# Semisimplicity of the Kontsevich-Zorich cocycle

In this post, I would like to discuss some of the ideas from my paper Semisimplicity and Rigidity of the Kontsevich-Zorich cocycle. It contains a number of somewhat disparate methods – Hodge theory, random walks and a bit of homogeneous and smooth dynamics. I will try to explain how these concepts interact and what the essential aspects are. Some familiarity with flat surfaces, as explained for example in the survey by Anton Zorich, will be assumed.

Below the fold I will discuss some of the analytic ideas from variations of Hodge structures. Then I’ll try to connect these to Teichmuller dynamics and explain how this leads to various semisimplicity results. For applications, I’ll discuss how this implies that measurable ${{\mathrm{SL}}_2{\mathbb R}}$-invariant bundles have to be real-analytic. I’ll also mention how real multiplication on factors of the Jacobians arises.

— 1. Background in variations of Hodge structure —

The analytic aspects of variations of Hodge structure were first developed by Griffiths. An introduction I particularly enjoyed is in the collection of articles Topics in Transcendental Algebraic Geometry. The first few chapters, written by Griffiths, are particularly illuminating. Note that the curvature calculations there correct some of the signs from the book of Griffiths and Harris.

A very clear and detailed exposition of these results is also available in Schmid’s article Singularities of the Period Mapping.

Without going into all the details, I would like to discuss some of the ideas that enter the arguments. Consider some compact complex manifold ${B}$.

Definition 1 A (polarized) variation of Hodge structure on ${B}$, of weight ${w}$, is the following data:

• A local system of free ${{\mathbb Z}}$-modules ${H_{\mathbb Z}}$ (think of ${{\mathbb Z}^n}$ with monodromy)
• A flat bilinear form on ${H_{\mathbb Z}}$ which is ${(-1)^w}$-symmetric
• A decreasing filtration by holomorphic subbundles ${F^\bullet}$ of the complexified local system ${H_{\mathbb C}:=H_{\mathbb Z}\otimes{\mathbb C}}$

$\displaystyle \cdots \subseteq F^{p+1}\subseteq F^p\subseteq \cdots \subseteq H_{\mathbb C}$

• The subbundles are required to satisfy ${H_{\mathbb C}=F^p\oplus\overline{F^{w-p+1}}}$. Denoting by ${\nabla}$ the flat connection on ${H_{\mathbb C}}$ induced from ${H_{\mathbb Z}}$ we must have for any vector field ${X}$ on ${B}$ that

$\displaystyle \nabla_X F^p \subseteq F^{p-1}$

The following observations appear in many proofs in the subject.

• Principle 1 On a compact complex manifold, a bounded subharmonic function has to be constant.
This is just the maximum principle. However, it extends to the case of quasi-projective varieties. Namely, one can take out proper complex-analytic subsets and bounded subharmonic functions still have to be constant. For Teichmuller dynamics, one cannot guarantee boundedness. But one has enough restriction on growth that a similar statement holds. I will discuss it in more detail below.

• Principle 2 If a holomorphic bundle has negative curvature, then the norm (and the log of the norm) of a holomorphic section is subharmonic.
This is a purely local statement. Combined with Principle 1, it implies there are no global holomorphic sections of bundles with negative curvature.

• Principle 3 Bundles that arise in variations of Hodge structure have unusual curvature properties.
This fact was discovered by Griffiths and has a lot of geometric consequences. While it
is not true that all bundles relevant to Hodge theory have negative curvature, they do have an alternating combination of positive and negative curvatures. Moreover, these pieces are so arranged that inductive arguments are possible.

One of the main consequences of Principle 3 is the Theorem of the Fixed Part.

Theorem 2 Suppose ${\phi}$ is a global flat section of a variation of Hodge structures over a compact complex manifold. Then each ${(p,q)}$-component of ${\phi}$ is also flat.

An important aspect of this theorem is that it applies to variations of arbitrary weight.

