Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

Abstract

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially on affine manifolds. All results apply to tensor powers of the cocycle and this implies that the measurable and real-analytic algebraic hulls coincide. We also prove that affine manifolds typically parametrize Jacobians with non-trivial endomorphisms. If the field of affine definition is larger than $\mathbb {Q}$, then a factor has real multiplication. The tools involve curvature properties of the Hodge bundles and estimates from random walks. In the appendix, we explain how methods from ergodic theory imply some of the global consequences of Schmid’s work on variations of Hodge structures. We also derive the Kontsevich-Forni formula using differential geometry.

Appendices

Appendix A: Connection to Schmid’s work

In this appendix, we explain the connection of our methods to Schmid’s work [34]. Namely, we show that methods from ergodic theory yield some of the global consequences of his results.

Setup. Consider a variation of Hodge structures E over a smooth quasi-projective base B. We do not assume B is compact. Since it is quasi-projective, through every point $b\in B$ there is at least one Riemann surface of finite type contained in B. In this setup, Theorems 7.22 through 7.25 from [34] hold. The main one, which implies the rest, is the Theorem of the Fixed Part—Theorem 7.22 in loc. cit. We explain how to prove it using ergodic theory.

Theorem A.1

(Theorem of the Fixed Part) With notation as above, if $\phi$ is a flat global section over B of $E_\mathbb {C}$, then each (p, q)-component of $\phi$ is also flat.

The case of curves. We first assume that B is one-dimensional, i.e. a compact Riemann surface with finitely many punctures. Recall that the universal cover of B maps to the classifying space of the variation. The appropriate version of the Schwartz lemma implies that if E is non-trivial, then B is necessarily a hyperbolic Riemann surface, of finite area by assumption. Moreover, the classifying map is a contraction for the appropriate metrics.

We can now consider the geodesic flow on B for the hyperbolic metric and the induced cocycle from the local system of E. Because the classifying map is a contraction, the boundedness assumption in the Oseledets theorem is satisfied. Moreover, we have not only the geodesic flow but also the full ${\text {SL}}_2\mathbb {R}$-action. The claims from Sects. 5 and 6 therefore apply verbatim.

This proves the claim in the case when the base is a finite-type Riemann surface.

The general case. To prove the general case, note we assumed the base is quasi-projective. In particular, it has lots of 1-dimensional subvarieties, to which the previous step applies. The main step in proving the theorem of the fixed part is showing that the subharmonic function coming from the norm of the section must be constant.

In the previous step, we established this when the base is 1-dimensional. Since B has such 1-dimensional subvarieties through every point, the subharmonic function is constant on all of them, so on all of B. The proof then proceeds as in Sect. 6, or [34, Section 7].

Appendix B: The Kontsevich-Forni formula

In this appendix, we provide a derivation of the Kontsevich-Forni formula. This was first stated by Kontsevich in [23], then proved by Forni in [19] (see also [17]). This appendix contains a proof in the formalism used in this paper.

Setup. Consider some complex manifold B of unspecified dimension, and consider over it a variation of weight-1 Hodge structure $H^1$. We have the decomposition $H^1_\mathbb {C}=H^{1,0}\oplus H^{0,1}$. Inside we have the real (flat) subbundle $H^1_\mathbb {R}$ with elements of the form $\overline{\alpha }\oplus \alpha$. We have the positive-definite Hodge norm, and all statements below are with respect to it. In a change of convention from Sect. 4, we take adjoints for the positive-definite metric now.

Proposition B.1

Suppose $c_1,\ldots ,c_k\in H^1_\mathbb {R}$ is a basis, at some point of B, of an isotropic subspace of $H^1_\mathbb {R}$. Extend $c_i$ using the Gauss–Manin connection to flat sections in a neighborhood. Denoting the second fundamental form of the Hodge bundle

$$\begin{aligned} \sigma :H^{1,0}\rightarrow \Omega ^1\otimes H^{0,1} \end{aligned}$$

we have the formula (notation explained below)

$$\begin{aligned} \partial \overline{\partial }\log \left\| \bigwedge _{i=1}^k c_i\right\| ^2={\text {tr}}(\sigma \wedge \sigma ^\dag )-{\text {tr}}\left( \sigma \wedge \pi _{\overline{\mathcal {C}}^\perp }\sigma ^\dag \pi _{\mathcal {C}^\perp }\right) \end{aligned}$$

(8.4)

We view $c_i$ as flat sections of $H^1_\mathbb {C}$ and project them to $H^{0,1}$ to get holomorphic sections $\phi _i$. Then the $\phi _i$ span a k-dimensional subbundle which we denote $\mathcal {C}$, and $\mathcal {C}^\perp$ is its Hodge-orthogonal inside $H^{0,1}$. The operator $\pi _{\mathcal {C}^\perp }$ is orthogonal projection to the space $\mathcal {C}^\perp$ and $\pi _{\overline{\mathcal {C}}^\perp }$ is orthogonal projection to its complex-conjugate.

