text{ when } q + p \gt n. }[/math]
See also
References
Original publications
- Akizuki, Yasuo; Nakano, Shigeo (1954). "Note on Kodaira-Spencer's proof of Lefschetz theorems" (in EN). Proceedings of the Japan Academy 30 (4): 266–272. doi:10.3792/pja/1195526105. ISSN 0021-4280. https://projecteuclid.org/euclid.pja/1195526105.
- 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. https://www.jstage.jst.go.jp/article/kyotoms1969/10/1/10_1_101/_article/-char/ja/.
Secondary sources
- ↑ Hitchin, N. J. (1981-07-01). "Kählerian Twistor Spaces" (in en). Proceedings of the London Mathematical Society s3-43 (1): 133–150. doi:10.1112/plms/s3-43.1.133. ISSN 1460-244X. http://www2.maths.ox.ac.uk/~hitchin/hitchinlist/Hitchin%20KAHLERIAN%20TWISTOR%20SPACES%20(PLMS%201981).pdf.
- ↑ 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) (in en). Differential Geometry of Complex Vector Bundles. Princeton University Press. pp. 68. ISBN 9781400858682. https://books.google.com/books?id=ePf_AwAAQBAJ.
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