Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, uses partial differential equations to study the cohomology groups of a smooth manifold M. The key tool is the Laplacian operator associated to a Riemannian metric on M.

The theory was developed by Hodge in the 1930s as an extension of de Rham cohomology. It has major applications in three settings:

Hodge theory has been particularly powerful in algebraic geometry. For a long time, some of the deepest results of algebraic geometry were only accessible through analytic methods such as Hodge theory. Since the 1980s, some results of Hodge theory for algebraic varieties have also been proved by arithmetic methods, known as p-adic Hodge theory.

Hodge theory for real manifolds

De Rham cohomology

The basic version of Hodge theory is about the de Rham complex. Let M be a closed smooth manifold. For a natural number k, let Ω^k(M) be the real vector space of smooth differential forms of degree k on M. The de Rham complex is the sequence of differential operators

0\rightarrow \Omega ^{0}(M){\xrightarrow {d_{0}}}\Omega ^{1}(M){\xrightarrow {d_{1}}}\cdots {\xrightarrow {d_{n-1}}}\Omega ^{n}(M){\xrightarrow {d_{n}}}0,

where d_k denotes the exterior derivative on Ω^k(M). This is a complex in the sense that d² = 0. De Rham's theorem says that the singular cohomology of X with real coefficients is computed by the de Rham complex:

H^{k}(M,{\bf {R}})\cong {\frac {\ker d_{k}}{\mathrm {im} \,d_{k-1}}}.

Now choose a Riemannian metric on X. That determines an L² metric on the vector spaces Ω^k(M), defined by integrating over M with respect to the volume form associated to the metric. Let δ: Ω^k+1(M) → Ω^k(M) be the adjoint operator of d with respect to these metrics. Then the Laplacian on forms is defined by

\Delta =d\delta +\delta d.

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:

\mathcal H_\Delta^k(M)=\{\alpha\in\Omega^k(M)\mid\Delta\alpha=0\}.

The Laplacian appeared first in mathematical physics. In particular, Maxwell's equations say that the electromagnetic field in a vacuum is a harmonic 2-form on spacetime, viewed as Minkowski space of dimension 4.

Every harmonic form α on a closed Riemannian manifold is closed, meaning that dα = 0. As a result, there is a canonical mapping $\varphi :{\mathcal {H}}_{\Delta }^{k}(M)\rightarrow H^{k}(M,{\bf {R}})$ . The Hodge theorem states that $\varphi$ is an isomorphism of vector spaces.^[1] 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 finite-dimensional. (Admittedly, there are other ways to prove this.) Indeed, the operators Δ are elliptic, and the kernel of an elliptic operator on a closed manifold is always a finite-dimensional vector space. Another consequence of the Hodge theorem is that a Riemannian metric on a closed manifold M determines a real-valued inner product on the integral cohomology of M modulo torsion. It follows, for example, that the image of the isometry group of M in the general linear group GL(H*(M,Z)) is finite (because the group of isometries of a lattice is finite).

A variant of the Hodge theorem is the Hodge decomposition. This says that any $k$ -form on a closed Riemannian manifold can be uniquely written as the sum of three parts:

\omega =d\alpha +\delta \beta +\gamma

where $γ$ is harmonic: $Δ γ = 0$ .^[2] In terms of the L² metric on differential forms, this gives an orthogonal direct sum decomposition:

\Omega ^{k}(M)\cong \mathrm {im} \,d_{k-1}\oplus \mathrm {im} \,\delta _{k+1}\oplus {\mathcal {H}}_{\Delta }^{k}(M).

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 $E_0,E_1,\dots,E_N$ be vector bundles, equipped with metrics, on a closed smooth manifold M with a volume form dV. Suppose that

L_i:\Gamma(E_i)\rightarrow\Gamma(E_{i+1})

are linear differential operators acting on C^∞ sections of these vector bundles, and that the induced sequence

0\to \Gamma (E_{0})\rightarrow \Gamma (E_{1})\rightarrow \cdots \rightarrow \Gamma (E_{N})\to 0

is an elliptic complex. Introduce the direct sums:

\mathcal E^\bullet=\bigoplus_i \Gamma(E_i)

L=\bigoplus L_i:\mathcal E^\bullet\rightarrow\mathcal E^\bullet

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

\mathcal H=\{e\in\mathcal E^\bullet\mid\Delta e=0\}.

Let $H:\mathcal E^\bullet\rightarrow\mathcal H$ be the orthogonal projection, and let G be the Green's operator for Δ. The Hodge theorem then asserts the following:^[3]

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, $H(E_j)\cong\mathcal H(E_j)$ , 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

Main article: Hodge structure

Let X be a smooth complex projective variety. That is, X is a closed complex submanifold of some complex projective space CP^N. (By Chow's theorem. such a submanifold is automatically an algebraic subvariety of CP^N, defined by homogenous polynomial equations.) Then the standard Riemannian metric on CP^N 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 C^∞ r-form on X (with complex coefficients) can be written uniquely as a sum of forms of type (p,q) with p+q = r, meaning forms that can locally be written as a finite sum

\sum f\,dz_{1}\wedge \cdots \wedge dz_{p}\wedge d{\overline {w_{1}}}\wedge \cdots \wedge d{\overline {w_{q}}}

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:^[4]

H^{r}(X,{\bf {C}})=\oplus _{p+q=r}H^{p,q}(X).

