hodge theory

In previous posts on this blog (for instance briefly in the post Prismatic Cohomology: An Overview) we have made mention of a correspondence between vector bundles with flat (or integrable) connection and local systems of complex vector spaces on complex analytic manifolds. This correspondence is an equivalence of categories, and the functor from the former to the latter can be thought of as assigning solutions to differential equations. This is a primitive version of what is known as the “Riemann-Hilbert correspondence“.

It is also worth noting that local systems give rise to representations (called monodromy representations) of the fundamental group of the complex analytic manifold they live on; this is a bijection, and is an important part of the formulation of geometric Langlands correspondence (see also The Global Langlands Correspondence for Function Fields over a Finite Field).

In this post we discuss another mathematical object related to vector bundles with flat connection and to local systems of complex vector spaces – Higgs bundles. They were first studied by Nigel Hitchin and the theory was then further developed by Carlos Simpson. Hitchin named these objects after the physicist Peter Higgs, apparently because of a physics-inspired motivation which we will not discuss further due to lack of knowledge of this particular history and analogy. However, we will briefly discuss how Higgs bundles also allow us to formulate a non-abelian generalization of the Hodge decomposition. This is just one of many applications these objects have found in mathematics.

Let $X$ be a smooth complex projective variety. A rank $n$ Higgs bundle on $X$ is a pair $(E,\phi)$ where $E$ is a rank $n$ holomorphic vector bundle and $\phi$ is a morphism from $E$ to $\mathrm{End}(E)\otimes \Omega$ satisfying the condition that $\phi\wedge \phi=0$ .

Let us see how these are related to vector bundles with flat connection and local systems. Let us start with an $n$ -dimensional representation of the fundamental group. Then as we have stated earlier, there is a local system that gives rise to such a representation. Therefore there is also a corresponding vector bundle with flat connection. Now suppose the representation of the fundamental group is reductive. Then a theorem of Kevin Corlette and Simon Donaldson says that we can equip the vector bundle with flat connection with a harmonic metric.

This harmonic metric splits the flat connection into an skew-Hermitian and a Hermitian part, which in turn gives rise to a holomorphic structure on the vector bundle, and an endomorphism-valued $1$ -form; these two objects are precisely what makes up a Higgs bundle, as stated in the previous paragraph. Work of Hitchin and Simpson then established that a Higgs bundle will admit a harmonic metric if it satifies the condition of polystability. Putting all of this together, we find a correspondence between reductive representations of the fundamental group and polystable Higgs bundles, which is called the Corlette-Simpson correspondence.

The moduli space of Higgs bundles has many fascinating properties and is therefore the object of study of much mathematical research. For instance, in the case that $X$ is a curve, the moduli of rank $n$ Higgs bundles on $X$ admits a map to a space (whose dimension is half the dimension of the moduli space) called the Hitchin base $B$ , defined as

$\displaystyle B=\bigoplus_{d=1}^{n} H^{0}(X,K^{d})$

where $K$ is the canonical bundle (i.e. top exterior power of the sheaf of differentials) of $X$ . The fibers of this map are abelian varieties, which may be viewed as the Jacobians of what are known as spectral curves. The theory of the moduli space of Higgs bundles is relevant to the proof of the fundamental lemma as proven by Ngo Bau Chao, and is also relevant to applications of mirror symmetry (see also An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry) to the geometric Langlands correspondence.

Since there is a correspondence between Higgs bundles, vector bundles with flat connections, and local systems, one might expect that their moduli spaces should be the same. While it is true that their moduli spaces are homeomorphic to each other, each of their moduli spaces have extra structure that is not preserved by these homeomorphisms! In fact, these moduli spaces are Kahler manifolds (again see An Intuitive Introduction to String Theory and (Homological) Mirror Symmetry), and each of them have different complex structures. These complex structures are related to each other in a way which is reminiscent of Hamilton’s quaternions, where we have elements $i$ , $j$ , and $k$ , satisfying $i^{2}=j^{2}=k^{2}=-1$ , and $ij=-ji=k$ . Kahler manifolds which have complex structures with this behavior are called hyperkahler manifolds. Once again this is an important aspect that allows one to apply mirror symmetry to the geometric Langlands correspondence.

