- Kodaira vanishing theorem
In

mathematics , the**Kodaira vanishing theorem**is a basic result ofcomplex manifold theory and complexalgebraic geometry , describing general conditions under whichsheaf 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 independentglobal section s — coincides with aholomorphic Euler characteristic that can be computed using theHirzebruch-Riemann-Roch theorem .**The complex analytic case**The statement of

Kunihiko Kodaira 's result is that if "M" is a compactKähler manifold , "L" anyholomorphic line bundle on "M" that is apositive line bundle , and "K" is thecanonical line bundle , then:"H"

^{"q"}("M", "KL") = {0}for "q" > 0. Here "KL" stands for the

tensor product of line bundles . By means ofSerre duality , "K" can be removed. There is a generalisation, the**Kodaira-Nakano vanishing theorem**, in which "K", the "n"thexterior power of theholomorphic cotangent bundle where "n" is thecomplex 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 $p+qmath>,:\; where\; the\; \u038fpdenote\; thesheavesof\; relative\; (algebraic)differential\; forms(seeK\xe4hler\; differential.\; This\; result\; does\; not\; always\; hold\; over\; fields\; of\; characteristic\; "p"\; 0;\; counterexamples\; are\; known.$Until 1987 the only known proof in characteristic zero was however based on the complex analytic proof and the

GAGA comparison theorems. However, in 1987Pierre Deligne andLuc Illusie gave a purely algebraic proof of the vanishing theorem in harv|Deligne|Illusie|1987. Their proof is based on showing thatHodge-de Rham spectral sequence foralgebraic 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 p^{2}et décomposition du complexe de de Rham

journal = Inventiones Mathematicae

volume = 89

pages = 247–270

year = 1987

doi = 10.1007/BF01389078

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