Hodge theory

Hodge theory

In mathematics, Hodge theory is one aspect of the study of the algebraic topology of a smooth manifold "M". More specifically, it works out the consequences for the cohomology groups of "M", with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on "M".

It was developed by W. V. D. Hodge in the 1930s as an extension of de Rham cohomology, and has major applications on three levels:

*Riemannian manifolds
*Kähler manifolds
*algebraic geometry of complex projective 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 by Kunihiko Kodaira (in Japan and later, partly under the influence of Hermann 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 forms of degree "k" on "M", then the de Rham complex is the sequence of differential operators

: 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 the Riesz 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 form Laplacian 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 a direct sum decomposition of W^{mathbb C} = W otimes mathbb C, the complexification 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 the Betti numbers 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 functions, 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 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 in monodromy questions.

References

*

ee also

*Hodge cycle
*Hodge conjecture
*Period mapping
*Torelli theorem
*Variation of Hodge structure
*Mixed Hodge structure
*Yoga of weights (algebraic geometry)


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