Nakano vanishing theorem explained

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 $H^q(M;\; \backslash Omega^p(F))$ equal zero. Here, $\backslash Omega^p(F)$ 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,Alternatively, if the first Chern class of F is positive,$$H^q(M;\; \backslash Omega^p(F))\; =\; 0\; \backslash text\; q\; +\; p\; >\; n.$$

See also

• Le Potier's vanishing theorem

References

Original publications

• Akizuki. Yasuo. Nakano. Shigeo. 1954. Note on Kodaira-Spencer's proof of Lefschetz theorems. Proceedings of the Japan Academy. EN. 30. 4. 266–272. 10.3792/pja/1195526105. 0021-4280. free.
• Book: Nakano, Shigeo. 1973. Number theory, algebraic geometry and commutative algebra — in honor of Yasuo Akizuki. Kinokuniya. Vanishing theorems for weakly 1-complete manifolds. 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. 10.2977/prims/1195192175. free.

Secondary sources

Notes and References

1. Hitchin. N. J.. 1981-07-01. Kählerian Twistor Spaces. Proceedings of the London Mathematical Society. en. s3-43. 1. 133–150. 10.1112/plms/s3-43.1.133. 1460-244X. 121623969.
2. Raufi. Hossein. 2012-12-18. The Nakano vanishing theorem and a vanishing theorem of Demailly-Nadel type for holomorphic vector bundles. 1212.4417. math.CV.
3. Book: Kobayashi, Shoshichi. Differential Geometry of Complex Vector Bundles. 2014-07-14. Princeton University Press. 9781400858682. 68. en.