Remark B.2

• (i) The Eq. 8.4 is an equality of (1, 1)-forms. The left-hand side can be interpreted as a Laplacian once a metric is introduced on B. For example, the hyperbolic metric on Teichmüller disks recovers the usual Kontsevich-Forni formula.

• (ii) The right-hand side of the formula is always a non-negative (1, 1)-form. This is because adjoints are for a positive-definite hermitian inner-product.

Notation. Write the $c_i$ in their Hodge decomposition

$$\begin{aligned} c_i=\overline{\phi _i}\oplus \phi _i \text { where } \phi _i \text { holomorphic section of } H^{0,1}=H^1_\mathbb {C}/H^{1,0} \end{aligned}$$

Proposition B.3

The isotropy condition on $c_i$ gives the pointwise on B equality of Hodge norms

$$\begin{aligned} \left\| \bigwedge _{1}^k c_i\right\| =2^k\left\| \bigwedge _{1}^k \phi _i\right\| \end{aligned}$$

Proof

If we apply a fixed real $k\times k$ matrix to the $c_i$ everywhere on B, then the claimed equality is not affected - both sides are rescaled by the determinant of the matrix. To check the equality at some given point of B, we can choose a real linear change of variables for the $c_i$ such that at the considered point, the $c_i$ are also Hodge-orthogonal.

Combined with the isotropy condition on $c_i$ we find that the $\phi _i$ must also be Hodge-orthogonal. Indeed, the $c_i$ being Hodge-orthogonal implies the real part of $\left\langle \phi _i,\phi _j\right\rangle$ has to vanish. The isotropy condition implies vanishing of the imaginary part.

But in this situation, the formula can be checked directly. Therefore, the asserted equality holds everywhere. $\square$

Proof of Proposition B.1

By the previous result, we need to compute $\overline{\partial }\partial \log \left\| \wedge \phi _i\right\| ^2$. Recall $\bigwedge _{i=1}^k \phi _i$ is a holomorhpic section of $\bigwedge ^k H^{0,1}$ and so we shall use Lemma 4.1 to compute the desired expression.

Recall from Sect. 4 the relation between the Gauss–Manin and Hodge connections on $H^1$

$$\begin{aligned} \nabla ^{GM}=\nabla ^{Hg}+\sigma -\sigma ^\dag \end{aligned}$$

Because the $c_i$ are flat for $\nabla ^{GM}$, looking at the component in $H^{0,1}$ we find

$$\begin{aligned} \nabla ^{Hg}\phi _i = -\sigma \overline{\phi _i} \end{aligned}$$

From now on, $\nabla$ denotes $\nabla ^{Hg}$ and we focus on the bundle $H^{0,1}$. We shall use the Leibniz rule for the connection and curvature

$$\begin{aligned} \nabla \bigwedge _{i=1}^k \phi _i&= \sum _{i=1}^k \phi _1\wedge \cdots \wedge \nabla \phi _i \wedge \cdots \wedge \phi _k\\ \Omega _{\wedge ^k H^{0,1}} \bigwedge _{i=1}^k \phi _i&= \sum _{i=1}^k \phi _1\wedge \cdots \wedge \Omega \phi _i \wedge \cdots \wedge \phi _k \end{aligned}$$

We abuse notation and denote by $\nabla$ the connection on both $H^{0,1}$ and its wedge powers, but we distinguish the curvatures. Denoting by $\phi :=\phi _1\wedge \cdots \wedge \phi _k$ Lemma 4.1 reduces the proof to evaluating

$$\begin{aligned} -\frac{\left\langle \Omega _{\wedge ^k} \phi ,\phi \right\rangle }{\left\| \phi \right\| ^2} - \frac{\left\langle \nabla \phi ,\phi \right\rangle \cdot \left\langle \phi ,\nabla \phi \right\rangle -\left\| \phi \right\| ^2\cdot \left\langle \nabla \phi ,\nabla \phi \right\rangle }{\left\| \phi \right\| ^4} \end{aligned}$$

(8.5)

We need to check the pointwise equality of the above (1, 1)-form and the right-hand side of Proposition B.1. The minus sign comes from the switched order of $\partial$ and $\overline{\partial }$ in Lemma 4.1 and the proposition we are proving.

