Kodaira Disappearance Law

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

1. what some of the higher cohomology groups of a smooth projective manifold look like and
2. under which circumstances a Kahler manifold can be embedded in the complex projective space (according to Kodaira's embedding theorem).

Kodaira's vanishing theorem is rather a surprising result, because it is generally difficult to find out the cohomology of a geometric object. In this case, however, a relatively large class of cohomologies is determined which even disappear, so that one can read off some properties in a long, exact sequence when they disappear .

The complex analytical case

Originally the theorem was proved by applying Hodge's theory to a compact Kahler manifold M of complex dimension n in the following form by Kunihiko Kodaira :

${\displaystyle H^{q}(M,K_{M}\otimes L)=0}$for , ${\displaystyle q>0}$

wherein the canonical line bundle of M and a positive holomorphic line bundle on M . (also written as) should be understood as the tensor product of two straight line bundles. With the help of the Serre duality one can easily infer the disappearance of other sheaf cohomology groups. The sheaf is isomorphic to , wherein the sheaf of holomophen (p, 0) shapes on M with values in L is. ${\displaystyle K_{M}}$ ${\displaystyle L\rightarrow M}$ ${\displaystyle K_{M}\otimes L}$ ${\displaystyle K_{M}+L}$ ${\displaystyle K_{M}\otimes L}$ ${\displaystyle \,\Omega ^{n}(L)}$ ${\displaystyle \,\Omega ^{p}(L)}$

This formulation was later generalized by Akizuki and Nakano as

${\displaystyle \,H^{q}(M,\Omega ^{p}(L))=0}$for , ${\displaystyle \,p+q>n}$

so that the sheaf has been replaced by. ${\displaystyle \,\Omega ^{n}(L)}$ ${\displaystyle \,\Omega ^{p}(L)}$

The algebraic case

In the context of algebraic geometry, whereby one always wants to translate analytical conditions into pure algebraic conditions in complex geometry, the requirement of the “positive bundle of lines” of the vanishing theorem has been replaced by “ample invertible sheaf” (ie with the help of the sheaf a projective embedding is possible) . So you have this statement:

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

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

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

Here is the sheaf of relative differential forms . A counterexample for bodies of characteristic was given in 1978 by Michel Raynaud . ${\displaystyle \Omega ^{p}}$ ${\displaystyle p>0}$

Until 1987, the above statements in characteristic 0 could only be proven by the original function-theoretical proof together with the application of Serre's GAGA principle. In 1987, however, a purely algebraic proof by Pierre Deligne and Luc Illusie appeared , in which they considered the Hodge-de-Rham spectral sequences of the algebraic De Rham cohomology and showed that these degenerate into grade 1.

Inference and application

Using the vanishing theorem , Kodaira proved the so-called embedding theorem of Kodaira , which says that a Kahler manifold can be embedded in a projective space and then, according to Chow's theorem, is an algebraic variety if there is a positive bundle of lines on it. In addition, the vanishing law is often used in the classification of compact complex manifolds, for example to determine the Hodge diamond .

Application in example

Let S be a del Pezzo surface (i.e. of complex dimension 2) for which the anti-canonical line bundle is positive by definition. With the short exact sequence you have ${\displaystyle K_{S}^{\ast }}$ ${\displaystyle \mathbb {Z} \rightarrow {\mathcal {O}}\rightarrow {\mathcal {O}}^{\ast }}$

${\displaystyle \cdots \rightarrow H^{1}(S,{\mathcal {O}})\rightarrow H^{1}(S,{\mathcal {O}}^{\ast })\rightarrow H^{2}(S,\mathbb {Z} )\rightarrow H^{2}(S,{\mathcal {O}})\rightarrow \dots }$.

According to the Kodaira Disappearance Theorem, are

${\displaystyle H^{1}(S,{\mathcal {O}})\cong H^{1}(S,K_{S}\otimes K_{S}^{\ast })=0}$ and

${\displaystyle H^{2}(S,{\mathcal {O}})\cong H^{2}(S,K_{S}\otimes K_{S}^{\ast })=0}$.

Therefore , what follows is what describes a correspondence between divisors and Chern classes on S ; refers to the Picard group of X here . In addition, you can take the Verschwindungssatz and using Poincaré duality and Hodge theory , the Hodge diamonds of S determine and gets it ${\displaystyle \operatorname {Pic} (X):=H^{1}(S,{\mathcal {O}}^{\ast })\cong H^{2}(S,\mathbb {Z} )}$ ${\displaystyle \operatorname {Pic} (X)}$

		1
	0		0
0		h 1.1		0
	0		0
		1

wherein the here h ^1.1 of S dependent.

generalization

• Disappearance Theorem from Kawamata Cattle Trail
• Disappearance law of needle

literature

• Pierre Deligne , Duc Illusie: Relèvements modulo p ² et décomposition du complexe de Rham. In: Inventiones Mathematicae, No. 89, 1987, pp. 247-270.
• Hélène Esnault , Eckart Viehweg : Lectures on vanishing theorems. Birkhäuser Verlag , Basel 1992.
• Friedrich Hirzebruch : New topological methods in algebraic geometry. 2nd Edition. Springer-Verlag, Berlin 1962.
• Phillip Griffiths , Joe Harris : Principles of Algebraic Geometry. 1st edition as Wiley Classics Library Edition . Wiley-Interscience, 1994, ISBN 0-471-05059-8 .
• Michel Raynaud : Contre-exemple au vanishing theorem en caractéristique p> 0. In: CP Ramanujam --- a tribute. Tata Institute of Fundamental Research , Studies in Mathematics, No. 8, Springer-Verlag, Berlin 1978, pp. 273-278.