As an example of the moduli space of Higgs bundles, let us consider the case where $X$ is a smooth projective curve (i.e. a Riemann surface) and the vector bundles are rank $1$ . Higgs bundles in this case consist of a line bundle $E$ on $X$ and an $\mathrm{End}(E)$ -valued $1$ -form, but in this case the endomorphisms of $E$ are trivial and $\phi$ just corresponds to a $1$ -form. The moduli space of line bundles on $X$ is the Jacobian $\mathrm{Jac}(X)$ and therefore the moduli space of rank $1$ Higgs bundles on $X$ is given by $\mathrm{Jac}(X)\times H^{0}(X,\Omega)$ which is also the cotangent bundle $T^{*}\mathrm{Jac}(X)$ of $\mathrm{Jac}(X)$ .

Let us now discuss how the theory of Higgs bundles and the Corlette-Simpson correspondence allows us to formulate a nonabelian generalization of the Hodge decomposition. We recall the classical Hodge decomposition for $H^{1}(X,\mathbb{C})$ :

$\displaystyle H^{1}(X,\mathbb{C})\cong H^{1}(X,\mathcal{O})\oplus H^{0}(X,\Omega)$

The cohomology group $H^{1}(X,\mathbb{C})$ is dual to the homology group $H_{1}(X,\mathbb{C})$ ; in turn, the latter is the abelianization of the fundamental group $\pi_{1}(X)$ , by the Hurewicz theorem. Therefore, we may also express the Hodge decomposition as

$\displaystyle \mathrm{Hom}(\pi_{1}(X),\mathbb{C})\cong H^{1}(X,\mathcal{O})\oplus H^{0}(X,\Omega)$

Now we consider Higgs bundles. Consider a Higgs bundle $(E,\phi)$ . We may think of the holomorphic vector bundle $E$ as an element of the nonabelian cohomology $\check{H}^{1}(X,\mathcal{GL}_{n}(\mathbb{C}))$ , and we may think of $\phi$ as an element of $\oplus H^{0}(X,\mathrm{End}(E)\otimes \Omega)$ . Combining this with the Corlette-Simpson correspondence between Higgs bundles and local systems (and hence representations of $\pi_{1}(X)$ ), we get

$\displaystyle \mathrm{Rep}(\pi_{1}(X),\mathrm{GL}_{n}(\mathbb{C}))\cong \check{H}^{1}(X,\mathcal{GL}_{n}(\mathbb{C}))\oplus H^{0}(X,\mathrm{End}(E)\otimes \Omega)$

This is the nonabelian generalization of the Hodge decomposition that we have alluded to throughout this post.

There are many other aspects of the theory of Higgs bundles that we have not yet discussed; for instance, the moduli space of Higgs bundles also arises as the space of solutions to the differential equations known as Hitchin’s equations. In another very different direction, since Hodge theory has a p-adic version (see also p-adic Hodge Theory: An Overview) one may also wonder if there is a p-adic version of the theory we have just discussed, and in fact there is! But we will leave all of this hopefully to future posts.

References:

Higgs bundle on Wikipedia

Nonabelian Hodge correspondence on Wikipedia

What is a… Higgs bundle by Steven Bradlow, Oscar Garcia-Prada, and Peter B. Gothen

Nonabelian Hodge Theory by Carlos T. Simpson

Perverse Sheaves and Fundamental Lemmas (lecture notes by Chao Li from a course by Wei Zhang)

In The Theory of Motives we discussed the notion of a Weil cohomology, and mentioned four “classical” examples, the singular (also known as Betti) cohomology, the de Rham cohomology, the $\ell$ -adic cohomology, and the crystalline cohomology.

Cohomology theories may be thought of as a way to study geometric objects using linear algebra, by associating vector spaces (or more generally, modules or abelian groups) to such a geometric object. But the four Weil cohomology theories above actually give more than just a vector space:

The singular cohomology has an action of complex conjugation.
The de Rham cohomology has a Hodge filtration.
The $\ell$ -adic cohomology has an action of the Galois group.
The crystalline cohomology has an action of Frobenius (and a Hodge filtration as well).

There are relations between these different cohomologies. For example, for a smooth projective variety $X$ over the complex numbers $\mathbb{C}$ , the singular cohomology of the corresponding complex analytic manifold $X(\mathbb{C})$ , with complex coefficients (this can be obtained from singular cohomology with integral coefficients by tensoring with $\mathbb{C}$ ) and the de Rham cohomology are isomorphic:

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H_{\mathrm{dR}}^{k}(X)$

The roots of this idea go back to de Rham’s work on complex manifolds, where chains in singular homology (which is dual to singular cohomology, see also Homology and Cohomology) can be paired with the differential forms of de Rham cohomology (see also Differential Forms), simply by integrating the differential forms along these chains. By the machinery developed by Alexander Grothendieck, this can be ported over into the world of algebraic geometry.

