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 H q ( M ; Ω p ( F ) ) {\textstyle H^{q}(M;\Omega ^{p}(F))} equal zero. Here, Ω p ( F ) {\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 ; Ω p ( F ) ) = 0 when q + p < n . {\displaystyle 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 ; Ω p ( F ) ) = 0 when q + p > n . {\displaystyle H^{q}(M;\Omega ^{p}(F))=0{\text{ when }}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. 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