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 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 K_M is the canonical line bundle, then

for q > 0. Here ${\displaystyle K_{M}\otimes L}$ stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of ${\displaystyle H^{q}(M,L^{\otimes -1})}$ for q < n. There is a generalisation, the Kodaira-Nakano vanishing theorem, in which ${\displaystyle K_{M}\otimes L\cong \Omega ^{n}(L)}$, where Ωⁿ(L) denotes the sheaf of holomorphic (n,0)-forms on M with values on L, is replaced by Ω^r(L), the sheaf of holomorphic (r,0)-forms with values on L. Then the cohomology group H^q(M, Ω^r(L)) vanishes whenever q + r > n.

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