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| 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 section]]s — coincides with a [[holomorphic Euler characteristic]] that can be computed using the [[Hirzebruch-Riemann-Roch theorem]].
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| == The complex analytic case ==
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| 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 form|positive]], and ''K<sub>M</sub>'' is the [[canonical bundle|canonical line bundle]], then
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| :::<math> H^q(M, K_M\otimes L) = 0 </math>
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| for ''q'' > 0. Here <math>K_M\otimes L</math> stands for the [[tensor product of line bundles]]. By means of [[Serre duality]], one obtains the vanishing of other cohomology group by removing ''K''. There is a generalisation, the '''Kodaira-Nakano vanishing theorem''', in which <math>K_M\otimes L\cong\Omega^n(L)</math>, where Ω<sup>''n''</sup>(''L'') denotes the sheaf of [[Dolbeault complex|holomorphic (''n'',0)-forms]] on ''M'' with values on ''L'', is replaced by Ω<sup>''r''</sup>(''L''), the sheaf of holomorphic (r,0)-forms with values on ''L''. Then the cohomology group H<sup>''q''</sup>(''M'', Ω<sup>''r''</sup>(''L'')) vanishes whenever ''q'' + ''r'' > ''n''.
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| == The algebraic case ==
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| 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 line bundle|ample]] (i.e., some tensor power gives a projective embedding). The algebraic Kodaira-Akizuki-Nakano vanishing theorem is the following statement:
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| : If ''k'' is a [[field (mathematics)|field]] of [[characteristic (algebra)|characteristic]] zero, ''X'' is a [[smooth morphism|smooth]] and [[projective morphism|projective]] ''k''-[[Scheme (mathematics)|scheme]] of dimension ''d'', and ''L'' is an ample invertible sheaf on ''X'', then
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| :::<math> H^q(X,L\otimes\Omega^p_{X/k}) = 0</math> for <math> p+q>d</math>, and | |
| :::<math> H^q(X,L^{\otimes-1}\otimes\Omega^p_{X/k}) = 0</math> for <math> p+q<d</math>,
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| : where the Ω<sup>p</sup> denote the [[Sheaf (mathematics)|sheaves]] of relative (algebraic) [[differential forms]] (see [[Kähler differential]]).
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| {{harvtxt|Raynaud|1978}} showed that this result does not always hold over fields of characteristic ''p'' > 0, and in particular fails for [[Raynaud surface]]s.
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| 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. 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.
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| ==Consequences and applications==
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| 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]].
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| ==See also==
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| * [[Kawamata–Viehweg vanishing theorem]]
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| * [[Mumford vanishing theorem]]
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| * [[Ramanujam vanishing theorem]]
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| ==References==
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| * {{Citation
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| | last = Deligne
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| | first = Pierre
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| | last2 = Illusie
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| | first2 = Luc
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| | title = Relèvements modulo p<sup>2</sup> et décomposition du complexe de de Rham
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| | journal = Inventiones Mathematicae
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| | volume = 89
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| | issue = 2
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| | pages = 247–270
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| | year = 1987
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| | doi = 10.1007/BF01389078 }}
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| *{{Citation | url = http://www.uni-due.de/%7Emat903/books/esvibuch.pdf|last1=Esnault | first1=Hélène | last2=Viehweg | first2=Eckart | title=Lectures on vanishing theorems | publisher=Birkhäuser Verlag | series=DMV Seminar | isbn=978-3-7643-2822-1 | id={{MathSciNet | id = 1193913}} | year=1992 | volume=20}}
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| *[[Phillip Griffiths]] and [[Joe Harris (mathematician)|Joseph Harris]], ''Principles of Algebraic Geometry''
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| *{{Citation | last1=Raynaud | first1=Michel | author1-link=Michel Raynaud | title=C. P. Ramanujam---a tribute | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Tata Inst. Fund. Res. Studies in Math. | id={{MathSciNet | id = 541027}} | year=1978 | volume=8 | chapter=Contre-exemple au vanishing theorem en caractéristique p>0 | pages=273–278}}
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| [[Category:Theorems in complex geometry]]
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| [[Category:Topological methods of algebraic geometry]]
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| [[Category:Theorems in algebraic geometry]]
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