Again borrowing from the world of complex manifolds, the machinery of Hodge theory gives us the following Hodge decomposition (see also Shimura Varieties):

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{C}}^{j})$

Now again for the case of smooth projective varieties over the complex numbers , $\ell$ -adic cohomology also has such an isomorphism with singular cohomology – but this time if it has $\ell$ -adic coefficients (i.e. in $\mathbb{Q}_{\ell}$ ).

$\displaystyle H_{\mathrm{sing}}^{k}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{Q}_{\ell}\simeq H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell})$

Such isomorphisms are also known as comparison isomorphisms (or comparison theorems).

More generally, if we have a field $B$ into which we can embed both $\mathbb{Q}_{\ell}$ and $\mathbb{C}$ (for instance $\mathbb{C})$ , we obtain the following comparison theorem:

$\displaystyle H_{\mathrm{et}}^{k}(X,\mathbb{Q}_{\ell}) \otimes_{\mathbb{Q_{\ell}}} B\simeq H_{\mathrm{dR}}^{k}(X) \otimes_{\mathbb{C}} B$

Here is a very interesting thing that these comparison theorems can give us. Let $X$ be a modular curve. Then the Hodge decomposition for the first cohomology gives us

$\displaystyle H_{\mathrm{sing}}^{1}(X(\mathbb{C}),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}=H^{1}(X,\Omega_{X/\mathbb{C}}^{0})\oplus H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$

But the $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$ is the cusp forms of weight $2$ as per the discussion in Modular Forms (see also Galois Representations Coming From Weight 2 Eigenforms). By the results of Hodge theory, the other summand $H^{1}(X,\Omega_{X/\mathbb{C}}^{0})$ is just the complex conjugate of $H^{0}(X,\Omega_{X/\mathbb{C}}^{1})$ . But we now also have a comparison with etale cohomology, which has a Galois representation! For this the modular form must lie in the cohomology with $\mathbb{Q}$ coefficients, which happens if it is a Hecke eigenform whose Hecke eigenvalues are in $\mathbb{Q}$ . So one of the great things that these comparison theorems gives us is this way of relating modular forms and Galois representations.

The comparison isomorphisms above work for smooth projective varieties over the complex numbers, but let us now go to the p-adic world, and let us consider smooth projective varieties over the p-adic numbers.

It was observed by John Tate (and later explored by Gerd Faltings) that the p-adic cohomology (i.e. the etale cohomology of a smooth projective variety over $\mathbb{Q}_{p}$ , or more generally some other p-adic field, with p-adic coefficients, distinguishing it from $\ell$ -adic cohomology where another prime $\ell$ different from $p$ must be brought in) can have a decomposition akin to the Hodge decomposition, after tensoring it with the p-adic complex numbers (this is the completion of the algebraic closure of the p-adic numbers):

$\displaystyle H^{k}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}}\mathbb{C}_{p}=\bigoplus_{i+j=k} H^{i}(X,\Omega_{X/\mathbb{Q}}^{j})\otimes_{\mathbb{Q}}\mathbb{C}_{p}(-j)$

The p-adic complex numbers here play the role of the complex numbers in the singular cohomology case above or the $\ell$ -adic numbers in the $\ell$ -adic case.

The ideas conjectured by Tate, and later completed by Faltings, was but the prototype of what is now known as p-adic Hodge theory. In its modern form, p-adic Hodge theory concerns comparison isomorphisms between different Weil cohomology theories on smooth projective varieties over the p-adic numbers. However, the role played by the complex numbers, $\ell$ -adic numbers (for the complex case), and p-adic complex numbers (for the p-adic case) must now be played by much more complicated objects called period rings, which were developed by Jean-Marc Fontaine. We will discuss the construction of the period rings at the end of this post, but first let us see how they work.

