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| In [[mathematics]], '''Hodge theory''', named after [[W. V. D. Hodge]], is one aspect of the study of [[differential form]]s of a [[smooth manifold]] ''M''. More specifically, it works out the consequences for the [[cohomology group]]s of ''M'', with real coefficients, of the [[partial differential equation]] theory of generalised [[Laplacian]] operators associated to a [[Riemannian metric]] on ''M''.
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| It was developed by Hodge in the 1930s as an extension of [[de Rham cohomology]], and has major applications on three levels:
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| *[[Riemannian manifold]]s
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| *[[Kähler manifold]]s
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| *[[algebraic geometry]] of complex [[projective varieties]], and even more broadly, [[motive (algebraic geometry)|motives]].
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| In the initial development, ''M'' was taken to be a [[closed manifold]] (that is, [[compact space|compact]] and without boundary). On all three levels, the theory was very influential on subsequent work, being taken up by [[Kunihiko Kodaira]] (in Japan and later, partly under the influence of [[Hermann Weyl]], at Princeton) and many others subsequently.
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| ==Applications and examples==
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| ===De Rham cohomology===
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| The original formulation of Hodge theory, due to W. V. D. Hodge, was for the [[de Rham cohomology|de Rham complex]]. If ''M'' is a compact orientable manifold equipped with a smooth metric ''g'', and Ω<sup>''k''</sup>(''M'') is the [[sheaf (mathematics)|sheaf]] of smooth [[differential form]]s of degree ''k'' on ''M'', then the de Rham complex is the sequence of [[differential operator]]s
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| :<math> 0\rightarrow \Omega^0(M) \xrightarrow{d_0} \Omega^1(M)\xrightarrow{d_1} \cdots\xrightarrow{d_{n-1}} \Omega^n(M)\xrightarrow{d_n} 0 </math>
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| where ''d<sub>k</sub>'' denotes the [[exterior derivative]] on Ω<sup>''k''</sup>(''M''). The de Rham cohomology is then the sequence of vector spaces defined by
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| :<math>H^k(M)=\frac{\ker d_k}{\mathrm{im}\,d_{k-1}}.</math>
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| One can define the formal adjoint of the exterior derivative ''d'', denoted δ, called [[codifferential]], as follows. For all α ∈ Ω<sup>''k''</sup>(''M'') and β ∈ Ω<sup>''k''+1</sup>(''M''), we require that
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| :<math>\int_M \langle d\alpha,\beta\rangle_{k+1} \,dV = \int_M\langle\alpha,\delta\beta\rangle_k \,dV</math>
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| where <math>\langle \ ,\ \rangle_k</math> is the metric induced on Ω<sup>''k''</sup>(''M''). The [[Laplacian]] form is then defined by Δ = dδ + δd. This allows one to define spaces of [[harmonic (mathematics)|harmonic]] forms
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| :<math>\mathcal H_\Delta^k(M)=\{\alpha\in\Omega^k(M)\mid\Delta\alpha=0\}.</math>
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| Since <math>d\mathcal H_\Delta^k(M)=0</math>, there is a canonical mapping <math>\varphi:\mathcal H_\Delta^k(M)\rightarrow H^k(M)</math>. The first part of Hodge's original theorem states that φ is an isomorphism of vector spaces. In other words, for each de Rham cohomology class on ''M'', there is a unique harmonic representative.
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| One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional. This follows since the operators Δ are [[elliptic operator|elliptic]], and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space.
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| ===Hodge theory of elliptic complexes===
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| In general, Hodge theory applies to any [[elliptic complex]] over a compact manifold.
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| Let <math>E_0,E_1,\dots,E_N</math> be [[vector bundles]], equipped with metrics, on a compact manifold ''M'' with a volume form ''dV''. Suppose that
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| :<math>L_i:\Gamma(E_i)\rightarrow\Gamma(E_{i+1})</math>
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| are [[differential operators]] acting on sections of these vector bundles, and that the induced sequence
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| :<math>\Gamma(E_0)\rightarrow \Gamma(E_1)\rightarrow\cdots\rightarrow\Gamma(E_N)</math>
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| is an elliptic complex. Introduce the [[direct sum]]s:
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| :<math>\mathcal E^\bullet=\bigoplus_i \Gamma(E_i)</math>
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| :<math>L=\bigoplus L_i:\mathcal E^\bullet\rightarrow\mathcal E^\bullet</math>
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| and let ''L*'' be the adjoint of ''L''. Define the elliptic operator Δ = ''LL*'' + ''L*L''. As in the de Rham case, this yields the vector space of harmonic sections
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| :<math>\mathcal H=\{e\in\mathcal E^\bullet\mid\Delta e=0\}.</math>
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| So let <math>H:\mathcal E^\bullet\rightarrow\mathcal H</math> be the orthogonal projection, and let ''G'' be the [[Green's function|Green's operator]] for Δ. The '''Hodge theorem''' then asserts the following:
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| #''H'' and ''G'' are well-defined.