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 H^r(X,C) depends only on the underlying topological space of X.

The piece H^p,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):^[5]

H^{p,q}(X)\cong H^{q}(X,\Omega ^{p}),

where Ω^p denotes the sheaf of holomorphic p-forms on X. For example, H^p,0(X) is the space of holomorphic p-forms on X. (If X is projective, Serre's GAGA theorem implies that a holomorphic p-form on all of X is in fact algebraic.)

The Hodge number h^p,q(X) means the dimension of the complex vector space H^p.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 h^p,q = h^q,p (because H^p,q(X) is the complex conjugate of H^q,p(X)) and h^p,q = h^n−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):

		h^2,2
	h^2,1		h^1,2
h^2,0		h^1,1		h^0,2
	h^1,0		h^0,1
		h^0,0

The Betti numbers of X are the sum of the Hodge numbers in a given row. 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

A basic application of Hodge theory is that the odd Betti numbers b_2a+1 of a smooth complex projective variety (or compact Kähler manifold) are even, by Hodge symmetry. This is not true for compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic to S¹×S³ and hence has b₁ = 1.

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 Lefschetz hyperplane theorem, the hard Lefschetz theorem, and the Hodge-Riemann bilinear relations.^[6] Hodge theory and extensions such as non-abelian Hodge theory also give strong restrictions on the possible fundamental groups of compact Kähler manifolds.

Algebraic cycles and the Hodge conjecture

Main article: Hodge conjecture

Let X be a smooth complex projective variety. A complex subvariety Y in X of codimension p defines an element of the cohomology group H^2p(X,Z). Moreover, the resulting class has a special property: its image in the complex cohomology H^2p(X,C) lies in the middle piece of the Hodge decomposition, H^p.p(X). The Hodge conjecture predicts a converse: every element of H^2p(X,Z) whose image in complex cohomology lies in the subspace H^p.p(X) should have a positive integral multiple which is a Z-linear combination of classes of complex subvarieties of X. (Such a linear combination is called an algebraic cycle on X.)

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

(H^{2p}(X,{\bf {Z}})/{\text{torsion}})\cap H^{p,p}(X)\subset H^{2p}(X,{\bf {C}})

may be much smaller than the whole group H^2p(X,Z)/torsion, even if the Hodge number h^p,p is big. In short, the Hodge conjecture predicts that the possible "shapes" of complex subvarieties of X (as described by cohomology) are determined by the Hodge structure of X (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 p = 1 (even integrally, that is, without the need for a positive integral multiple in the statement).

The Hodge structure of a variety X describes the integrals of algebraic differential forms on X over homology classes in X. In this sense, Hodge theory is related to a basic issue in calculus: there is in general no "formula" for the integral of an algebraic function. In particular, definite integrals of algebraic functions, known as periods, can be transcendental numbers. The difficulty of the Hodge conjecture reflects the lack of understanding of such integrals in general.

Example: For a smooth complex projective K3 surface X, the group H²(X,Z) is isomorphic to Z²², and H^1,1(X) is isomorphic to C²⁰. Their intersection can have rank anywhere between 1 and 20; this rank is called the Picard number of X. The moduli space of all projective K3 surfaces has a countably infinite set of components, each of complex dimension 19. The subspace of K3 surfaces with Picard number a has dimension 20−a.^[7] (Thus, for most projective K3 surfaces, the intersection of H²(X,Z) with H^1,1(X) is isomorphic to Z, but for "special" K3 surfaces the intersection can be bigger.)

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 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). Griffiths's notion of a variation of Hodge structure describes how the Hodge structure of a smooth complex projective variety X varies when X varies. In geometric terms, this amounts to studying the period mapping associated to a family of varieties. Saito's theory of Hodge modules is a generalization. Roughly speaking, a mixed Hodge module on a variety X is a sheaf of mixed Hodge structures over X, as would arise from a family of varieties which need not be smooth or compact.

Notes

↑ 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.

References

Griffiths, Phillip; Harris, Joseph (1994) [1978]. Principles of Algebraic Geometry. Wiley Classics Library. Wiley Interscience. ISBN 0-471-05059-8. MR 0507725.
Hodge, W. V. D. (1941), The Theory and Applications of Harmonic Integrals, Cambridge University Press, ISBN 978-0-521-35881-1, MR 0003947
Huybrechts, Daniel (2005), Complex Geometry: An Introduction, Springer, ISBN 3-540-21290-6, MR 2093043
Voisin, Claire (2007) [2002], Hodge Theory and Complex Algebraic Geometry (2 vols.), Cambridge University Press, ISBN 978-0-521-71801-1, MR 1967689
Warner, Frank (1983) [1971], Foundations of Differentiable Manifolds and Lie Groups, Springer, ISBN 0-387-90894-3, MR 0722297
Wells, Jr., Raymond O. (2008) [1973], Differential Analysis on Complex Manifolds (3rd ed.), Springer, ISBN 978-0-387-73891-8, MR 2359489

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