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,

$${\displaystyle H^{q}(M;\Omega ^{p}(F))=0{\text{ when }}q+p<n.}$$

Alternatively, if the first Chern class of F is positive,

$${\displaystyle H^{q}(M;\Omega ^{p}(F))=0{\text{ when }}q+p>n.}$$

See also

• Le Potier's vanishing theorem

References