Kodaira vanishing theorem

The vanishing of Kodaira is a set of the complex geometry and algebraic geometry. It deals with the questions:

The vanishing of Kodaira is more of an over raschenes result, since it is generally quite difficult to figure out the cohomology of a geometric object. In the case, however, a relatively large class of Kohomologien be determined that even disappear, so you can read some properties in a long exact sequence with the disappearance.

The complex analytic case

Originally the sentence by applying the Hodge theory on a compact Kählermannigfaltigkeit M of complex dimension n proved in the form of Kunihiko Kodaira:

Where the canonical line bundle of M and a positive holomorphic line bundle over M. ( as written ) is to be understood as a tensor product of two line bundles. Using the Serre duality can be easily closed to the disappearance of other Garbenkohomologiegruppen. The sheaf is isomorphic to, the sheaf of the holomophen (p, 0) - form on M with values ​​in L is.

This formulation was later generalized by Akizuki and Nakano as

So that the sheaf has been replaced by.

The algebraic case

In the context of algebraic geometry, in which you want to always translate analytical conditions in pure algebraic conditions in complex geometry, the condition of " positive line bundle " of Verschwindungssatzes by " ample invertible sheaf " (ie, using the sheaf is a projective embedding possible) replaced. So you have this statement:

Let k be a field of characteristic 0, X n is a non- singular projective k- scheme of dimension and L is an ample invertible sheaf on X, then applies

Here is the sheaf of relative differential forms. A counter-example for the body of characteristics was given in 1978 by Michel Raynaud.

Until 1987, one could prove the above statements in characteristic 0 only by the original function-theoretic proof along with the application of the GAGA principle of Serre. 1987 but appeared a purely algebraic proof of Deligne Pierre and Luc Illusie, in which they looked at the Hodge -de Rham spectral sequences of algebraic de Rham cohomology and showed that these degenerate into grade 1.

Conclusion and application

Using the Verschwindungssatzes Kodaira proved the so-called embedding theorem of Kodaira, which states that a Kählermannigfaltigkeit can be embedded in a projective space and then by the theorem of Chow is an algebraic variety, if it exists a positive line bundle. Moreover, the vanishing is often used in the classification of compact complex manifolds, for example, to determine the Hodge diamond.

Application Example

Let S be a del- Pezzo surface ( ie complex dimension 2), for which the anti- canonical line bundles according to the definition is positive. With the short exact sequence one has

After the Kodaira - vanishing are

Hence, it follows that describes a correspondence between divisors and Chern classes of S; denotes the Picardgruppe of X. In addition, one can determine the vanishing and with the help of Poincaré duality and Hodge decomposition, the Hodge diamond of S, while preserving

In which case the h1, 1 are dependent on S.

Generalization

• Vanishing of Kawamata - Viehweg
• Vanishing of needle