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| #Id = ''H'' + Δ''G'' = ''H'' + ''G''Δ
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| #''LG'' = ''GL'', ''L*G'' = ''GL*''
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| #The cohomology of the complex is canonically isomorphic to the space of harmonic sections, <math>H(E_j)\cong\mathcal H(E_j)</math>, in the sense that each cohomology class has a unique harmonic representative.
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| ==Hodge structures==
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| {{main|Hodge structure}}
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| An abstract definition of (real) '''Hodge structure''' is now given: for a real [[vector space]] ''W'', a Hodge structure of integer weight ''k'' on ''W'' is a [[direct sum of vector spaces|direct sum]] decomposition of ''W''<sup>'''C'''</sup> = ''W'' ⊗ '''C''', the [[complexification]] of ''W'', into graded pieces ''W<sup>p, q</sup>'' where ''k'' = ''p'' + ''q'', and the [[complexification|complex conjugation]] of ''W''<sup>'''C'''</sup> interchanges this subspace with ''W<sup>q, p</sup>''.
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| The basic statement in algebraic geometry is then that the [[singular cohomology]] groups with real coefficients of a non-singular complex projective variety ''V'' carry such a Hodge structure, with <math> H^k (V) </math> having the required decomposition into complex subspaces ''H<sup>p, q</sup>''. The consequence for the [[Betti number]]s is that, taking dimensions
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| :<math> b_{k} = \dim H^{k} (V) = \sum_{p+q=k} h^{p,q},\, </math>
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| where the sum runs over all pairs ''p'', ''q'' with ''p'' + ''q'' = ''k'' and where
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| :<math> h^{p,q} = \dim H^{p,q}.\, </math>
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| The sequence of Betti numbers becomes a '''Hodge diamond''' of '''Hodge numbers''' spread out into two dimensions.
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| This grading comes initially from the theory of '''harmonic forms''', that are privileged representatives in a de Rham cohomology class picked out by the Hodge Laplacian (generalising [[harmonic function]]s, which must be [[locally constant]] on compact manifolds by their ''maximum principle''). In later work (Dolbeault) it was shown that the Hodge decomposition above can also be found by means of the [[sheaf cohomology]] groups <math>H^{q} (V,\Omega^{p})</math> in which Ω<sup>''p''</sup> is the [[sheaf (mathematics)|sheaf]] of holomorphic ''p''-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.
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| In the case of singularities or noncompact varieties, the Hodge structure has to be modified to a [[mixed Hodge structure]], where the double-graded direct sum decomposition is replaced by a pair of [[filtration (abstract algebra)|filtrations]]. This case is much used, for example in [[monodromy]] questions.
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| ==See also==
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| *[[Hodge–Arakelov theory]]
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| *[[Hodge cycle]]
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| *[[Hodge conjecture]]
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| *[[Period mapping]]
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| *[[Torelli theorem]]
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| *[[Variation of Hodge structure]]
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| *[[Mixed Hodge structure]]
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| *[[Logarithmic form]]
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| ==References==
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| * {{cite book | author=P. Griffiths | authorlink=Phillip Griffiths | coauthors=[[Joe Harris (mathematician)|J. Harris]] | title=Principles of Algebraic Geometry | series=Wiley Classics Library | publisher=Wiley Interscience | year=1994 | isbn=0-471-05059-8 | page=117 }}
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| *{{Citation | last1=Hodge | first1=W. V. D. | author1-link=W. V. D. Hodge | title=The Theory and Applications of Harmonic Integrals | url=http://books.google.com/books?id=-8k8AAAAIAAJ | publisher=[[Cambridge University Press]] | isbn= 978-0-521-35881-1 | id={{MR|0003947}} | year=1941}}
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| * [[Ofer Gabber]], [[Lorenzo Ramero]] (2009). ''[http://arxiv.org/abs/math.AG/0409584 Foundations for almost ring theory]''.
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| [[Category:Hodge theory| ]]
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