Hodge theory

Hodge theoryMathworldPlanetmath is a branch of algebraic geometryMathworldPlanetmathPlanetmath and complex manifold theory that deals with the decomposition of the cohomology groupsPlanetmathPlanetmath of an complex projective algebraic variety.

Let Ωk(M) denote the space of differential k-forms. We know that there is a map, called the exterior derivativeMathworldPlanetmath


and that d2=0. This forms a complex and the cohomology of this complex, HDR*(M) is called the de Rham cohomologyMathworldPlanetmath of M. It is isomorphic to the singular cohomology of M.

For a complex manifold, it is more natural to have complex coordinates for our differential forms. Writing zj=xj+iyj we obtain two 1-forms, dzj and dzj¯. These forms serve as a basis for all k-forms, dzi1dzi2dzipdzj1¯dzjq¯, where p+q=k. We call forms written like this to be (p,q)-forms. Let Ωp,q be the space of (p,q)-forms. Then we have a natural decomposition


The question then becomes , does this decomposition pass to the level of cohomology? The answer in general is no. If it were true in general, it would force all manifolds to have even first cohomology, which is obviously not true. But fortunately there are some nice classes of manifolds for which is does hold.

The Laplacian opererator Δ=dϕ+ϕd where ϕ is the adjoint operator (with respect to the Riemannian metric) to d. A form is called harmonic if its Laplacian is 0. The motivation for considering the Laplacian comes from looking for differential forms of minimalPlanetmathPlanetmath length. Because a basis on TM induces a basis on T*M then we have an metric on kT*M Due to this, we can define a norm on Ωk. It turns out that for ω to be minimal, then (ω,dγ)=0 for all γΩk-1. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to (ϕω,γ)=0. So a closed form with minimal norm is harmonic.

The most fundamental theorem of Hodge theory states that the space of harmonic k-forms, HΔk(M)HDRk(M).

Now for local coordinates, a 1-form is written as ω=fidzi+gjdzj¯. We define two new operators and ¯ such that


These operators also satisfy the relationsMathworldPlanetmathPlanetmath 2=2¯=0 and ¯+¯=0. These operators also have adjoints and they as well have the relation that d=*+¯*. Writing these in terms of the laplacian we obtain that


where 𝕆 denotes some cross terms. For this to be of any interest to us in terms on Hodge theory, these cross terms must vanish. Assuming they do, then if a form is harmonic, then each of the components of the (p,q)-form are harmonic. Therefore, we obtain a decomposition of the space of harmonic forms


. This gives us a Hodge decomposition on the level of cohomology due to the isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath between the space of harmonic forms and the de Rham cohomology groups.

This has some immediate and interesting consequences. It tells us that any smooth complex projective variety has an even first betti numberMathworldPlanetmath. With some work, one could also show that the second cohomology group is non-empty.

We worked under the assumptionPlanetmathPlanetmath that the cross terms in the above expression vanished. A example of a class of manifolds where this holds true is Kahler ManifoldsMathworldPlanetmath, which are complex manifolds with a Riemannian metric that is compatible with the complex structureMathworldPlanetmath.

This is just a small portion of the large topic of Hodge theory.

Title Hodge theory
Canonical name HodgeTheory
Date of creation 2013-03-22 15:21:39
Last modified on 2013-03-22 15:21:39
Owner bwebste (988)
Last modified by bwebste (988)
Numerical id 12
Author bwebste (988)
Entry type Definition
Classification msc 14N05
Classification msc 58A14