Example 3 Suppose ${H}$ is a variation of Hodge structures and ${V\subseteq H_{\mathbb C}}$ is some local subsystem. Suppose it has a complementary local system ${V'\subseteq H_{\mathbb C}}$ such that ${V\oplus V'=H_{\mathbb C}}$. Note that there are no assumptions on how ${V}$ or ${V'}$ are related to the Hodge structure on ${H_{\mathbb C}}$. The conclusion is that ${V}$ must be itself a variation of Hodge structure. To prove it, consider the operator ${\pi_V\in {\mathrm{End}}(H_{\mathbb C})}$ which is the projection to ${V}$ along ${V'}$. This is a global flat section of the variation of Hodge structure ${{\mathrm{End}}(H_{\mathbb C})}$ and the Theorem of the Fixed Part applies to it. Unraveling this gives the desired conclusion.

Remark 4 One of the difficulties in applying this result is finding the complement. In my paper, I need to refer to results of Avila, Eskin and Moller. This provides complements in the case of the Hodge bundle. To find complements in all tensor powers, one needs a certain statement about “algebraic hulls” of a cocycle. Section 2 of my paper proves the needed result.

— 2. Teichmuller dynamics —

One would like to apply the techniques described above to the variations of Hodge structure that arise on Teichmuller disks. Because this is no longer a compact complex manifold, Principle 1 as stated above does not apply.

However, one can use the ${{\mathrm{SL}}_2{\mathbb R}}$-action and the finiteness of an invariant measure. Most points on an ${{\mathrm{SL}}_2{\mathbb R}}$-orbit are in a compact part and this suffices to establish Principle 1 in this setting.

Example 5 Suppose given a positive function ${f:{\mathbb R}{\rightarrow} {\mathbb R}_{\geq 0}}$ and assume

$\displaystyle \lim_{t{\rightarrow} +\infty}\frac{f(t)}{t}{\rightarrow} 0$

This doesn’t restrict ${f}$ too much, but if it is convex then we must have

$\displaystyle f(t){\rightarrow} \text{constant}$

Subharmonic functions are the complex analogue of convex functions. In section 4 of the paper, I prove an analogue of the above fact for functions subharmonic along a random walk.

The idea of the proof is an integration by parts argument. One proves that a subharmonic function either has to grow at a definite rate along the random walk, or it has to be constant. The proof is by contradiction – assuming that the function grows sublinearly one proves that it is constant.

Incidentally, the same type of integration by parts proves the formula for the sum of Lyapunov exponents due to Kontsevich and Forni.

— 3. Semisimplicity —

With the result on subharmonic functions available, one proves the Theorem of the Fixed Part in the setting of Teichmuller dynamics. Note that one has to ensure the sublinear growth of the functions considered. This follows because any global section of a cocycle has this growth bound (it has to be in the zero Lyapunov subspace).

To prove semisimplicity for the Kontsevich-Zorich cocycle, one follows a standard strategy. Some difficulties arise because several copies (i.e. isotypical components) of the same bundle might arise. One deals with them in analogy with the following.

Example 6 Suppose ${G}$ is some finite group and ${V}$ is some representation of ${G}$ on a vector space. Denote by ${V_1,\ldots,V_k}$ the irreducible representations of ${G}$. By general theory we know that we have an isomorphism

$\displaystyle V\cong \oplus_i V_i^{m_i}$

However, this is not canonical. The different copies of ${V_i}$ inside ${V}$ do not arise in a canonical manner. A better way to say this is that we have the canonical isomorphism

$\displaystyle V\cong \oplus V_i\otimes W_i$

where ${W_i}$ are vector spaces of dimension ${m_i}$. To describe the ${W_i}$, note that they can be defined as the space of maps between ${G}$-representations:

$\displaystyle W_i:={\mathrm{Hom}}_G(V_i,V)$

We have a natural map ${V_i\otimes W_i {\rightarrow} V}$ given by ${v_i\otimes \phi_i\mapsto \phi_i(v_i)}$. The above decomposition behaves much better when the spaces have some extra information, such as a Hodge structure.

To deal with invariant subbundles of the Kontsevich-Zorich cocycle, one needs to use the same strategy as above. Then one recovers all the extra information that is available.