Remark that we are proving a pointwise equality. In particular, if we apply any fixed $k\times k$ complex matrix to the sections $\phi _i$ the value given by 8.5 does not change. Thus, to prove the claimed equality at a point of B we can apply a matrix to assume that the $\phi _i$ are mutually orthogonal and of unit norm at the considered point. For the calculation, we also complete them to an orthonormal basis $\{\phi _i\}_{i=1}^g$ of the fiber considered.

Finally, Eq. (4.13) gives $\Omega _{H^{0,1}}=-\sigma \wedge \sigma ^\dag$ (recall we are taking adjoints for the positive-definite metric now, hence the minus sign). Denote the entries of 1-form valued maps $\sigma$ and $\sigma ^\dag$ by

$$\begin{aligned} \sigma \overline{\phi _i}&= \sum _{j=1}^g \sigma _i^j \phi _j\\ \sigma ^\dag \phi _k&= \sum _{l=1}^g (\sigma ^\dag )_{k}^l \overline{\phi }_l \end{aligned}$$

We then have, using orthonormality of $\{\phi _i\}_{i=1}^g$

$$\begin{aligned} -\left\langle \Omega _{\wedge ^k} \phi ,\phi \right\rangle&= \left\langle \sum _{i=1}^k \phi _1\wedge \cdots \wedge (\sigma \sigma ^\dag \phi _i)\wedge \cdots \wedge \phi _k , \phi _1 \wedge \cdots \wedge \phi _k\right\rangle \\&=\sum _{i=1}^k\sum _{j=1}^g \sigma ^i_j\wedge \left( \sigma ^\dag \right) _i^j \end{aligned}$$

We next have

$$\begin{aligned} \left\langle \nabla \phi ,\phi \right\rangle&=\left\langle \sum _{i=1}^k \phi _1\wedge \cdots \wedge (-\sigma \overline{\phi }_i)\wedge \cdots \wedge \phi _k,\phi _1\wedge \cdots \wedge \phi _k\right\rangle \\&=-\sum _{i=1}^k \sigma _i^i \end{aligned}$$

We then find

$$\begin{aligned} \left\langle \nabla \phi ,\phi \right\rangle \cdot \left\langle \phi ,\nabla \phi \right\rangle = \left( \sum _{i=1}^k \sigma _i^i\right) \cdot \left( \sum _{i=1}^k \overline{\sigma }_i^i\right) \end{aligned}$$

We also have

$$\begin{aligned} \left\langle \nabla \phi ,\nabla \phi \right\rangle&= \left\langle \sum _{i=1}^k \phi _1\wedge \cdots \sigma ^i_i \phi _i\cdots \wedge \phi _k + \sum _{k<l}\phi _1\wedge \cdots \sigma _i^l \phi _l\cdots \wedge \phi _k,\right. \nonumber \\&\qquad \left. \sum _{i=1}^k \phi _1\wedge \cdots \sigma ^i_i \phi _i\cdots \wedge \phi _k + \sum _{k<l}\phi _1\wedge \cdots \sigma _i^l \phi _l\cdots \wedge \phi _k\right\rangle \nonumber \\&=\left( \sum _{i=1}^k \sigma _i^i\right) \left( \sum _{i=1}^k \overline{\sigma }_i^i\right) + \sum _{i=1}^k \sum _{l=k+1}^g \sigma _i^l \overline{\sigma }_i^l \end{aligned}$$

(8.6)

Note that $\left\| \phi \right\| =1$ by our normalization and we also have that $\left( \sigma ^\dag \right) ^j_i=\overline{\sigma }^i_j$. We can now combine all three terms to get the claimed formula. The first term in Eq. (8.6) cancels the $\left\langle \nabla \phi ,\phi \right\rangle$ term. The second term of the same equation provides the needed contribution to the desired formula. Indeed, summing up all the terms we obtain

$$\begin{aligned} \sum _{i=1}^g \sum _{l=1}^g \sigma _i^l \overline{\sigma }_i^l - \sum _{i=k+1}^g \sum _{l=k+1}^g \sigma _i^l \overline{\sigma }_i^l \end{aligned}$$

and this corresponds to the right-hand side of (8.4) written out explicitly. $\square$