Kodaira vanishing theorem


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, "L" any holomorphic line bundle on "M" that is a positive line bundle, and "K" is the canonical line bundle, then

:"H""q"("M", "KL") = {0}

for "q" > 0. Here "KL" stands for the tensor product of line bundles. By means of Serre duality, "K" can be removed. There is a generalisation, the Kodaira-Nakano vanishing theorem, in which "K", the "n"th exterior power of the holomorphic cotangent bundle where "n" is the complex dimension of "M", is replaced by the "r"th exterior power. Then the cohomology group 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,LotimesOmega^p_{X/k}) = 0 for p+q>d, and::: H^q(X,L^{otimes-1}otimesOmega^p_{X/k}) = 0 for p+q,: where the Ώp denote the sheaves of relative (algebraic) differential forms (see Kähler differential. This result does not always hold over fields of characteristic "p" > 0; counterexamples are known.

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 harv|Deligne|Illusie|1987. Their proof is based on showing that Hodge-de Rham spectral sequence for algebraic de Rham cohomology degenerates in degree 1. It is remarkable that 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.

References

*Phillip Griffiths and Joseph Harris, "Principles of Algebraic Geometry", Chapter 1.2
* Citation
last = Deligne
first = Pierre
last2 = Illusie
first2 = Luc
title = Relèvements modulo p2 et décomposition du complexe de de Rham
journal = Inventiones Mathematicae
volume = 89
pages = 247–270
year = 1987
doi = 10.1007/BF01389078


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Mumford vanishing theorem — In algebraic geometry, the Mumford vanishing theorem Mumford (1967) states that if L is a semi ample invertible sheaf with Iitaka dimension at least 2 on a complex projective manifold, then The Mumford vanishing theorem is related to the… …   Wikipedia

  • Kunihiko Kodaira — Infobox Scientist name =Kunihiko Kodaira box width = image width = caption = birth date =Birth date|1915|03|16 birth place =Tokyo, Japan death date =Death date and age|1997|07|26|1915|03|16 death place =Kōfu, Japan residence = citizenship =JPN… …   Wikipedia

  • Verschwindungssatz von Kodaira — Der Verschwindungssatz von Kodaira ist ein Satz aus der komplexen Geometrie und algebraischen Geometrie. Er beschäftigt sich mit den Fragen: wie einige der höheren Kohomologiegruppen einer glatten projektiven Mannigfaltigkeiten aussehen und unter …   Deutsch Wikipedia

  • Riemann–Roch theorem — In mathematics, specifically in complex analysis and algebraic geometry, the Riemann–Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates… …   Wikipedia

  • Uniformization theorem — In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. In fact, one can find a metric with constant Gaussian curvature in any given conformal class.In other words… …   Wikipedia

  • David Mumford — in 1975 Born 11 June 1937 (1937 06 11) (age 74) …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • Positive form — In complex geometry, the term positive form refers to several classes of real differential formsof Hodge type (p, p) . (1,1) forms Real ( p , p ) forms on a complex manifold M are forms which are of type ( p , p ) and real,that is, lie in the… …   Wikipedia

  • Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… …   Wikipedia

  • Surface of class VII — In mathematics, surfaces of class VII are non algebraic complex surfaces studied by (Kodaira 1964, 1968) that have Kodaira dimension −∞ and first Betti number 1. Minimal surfaces of class VII (those with no rational curves with self… …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.