Remark 7 To deal with the linear algebra of Hodge structures, it is often convenient to introduce the Deligne torus ${{\mathbb S}}$. This is a real-algebraic group defined by

$\displaystyle {\mathbb S}= \left\{ \begin{bmatrix} a & b\\ -b & a \end{bmatrix} \in {\mathrm{GL}}_2{\mathbb R} \right\}$

A Hodge structure on a real vector space ${H_{\mathbb R}}$ is then just an algebraic representation ${{\mathbb S}{\rightarrow} {\mathrm{GL}}(H_{\mathbb R})}$. This is the same as a splitting of the complexification ${H_{\mathbb C}=\oplus H^{p,q}}$ where ${z=a+\sqrt{-1}b\in {\mathbb S}}$ acts by ${z^p\overline{z}^q}$ on ${H^{p,q}}$. Because the representation is real, we necessarily have ${H^{p,q}=\overline{H^{q,p}}}$. Note also that the usual Hodge-star operator is just

$\displaystyle \begin{bmatrix} 0 & 1\\ -1 & 0 \end{bmatrix} \in {\mathbb S}$

In the Hodge theory literature, this is usually called the Weil operator and it makes sense for any Hodge structure.

The above abstraction might seem unnecessary, but it does simplify some arguments. For example, the following result due to Deligne is non-trivial to prove from basic principles.

Lemma 8 Suppose ${W}$ is an ${{\mathbb R}}$-vector space and assume ${{\mathrm{End}}(W)}$ has a Hodge structure compatible with the algebra structure. Then ${W}$ itself carries a natural Hodge structure inducing the one on ${{\mathrm{End}}(W)}$.

Proof: Viewing a Hodge structure as a representation of ${{\mathbb S}}$, we have a map ${{\mathbb S}{\rightarrow} {\mathrm{PGL}}(W)}$. Indeed, ${{\mathrm{PGL}}(W)}$ is the automorphism group of the algebra ${{\mathrm{End}}(W)}$. But such a map always lifts to ${{\mathbb S}{\rightarrow} {\mathrm{GL}}(W)}$ whence the Hodge structure on ${W}$. $\Box$

— 4. Rigidity —

Once the semisimplicity results are established, one can apply them to the measurable ${{\mathrm{SL}}_2{\mathbb R}}$-dynamics. One consequence is that ${{\mathrm{SL}}_2{\mathbb R}}$-invariant bundles which are apriori only measurable have to be in fact real-analytic.

This is used by Chaika and Eskin to show that Every flat surface is Birkhoff and Osceledets generic in almost every direction. In other words, the Oseledets Multiplicative Ergodic theorem holds on every, rather than almost every, flat surface in a.e. direction. Of course the key inputs for this result are the measure rigidity theorems of Eskin and Mirzkhani as well as the isolation and orbit closure results of Eskin, Mirzkhani and Mohammadi.

To prove that measurable ${{\mathrm{SL}}_2{\mathbb R}}$-invariant bundles are real-analytic, one proceeds in two stages. First, using that invariant bundles are Hodge-orthogonal, one finds that on stable and unstable leaves these have to vary real-analytically.

To see this, recall that we have Lyapunov filtrations by order of growth in the future (resp. past) of the geodesic flow. These vary measurably in general, but on stable (resp. unstable) leaves these filtrations are flat.

Using Hodge-orthogonality of invariant bundles, one can inductively “locate” the invariant bundles as Hodge-orthogonal to the previous subbundles. The basis of the induction is provided by the “tautological” subbundle responsible for the ${{\mathrm{SL}}_2{\mathbb R}}$-action, which is known to vary real-analytically.

The second stage of the argument is to prove that the bundles vary polynomially, not just real-analytically. This follows using a contraction-expansion argument which was probably first introduced by Kontsevich. He showed that any ${{\mathrm{GL}}_2{\mathbb R}}$-invariant manifold has to be linear in period coordinates.

Once we know that invariant bundles vary polynomially on stable (resp. unstable) leaves, a short argument proves they must vary polynomially in all directions.

