Hodge theory - Definition

In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M 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.

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.

An abstract definition of Hodge structure is now given: for a real vector space W, a Hodge structure on W is a direct sum decomposition of

<math> W \otimes \mathcal C,<math>

the complexification of W, into graded pieces

W^p,q

where k = p+q is fixed, and such that complex conjugation 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) = Σ h^p,q,

with

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 (Dolbeaut) it was shown that the Hodge decomposition above can also be found by means of the sheaf cohomology groups

H^p(V,Ω^q)

in which

Ω^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, 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.

See also

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