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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 complex analytic case

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

H^{q}(M,K_{M}\otimes L)=0

for q > 0. Here $K_{M}\otimes L$ stands for the tensor product of line bundles. By means of Serre duality, one also obtains the vanishing of $H^{q}(M,L^{\otimes -1})$ for q < n. There is a generalisation, the Kodaira–Nakano vanishing theorem, in which $K_{M}\otimes L\cong \Omega ^{n}(L)$ , 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.

The algebraic case

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). The algebraic Kodaira–Akizuki–Nakano vanishing theorem is the following statement:

If k is a field of characteristic zero, X is a smooth and projective k-scheme of dimension d, and L is an ample invertible sheaf on X, then

H^{q}(X,L\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q>d,{\text{ and}}

H^{q}(X,L^{\otimes -1}\otimes \Omega _{X/k}^{p})=0{\text{ for }}p+q<d,

where the Ω^p denote the sheaves of relative (algebraic) differential forms (see Kähler differential).

Raynaud (1978) showed that this result does not always hold over fields of characteristic p > 0, and in particular fails for Raynaud surfaces. 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.

Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the GAGA comparison theorems. However, in 1987 Pierre Deligne and Luc Illusie gave a purely algebraic proof of the vanishing theorem in (Deligne & Illusie 1987). Their proof is based on showing that the Hodge–de Rham spectral sequence for algebraic de Rham cohomology degenerates in degree 1. This is shown by lifting a corresponding more specific result from characteristic p > 0 — the positive-characteristic result does not hold without limitations but can be lifted to provide the full result.

Consequences and applications

Historically, the Kodaira embedding theorem was derived with the help of the vanishing theorem. With application of Serre duality, the vanishing of various sheaf cohomology groups (usually related to the canonical line bundle) of curves and surfaces help with the classification of complex manifolds, e.g. Enriques–Kodaira classification.

Note

^ (Fujino 2009, Proposition 2.64)

References

Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo p² et décomposition du complexe de de Rham", Inventiones Mathematicae, 89 (2): 247–270, Bibcode:1987InMat..89..247D, doi:10.1007/BF01389078, S2CID 119635574
Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems (PDF), DMV Seminar, vol. 20, Birkhäuser Verlag, ISBN 978-3-7643-2822-1, MR 1193913
Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry
Kodaira, Kunihiko (1953), "On a differential-geometric method in the theory of analytic stacks", Proc. Natl. Acad. Sci. USA, 39 (12): 1268–1273, Bibcode:1953PNAS...39.1268K, doi:10.1073/pnas.39.12.1268, PMC 1063947, PMID 16589409
Lauritzen, Niels; Rao, Prabhakar (1997), "Elementary counterexamples to Kodaira vanishing in prime characteristic", Proc. Indian Acad. Sci. Math. Sci., 107, Springer Verlag: 21–25, arXiv:alg-geom/9604012, doi:10.1007/BF02840470, S2CID 16736679
Raynaud, Michel (1978), "Contre-exemple au vanishing theorem en caractéristique p>0", C. P. Ramanujam---a tribute, Tata Inst. Fund. Res. Studies in Math., vol. 8, Berlin, New York: Springer-Verlag, pp. 273–278, MR 0541027
Fujino, Osamu (2009). "Introduction to the log minimal model program for log canonical pairs". arXiv:0907.1506 [math.AG].
Sommese, Andrew John (1986). "On the adjunction theoretic structure of projective varieties". Complex Analysis and Algebraic Geometry. Lecture Notes in Mathematics. Vol. 1194. pp. 175–213. doi:10.1007/BFb0077004. ISBN 978-3-540-16490-6.