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 implications for the group with index q = 0 is usually that its dimension — the number of independent global sections — coincides with a holomorphic Euler characteristic that can be computed using the Hirzebruch–Riemann–Roch theorem.
The statement of Kunihiko Kodaira's result is that if M is a compact Kähler manifold of complex dimension n, L any holomorphic line bundle on M that is positive, and KM is the canonical line bundle, then for q > 0.
, where Ωn(L) denotes the sheaf of holomorphic (n,0)-forms on M with values on L, is replaced by Ωr(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group Hq(M, Ωr(L)) vanishes whenever q + r > n. The Kodaira vanishing theorem can be formulated within the language of algebraic geometry without any reference to transcendental methods such as Kähler metrics.
Positivity of the line bundle L translates into the corresponding invertible sheaf being ample (i.e., some tensor power gives a projective embedding).
Later Sommese (1986) give a counterexample for singular varieties with non-log canonical singularities,[1] and also,Lauritzen & Rao (1997) gave elementary counterexamples inspired by proper homogeneous spaces with non-reduced stabilizers.