Hodge theory
In
The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings: Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles.
While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.
History
The field of
As originally stated,
De Rham's original statement is then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology.
Separately, a 1927 paper of
Hodge felt that these techniques should be applicable to higher dimensional varieties as well. His colleague Peter Fraser recommended de Rham's thesis to him. In reading de Rham's thesis, Hodge realized that the real and imaginary parts of a holomorphic 1-form on a Riemann surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now known as the Hodge star operator. He further conjectured that each cohomology class should have a distinguished representative with the property that both it and its dual vanish under the exterior derivative operator; these are now called harmonic forms. Hodge devoted most of the 1930s to this problem. His earliest published attempt at a proof appeared in 1933, but he considered it "crude in the extreme". Hermann Weyl, one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not. In 1936, Hodge published a new proof. While Hodge considered the new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, Hermann Weyl and Kunihiko Kodaira modified Hodge's proof to repair the error. This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.
In retrospect it is clear that the technical difficulties in the existence theorem did not really require any significant new ideas, but merely a careful extension of classical methods. The real novelty, which was Hodge’s major contribution, was in the conception of harmonic integrals and their relevance to algebraic geometry. This triumph of concept over technique is reminiscent of a similar episode in the work of Hodge’s great predecessor Bernhard Riemann.
—M. F. Atiyah, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, Biographical Memoirs of Fellows of the Royal Society, vol. 22, 1976, pp. 169–192.
Hodge theory for real manifolds
De Rham cohomology
The Hodge theory references the
where dk denotes the
Operators in Hodge theory
Choose a Riemannian metric g on M and recall that:
The metric yields an
Naturally the above inner product induces a norm, when that norm is finite on some fixed k-form:
then the integrand is a real valued, square integrable function on M, evaluated at a given point via its point-wise norms,
Consider the
Then the
This is a second-order linear differential operator, generalizing the Laplacian for functions on Rn. By definition, a form on M is harmonic if its Laplacian is zero:
The Laplacian appeared first in
Every harmonic form α on a closed Riemannian manifold is closed, meaning that dα = 0. As a result, there is a canonical mapping . The Hodge theorem states that is an isomorphism of vector spaces.[4] In other words, each real cohomology class on M has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum L2 norm that represents a given cohomology class. The Hodge theorem was proved using the theory of elliptic partial differential equations, with Hodge's initial arguments completed by Kodaira and others in the 1940s.
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are
A variant of the Hodge theorem is the Hodge decomposition. This says that there is a unique decomposition of any differential form ω on a closed Riemannian manifold as a sum of three parts in the form
in which γ is harmonic: Δγ = 0.[5] In terms of the L2 metric on differential forms, this gives an orthogonal direct sum decomposition:
The Hodge decomposition is a generalization of the Helmholtz decomposition for the de Rham complex.
Hodge theory of elliptic complexes
Atiyah and Bott defined elliptic complexes as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let be
are linear
is an elliptic complex. Introduce the direct sums:
and let L∗ be the adjoint of L. Define the elliptic operator Δ = LL∗ + L∗L. As in the de Rham case, this yields the vector space of harmonic sections
Let be the orthogonal projection, and let G be the Green's operator for Δ. The Hodge theorem then asserts the following:[6]
- H and G are well-defined.
- Id = H + ΔG = H + GΔ
- LG = GL, L∗G = GL∗
- The cohomology of the complex is canonically isomorphic to the space of harmonic sections, , in the sense that each cohomology class has a unique harmonic representative.
There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.
Hodge theory for complex projective varieties
Let X be a smooth complex projective manifold, meaning that X is a closed complex submanifold of some complex projective space CPN. By Chow's theorem, complex projective manifolds are automatically algebraic: they are defined by the vanishing of homogeneous polynomial equations on CPN. The standard Riemannian metric on CPN induces a Riemannian metric on X which has a strong compatibility with the complex structure, making X a Kähler manifold.
For a complex manifold X and a natural number r, every
with f a C∞ function and the zs and ws holomorphic functions. On a Kähler manifold, the (p, q) components of a harmonic form are again harmonic. Therefore, for any compact Kähler manifold X, the Hodge theorem gives a decomposition of the cohomology of X with complex coefficients as a direct sum of complex vector spaces:[7]
This decomposition is in fact independent of the choice of Kähler metric (but there is no analogous decomposition for a general compact complex manifold). On the other hand, the Hodge decomposition genuinely depends on the structure of X as a complex manifold, whereas the group Hr(X, C) depends only on the underlying topological space of X.
Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product with complex coefficients is compatible with the Hodge decomposition:
The piece Hp,q(X) of the Hodge decomposition can be identified with a coherent sheaf cohomology group, which depends only on X as a complex manifold (not on the choice of Kähler metric):[8]
where Ωp denotes the
On the other hand, the integral can be written as the cap product of the homology class of Z[clarification needed] and the cohomology class represented by . By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call [Z], and the cap product can be computed by taking the cup product of [Z] and α and capping with the fundamental class of X.
