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 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 KM is the canonical line bundle, then
for q > 0. Here stands for the tensor product of line bundles. By means of Serre duality, one obtains the vanishing of some other cohomology group by removing K.[clarification needed] There is a generalisation, the Kodaira-Nakano vanishing theorem, in which , 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 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
- for , and
- for ,
- where the Ωp denote the sheaves of relative (algebraic) differential forms (see Kähler differential).
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 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, 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.
- Deligne, Pierre; Illusie, Luc (1987), "Relèvements modulo p2 et décomposition du complexe de de Rham", Inventiones Mathematicae 89 (2): 247–270, doi:10.1007/BF01389078
- Esnault, Hélène; Viehweg, Eckart (1992), Lectures on vanishing theorems (PDF), DMV Seminar 20, Birkhäuser Verlag, ISBN 978-3-7643-2822-1, MR 1193913
- Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry
- Raynaud, Michel (1978), "Contre-exemple au vanishing theorem en caractéristique p>0", C. P. Ramanujam---a tribute, Tata Inst. Fund. Res. Studies in Math. 8, Berlin, New York: Springer-Verlag, pp. 273–278, MR 541027