- Hodge theory
In
mathematics , Hodge theory is one aspect of the study of thealgebraic topology of asmooth manifold "M". More specifically, it works out the consequences for thecohomology group s of "M", with real coefficients, of thepartial differential equation theory of generalisedLaplacian operators associated to aRiemannian metric on "M".It was developed by
W. V. D. Hodge in the 1930s as an extension ofde Rham cohomology , and has major applications on three levels:*
Riemannian manifold s
*Kähler manifold s
*algebraic geometry of complexprojective varieties , and even more broadly, motives.In the initial development, "M" was taken to be
compact and without boundary. On all three levels the theory was very influential on subsequent work, being taken up byKunihiko Kodaira (in Japan and later, partly under the influence ofHermann Weyl , at Princeton) and many others subsequently.Applications and examples
De Rham cohomology
The original formulation of Hodge theory, due to W. V. D. Hodge, was for the de Rham complex. If "M" is a compact orientable manifold equipped with a smooth metric "g", and Omega^k(M) is the sheaf of
differential form s of degree "k" on "M", then the de Rham complex is the sequence ofdifferential operator s:0 ightarrow mathbb Rxrightarrow{d_{-1 Omega^0(M) xrightarrow{d_{0 Omega^1(M)xrightarrow{d_{1 dotsxrightarrow{d_{n-1 Omega^n(M)xrightarrow{d_{n 0 where d_k denotes the
exterior derivative on Omega^k(M). The de Rham cohomology is then the sequence of vector spaces defined by:H^k(M)=frac{ker d_k}{mathrm{im},d_{k-1.
One can define the
Hilbert space adjoint of the exterior derivative "d", denoted delta by means of theRiesz representation theorem as follows. For all alphainOmega^k(M) and etainOmega^{k+1}(M), we require that:int_M langle dalpha,eta angle_{k+1} dV=int_Mlanglealpha,deltaeta angle_k dVwhere langle , angle_k is the metric induced on Omega^k(M). The formLaplacian is then defined by Delta=ddelta+delta d. This allows one to define spaces of harmonic forms:mathcal H_Delta^k(M)={alphainOmega^k(M)midDeltaalpha=0}One can easily show that dmathcal H_Delta^k(M)=0, so there is a canonical mapping varphi:mathcal H_Delta^k(M) ightarrow H^k(M). The first part of Hodge's original theorem states that phi is an isomorphism of vector spaces. In other words, for each de Rham cohomology class on "M", there is a unique harmonic representative.
One major consequence of this is that the de Rham cohomology groups on a compact manifold are finite-dimensional. This follows since the operators Delta are elliptic, and the kernel of an elliptic operator on a compact manifold is always a finite-dimensional vector space. However, Hodge theory actually yields an even greater abundance of riches, as we shall see in the sequel.
Hodge theory of elliptic complexes
In general, Hodge theory applies to any
elliptic complex over a compact manifold.Let E_0,E_1,dots,E_N be
vector bundles , equipped with metrics, on a compact manifold "M" with a volume form "dV". Suppose that:L_i:Gamma(E_i) ightarrowGamma(E_{i+1})
are
differential operators acting on sections of these vector bundles, and that the induced sequence:Gamma(E_0) ightarrow Gamma(E_1) ightarrowdots ightarrowGamma(E_N)
is an elliptic complex. It is convenient to introduce the direct sum mathcal E^cdot=igoplus_i Gamma(E_i). Let L=igoplus L_i:mathcal E^cdot ightarrowmathcal E^cdot, and let L^* be the adjoint of "L". Define the elliptic operator Delta=LL^*+L^*L. As in the de Rham case, this yields the vector space of harmonic sections
:mathcal H={einmathcal E^cdotmidDelta e=0}.
So let H:mathcal E^cdot ightarrowmathcal H be the orthogonal projection, and let "G" be the Green's operator for Delta. The Hodge theorem then asserts the following:
#"H" and "G" are well-defined.
#Id=H+Delta G=H+GDelta
#LG=GL, L^*G=GL^*
#The cohomology of the complex is canonically isomorphic to the space of harmonic sections, H(E_j)congmathcal H(E_j), in the sense that each cohomology class has a unique harmonic representative.Hodge structures
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 adirect sum decomposition of W^{mathbb C} = W otimes mathbb C, thecomplexification of W, into graded pieces W^{p,q} where k = p+q , and the complex conjugation of W^{mathbb C} interchanges this subspace with W^{q,p} .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 H^{k}(V) having the required decomposition into complex subspaces H^{p,q} . The consequence for theBetti number s is that, taking dimensions:b_{k} = dim H^{k} (V) = sum_{p+q=k} h^{p,q}, ,
where the sum runs over all pairs p,q with p+q=k and where
:h^{p,q} = dim H^{p,q} .
The sequence of Betti numbers becomes a Hodge diamond of Hodge numbers spread out into two dimensions.
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 belocally 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 thesheaf cohomology groups H^{p} (V,Omega^{q}) in which Omega^{q} is the sheaf of holomorphic q-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.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 filtrations. This case is much used, for example inmonodromy questions.References
*
ee also
*
Hodge cycle
*Hodge conjecture
*Period mapping
*Torelli theorem
*Variation of Hodge structure
*Mixed Hodge structure
*Yoga of weights (algebraic geometry)
Wikimedia Foundation. 2010.