Because [Z] is a cohomology class, it has a Hodge decomposition. By the computation we did above, if we cup this class with any class of type , then we get zero. Because , we conclude that [Z] must lie in .
The Hodge number hp,q(X) means the dimension of the complex vector space Hp.q(X). These are important invariants of a smooth complex projective variety; they do not change when the complex structure of X is varied continuously, and yet they are in general not topological invariants. Among the properties of Hodge numbers are Hodge symmetry hp,q = hq,p (because Hp,q(X) is the complex conjugate of Hq,p(X)) and hp,q = hn−p,n−q (by Serre duality).
The Hodge numbers of a smooth complex projective variety (or compact Kähler manifold) can be listed in the Hodge diamond (shown in the case of complex dimension 2):
h2,2 | ||||
h2,1 | h1,2 | |||
h2,0 | h1,1 | h0,2 | ||
h1,0 | h0,1 | |||
h0,0 |
For example, every smooth projective curve of genus g has Hodge diamond
1 | ||
g | g | |
1 |
For another example, every K3 surface has Hodge diamond
1 | ||||
0 | 0 | |||
1 | 20 | 1 | ||
0 | 0 | |||
1 |
The
The "Kähler package" is a powerful set of restrictions on the cohomology of smooth complex projective varieties (or compact Kähler manifolds), building on Hodge theory. The results include the
Hodge theory and extensions such as
Algebraic cycles and the Hodge conjecture
Let be a smooth complex projective variety. A complex subvariety in of codimension defines an element of the cohomology group . Moreover, the resulting class has a special property: its image in the complex cohomology lies in the middle piece of the Hodge decomposition, . The Hodge conjecture predicts a converse: every element of whose image in complex cohomology lies in the subspace should have a positive integral multiple that is a -linear combination of classes of complex subvarieties of . (Such a linear combination is called an algebraic cycle on .)
A crucial point is that the Hodge decomposition is a decomposition of cohomology with complex coefficients that usually does not come from a decomposition of cohomology with integral (or rational) coefficients. As a result, the intersection
may be much smaller than the whole group , even if the Hodge number is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of (as described by cohomology) are determined by the Hodge structure of (the combination of integral cohomology with the Hodge decomposition of complex cohomology).
The Lefschetz (1,1)-theorem says that the Hodge conjecture is true for (even integrally, that is, without the need for a positive integral multiple in the statement).
The Hodge structure of a variety describes the integrals of algebraic differential forms on over homology classes in . In this sense, Hodge theory is related to a basic issue in
Example: For a smooth complex projective K3 surface , the group is isomorphic to , and is isomorphic to . Their intersection can have rank anywhere between 1 and 20; this rank is called the
This example suggests several different roles played by Hodge theory in complex algebraic geometry. First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety. Second, Hodge theory gives information about the moduli space of smooth complex projective varieties with a given topological type. The best case is when the Torelli theorem holds, meaning that the variety is determined up to isomorphism by its Hodge structure. Finally, Hodge theory gives information about the Chow group of algebraic cycles on a given variety. The Hodge conjecture is about the image of the cycle map from Chow groups to ordinary cohomology, but Hodge theory also gives information about the kernel of the cycle map, for example using the intermediate Jacobians which are built from the Hodge structure.
Generalizations
Mixed Hodge theory, developed by Pierre Deligne, extends Hodge theory to all complex algebraic varieties, not necessarily smooth or compact. Namely, the cohomology of any complex algebraic variety has a more general type of decomposition, a mixed Hodge structure.
A different generalization of Hodge theory to singular varieties is provided by intersection homology. Namely, Morihiko Saito showed that the intersection homology of any complex projective variety (not necessarily smooth) has a pure Hodge structure, just as in the smooth case. In fact, the whole Kähler package extends to intersection homology.
A fundamental aspect of complex geometry is that there are continuous families of non-isomorphic complex manifolds (which are all diffeomorphic as real manifolds).
See also
- Potential theory
- Serre duality
- Helmholtz decomposition
- Local invariant cycle theorem
- Arakelov theory
- Hodge-Arakelov theory
- ddbar lemma, a key consequence of Hodge theory for compact Kähler manifolds.
Notes
- ^ Chatterji, Srishti; Ojanguren, Manuel (2010), A glimpse of the de Rham era (PDF), working paper, EPFL
- JSTOR 1968379.
- ^ Michael Atiyah, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, Biogr. Mem. Fellows R. Soc., 1976, vol. 22, pp. 169–192.
- ^ Warner (1983), Theorem 6.11.
- ^ Warner (1983), Theorem 6.8.
- ^ Wells (2008), Theorem IV.5.2.
- ^ Huybrechts (2005), Corollary 3.2.12.
- ^ Huybrechts (2005), Corollary 2.6.21.
- ^ Huybrechts (2005), sections 3.3 and 5.2; Griffiths & Harris (1994), sections 0.7 and 1.2; Voisin (2007), v. 1, ch. 6, and v. 2, ch. 1.
- ^ Griffiths & Harris (1994), p. 594.