Hodge theory
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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:
In the initial development, M was taken to be compact and without boundary. On all three levels the theory was most influential on subsequent work, being taken up by Kunihiko Kodaira (probably partly under the influence of Hermann Weyl at Princeton) and many others subsequently.
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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 <math>\Omega^k(M)<math> is the sheaf of differential forms of degree k on M, then the de Rham complex is the sequence of differential operators
- <math>d:0\rightarrow\mathbb R\rightarrow \Omega^0(M) \rightarrow \Omega^1(M)\rightarrow \dots\rightarrow \Omega^n(M)\rightarrow 0<math>
The de Rham cohomology is then the sequence of vector spaces defined by <math>H^k(M)=\frac{\ker(d:\Omega^k(M)\rightarrow\Omega^{k+1}(M))}{d\Omega^{k-1}(M)}<math>
One can define the Hilbert space adjoint of d, denoted <math>\delta<math> by means of the Riesz representation theorem as follows. For all <math>\alpha\in\Omega^k(M)<math> and <math>\beta\in\Omega^{k+1}(M)<math>, we require that
- <math>\int_M \langle d\alpha,\beta\rangle dV=\int_M\langle\alpha,\delta\beta\rangle dV<math>
where <math>\langle -,-\rangle<math> is the metric induced on <math>\Omega^k(M)<math>. The form Laplacian is then defined by <math>\Delta=d\delta+\delta d<math>. This allows one to define spaces of harmonic (mathematics) forms
- <math>\mathcal H_\Delta^k(M)=\{\alpha\in\Omega^k(M)\mid\Delta\alpha=0\}<math>
One can easily show that <math>d\mathcal H_\Delta^k(M)=0<math>, so there is a canonical mapping <math>\phi:\mathcal H_\Delta^k(M)\rightarrow H^k(M)<math>. The first part of Hodge's original theorem states that <math>\phi<math> 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 <math>\Delta<math> 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 <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 <math>L_i:\Gamma(E_i)\rightarrow\Gamma(E_{i+1})<math> are differential operators acting on sections of these vector bundles, and that the induced sequence
- <math>\Gamma(E_0)\rightarrow \Gamma(E_1)\rightarrow\dots\rightarrow\Gamma(E_N)<math>
is an elliptic complex. It is convenient to introduce the direct sum <math>\mathcal E^\cdot=\bigoplus_i \Gamma(E_i)<math>. Let <math>L=\bigoplus L_i:\mathcal E^\cdot\rightarrow\mathcal E^\cdot<math>, and let <math>L^*<math> be the adjoint of L. Define the elliptic operator <math>\Delta=LL^*+L^*L<math>. As in the de Rham case, this yields the vector space of harmonic sections
- <math>\mathcal H=\{e\in\mathcal E^\cdot\mid\Delta e=0\}.<math>
So let <math>H:\mathcal E^\cdot\rightarrow\mathcal H<math> be the orthogonal projection, and let G be the Green's operator for <math>\Delta<math>. The Hodge theorem then asserts the following:
- H and G are well-defined.
- <math>Id=H+\Delta G=H+G\Delta<math>
- <math>LG=GL, L^*G=GL^*<math>
- 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.
Hodge structures
An abstract definition of (real) Hodge structure is now given: for a real vector space W, a Hodge structure of weight <math>k\in\mathbb{Z}<math> on W is a direct sum decomposition of
- <math> W^C = W \otimes \mathbb C,<math>
the complexification of W, into graded pieces
- Wp,q
where k = p+q is fixed, and such that the complex conjugation of WC interchanges this subspace with
- Wq,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
- Hk(V)
having the required decomposition into complex subspaces
- Hp,q.
The consequence for the Betti numbers is that, taking dimensions
- bk = dim Hk(V) = Σ hp,q,
with
- hp,q = dim Hp,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 (Dolbeaut) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology groups
- Hp(V,Ωq)
in which
- Ωq
is the sheaf of holomorphic q-forms. This gives a more directly algebraic interpretation, without Laplacians, for this case.
In the case of singularities, the Hodge structure has to be modified to a mixed Hodge structure, where what survives is a filtration rather than a direct sum decomposition. This case is much used, for example in monodromy questions.