In mathematics, specifically in the study of

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to eve ...

s over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the

Kodaira vanishing theorem In mathematics, the Kodaira vanishing theorem is a basic result of complex manifold theory and complex algebraic geometry, describing general conditions under which sheaf cohomology groups with indices ''q'' > 0 are automatically zero. The implicat ...

. Given a compact complex manifold ''M'' with a

holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

''F'' over ''M'', the Nakano vanishing theorem provides a condition on when the cohomology groups

H^q(M; \Omega^p(F))

equal zero. Here,

\Omega^p(F)

denotes the

sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics) In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open s ...

of holomorphic (''p'',0)-forms taking values on ''F''. The theorem states that, if the first

Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches ...

of ''F'' is negative,

H^q(M; \Omega^p(F)) = 0 \text q + p < n.

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

H^q(M; \Omega^p(F)) = 0 \text q + p > n.

References

Original publications

* * *

Secondary sources

Theorems in complex geometry Topological methods of algebraic geometry Theorems in algebraic geometry {{analysis-stub

See also

References

Original publications

Secondary sources