Let $X$ be a smooth projective variety over $\mathbb{Q}_{p}$ (or more generally some other p-adic field). Let $H_{\mathrm{dR}}^{i}(X)$ and $H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})$ be its de Rham cohomology and the p-adic etale cohomology of its base change to the algebraic closure $\overline{\mathbb{Q}}_{p}$ respectively. The comparison isomorphism at the center of p-adic Hodge theory is the following:

$\displaystyle H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}=H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}}$

The object denoted $B_{\mathrm{dR}}$ here is the aforementioned period ring. It is equipped with both a Galois action and a filtration akin to the Hodge filtration. More than just that isomorphism above, we also have a way of obtaining the de Rham cohomology if we are given the p-adic etale cohomology, simply by taking the part that is invariant under the Galois action:

$\displaystyle \displaystyle H_{\mathrm{dR}}^{i}(X)=(H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})\otimes_{\mathbb{Q}_{p}} B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

To go the other way, i.e. to recover the p-adic etale cohomology from the de Rham cohomology, we will need a different kind of period ring. This period ring is $B_{\mathrm{cris}}$ , which aside from having a Galois action and a filtration also has an action of Frobenius. Aside from providing us the same isomorphism between de Rham and p-adic etale cohomology upon tensoring, it also provides us with a solution to our earlier problem (as long as $X$ has a smooth proper integral model) as follows:

$\displaystyle H_{\mathrm{et}}^{i}(X_{\overline{\mathbb{Q}}_{p}},\mathbb{Q}_{p})= \mathrm{Fil}^{0}(H_{\mathrm{dR}}^{i}(X)\otimes_{\mathbb{Q}_{p}} B_{\mathrm{cris}})^{\varphi=1}$

This idea can be further abstracted – since etale cohomology provides Galois representations, we can just take some p-adic Galois representation instead, without caring whether it comes from etale cohomology or not, and tensor it with a period ring, then take Galois invariants. For instance let $V$ be some p-adic Galois representation. Then we can take the tensor product

$V_{\mathrm{dR}}=(V\otimes B_{\mathrm{dR}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If the dimension of $V_{\mathrm{dR}}$ is equal to the dimension of $V$ , then we say that the Galois representation $V$ is de Rham. Similarly we can tensor with $B_{\mathrm{cris}}$ :

$V_{\mathrm{cris}}=(V\otimes B_{\mathrm{cris}})^{\mathrm{Gal}_{\mathbb{Q}_{p}}}$

If its $V_{\mathrm{cris}}$ is equal to the dimension of $V$ , we say that $V$ is crystalline.

The idea of these “de Rham” and “crystalline” Galois representations is that if they come from the corresponding cohomologies then they will have these properties. But does the converse hold? If they are “de Rham” and “crystalline” does that mean that they come from the corresponding cohomologies (i.e. they “come from geometry”)? This is roughly the content of the Fontaine-Mazur conjecture.

Now let us say a few things about the construction of these period rings. These constructions make use of the concepts we discussed in Perfectoid Fields. We start with the ring $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$ , which, as we recall from Perfectoid Fields, is the ring of Witt vectors of the tilt of $\mathcal{O}_{\mathbb{C}_{p}}$ . By inverting $p$ and taking the completion with respect to the canonical map $\theta: A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}}) \to\mathcal{O}_{\mathbb{C}_{p}}$ , we obtain a ring which we suggestively denote by $B_{\mathrm{dR}}^{+}$ .

There is a special element $t$ of $B_{\mathrm{dR}}^{+}$ which we think of as the logarithm of the element $(1, \zeta^{1/p},\zeta^{1/p},\ldots)$ . Upon inverting this element $t$ , we obtain the field $B_{\mathrm{dR}}$ .

The field $B_{\mathrm{dR}}$ is equipped with a Galois action, carried over from the fields involved in its construction, and a filtration, given by $\mathrm{Fil}^{i}B_{\mathrm{dR}}=t^{i}B_{\mathrm{dR}}$ .

To construct $B_{\mathrm{cris}}$ , we once again start with $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})$ and invert $p$ . However, to have a Frobenius, instead of completing with respect to the kernel of the map $\theta$ , we take a generator of this kernel (which we shall denote by $\omega$ ). Then we denote by $B_{\mathrm{cris}}^{+}$ the ring formed by all the power series of the form $\sum_{n=0}^{\infty} a_{n}\omega^{n}/n!$ where the $a_{n}$ ‘s are elements of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ which converge as $n\to\infty$ , under the topology of $A_{\mathrm{inf}}(\mathcal{O}_{\mathbb{C}_{p}})[1/p]$ (which is not the p-adic topology!). Once again there will be an element $t$ like before; we invert $t$ to obtain $B_{\mathrm{cris}}$ .

There is yet another period ring called $B_{\mathrm{st}}$ , where the subscript stands for semistable; in addition to a Galois action, filtration, and Frobenius, it has a monodromy operator. Since this is less extensively discussed in introductory literature, we follow this lead and leave this topic, and the many other wonderful topics related to p-adic Hodge theory, to future posts on this blog.