Example 9 I would like to sketch an argument that a ${{\mathrm{GL}}_2{\mathbb R}}$-invariant manifold ${{\mathcal M}}$ has to be affine in period coordinates. Assume also that it carries a Lebesgue-class ${{\mathrm{GL}}_2{\mathbb R}}$-invariant measure ${\mu}$ which gives a probability measure on surfaces of area less than one. Viewing the manifold inside the stratum ${{\mathcal M}\subseteq {\mathcal H}}$, consider the inclusion of tangent spaces ${T{\mathcal M}\subseteq T{\mathcal H}}$. The geodesic flow ${g_t}$ on ${T{\mathcal M}}$ will have a symmetry in the Lyapunov spectrum (like in the case of a stratum). This follows from results of Forni and uses the ${{\mathrm{SL}}_2{\mathbb R}}$-invariance. Looking at the Lyapunov decomposition of ${T{\mathcal M}}$, we see that we must have ${T{\mathcal M}=T{\mathcal M}^+\oplus T{\mathcal M}^-}$ where the stable and unstable spaces have to come from the corresponding pieces in the stratum. This implies that the intersection of ${{\mathcal M}}$ with the stable and unstable foliations on ${{\mathcal H}}$ is what one would expect of an affine manifold. With some more work, the above can be made into an argument. This is done, for instance, in this paper of Avila and Gouezel.

— 5. Real Multiplication —

Martin Moller has shown that Teichmuller curves parametrize curves whose Jacobians have (a factor with) real multiplication by a totally real number field. This essentially says that the Hodge structures that appear have a non-trivial splitting. Moreover the factors of the splitting are parametrized by embeddings of a totally real number field. This allows one to define an action of the number field by scaling which is Galois-equivariant.

Let us see the simplest instance of a Hodge structure with real multiplication.

Example 10 Consider the quadratic field ${k={\mathbb Q}[\alpha]/(\alpha^2-D)}$ where ${D}$ is some fixed positive (square-free) integer. Viewing ${k}$ as an abstract field, it has two real embeddings ${\iota_+,\iota_-}$ into ${{\mathbb R}}$. Namely ${\iota_\pm(\alpha)=\pm \sqrt{D}}$. Consider now two Hodge structures ${H_+}$ and ${H_-}$ described as follows. The spaces ${H_\pm}$ are spanned by vectors ${e,f}$, namely ${H_{\pm}({\mathbb Z})=\langle e_\pm,f_\pm\rangle}$. We also have ${H^{1,0}_{\pm}\subset H_{\pm}({\mathbb C})}$ given by

$\displaystyle H^{1,0}_{\pm}=e_\pm + \tau_\pm f_\pm$

where ${\tau_\pm\in{\mathbb C}}$ have positive imaginary part. We can now let ${H=H_+\oplus H_-}$ and have ${\alpha\in {\mathbb Q}[\alpha]/(\alpha^2-D)}$ act in this decomposition by ${\rho(\alpha):=\iota_+(\alpha)\oplus \iota_-(\alpha)}$, i.e. as ${\sqrt{D}\oplus(-\sqrt{D})}$. Let us now describe a ${{\mathbb Z}}$-structure on ${H=H_+\oplus H_-}$ for which the action of ${\alpha}$ is by integral matrices. Take the basis defined by

$\displaystyle \begin{array}{rcl} a_1 =& e_+ +\sqrt{D} e_-\\ a_2 =& (e_+-\sqrt{D}e_-)\sqrt{D}\\ b_1 =&f_+ +\sqrt{D} f_-\\ b_2 =& (f_+-\sqrt{D}f_-)\sqrt{D} \end{array}$

Then we have

$\displaystyle \begin{array}{rcl} \rho(\alpha)a_1 = a_2 & \rho(\alpha) a_2 = Da_1\\ \rho(\alpha)b_1 = b_2 & \rho(\alpha) b_2 = Db_1 \end{array}$

Note that we can also put a symplectic structure such that the action ${\rho}$ respects it.

The above example generalizes to any totally real number field. The theorem then is that on an affine invariant manifold, a similar decomposition occurs for some factor of the cohomology of the Riemann surfaces. Moreover, the ${1}$-form ${\omega}$ giving the flat structure has to be in one of these “eigenspaces”.

The proof has two ingredients. First, a theorem of Alex Wright proved in this paper implies that at the level of local systems, a desired decomposition is available. Second, using the semisimplicity one finds that this decomposition must respect the Hodge structures. This gives the claim about real multiplication.