Posted by: matheuscmss | December 5, 2011

## Lyapunov spectrum of equivariant subbundles of Hodge bundle

Last Friday G. Forni, A. Zorich and myself uploaded to ArXiv the article Lyapunov spectrum of invariant subbundles of Hodge bundle. In this paper (partly announced here), we study the behavior of Kontsevich-Zorich cocycle restricted to (Teichmuller flow and/or $SL(2,\mathbb{R})$) equivariant subbundles of the Hodge bundle.

Firstly, let me say that we mostly consider this article as a survey. Indeed, a large portion of this article is dedicated to revisit some variational formulas in G. Forni’s paper using the perspective of Differential Geometry: more precisely, we re-interpret Forni’s variational formulas for the growth of Hodge norms of vectors and isotropic subspaces of the Hodge bundle in terms of features of the second fundamental form (a.k.a. Kodaira-Spencer map) and the curvature form of the Gauss-Manin connection on the Hodge bundle.

Then, we use this point of view to detect the minimal set of assumptions on a subbundle $V$ of the (real) Hodge bundle under which a “Kontsevich-Zorich-Forni like formula” for the sum of Lyapunov exponents holds: whenever $V$ is a $SL(2,\mathbb{R})$-equivariant and Hodge-star invariant, the sum of the non-negative Lyapunov exponents related to $V$ is described by the average of the eigenvalues of a certain “curvature form” (see Corollary 3.5).

Furthermore, also by using this point of view, we deduce a new result (Theorem 3) relating the neutral Oseledets bundle $E^0$ of the Kontsevich-Zorich cocycle (i.e., the Oseledets subspace associated to the zero Lyapunov exponents) and the annihilator $Ann(B^{\mathbb{R}})$ of the second fundamental form $B^{\mathbb{R}}$ of the Gauss-Manin connection. In particular, we show that, if $Ann(B^{\mathbb{R}})$ is Teichmuller-flow invariant then $Ann(B^{\mathbb{R}})\subset E^0$, and, if $E^0$ is $SO(2,\mathbb{R})$-invariant, then $E^0\subset Ann(B^{\mathbb{R}})$, and hence $E^0$ and $Ann(B^{\mathbb{R}})$ coincide whenever they are $SL(2,\mathbb{R})$-invariant, i.e., the annihilator of the second fundamental form is a natural candidate for the neutral Oseledets bundles of the Kontsevich-Zorich cocycle (at least under the appropriate invariance assumptions). An interesting corollary of this (and the fact that the infinitesimal variation of the Hodge norm $\|.\|$ along the Kontsevich-Zorich cocycle is measured by the second funamental form, i.e., $\frac{d}{dt}\|c\| = -2 Re B^{\mathbb{R}}(c,c)$, see Lemma 2.4) is the fact that the Kontsevich-Zorich cocycle acts by isometries (with respect to the Hodge norm) along $E^0$ whenever this subbundle is $SL(2,\mathbb{R})$-invariant.

Finally, we “test” this new result against two classes of examples (presented in Appendix A and B). In the first class of examples, namely, square-tiled cyclic covers, we verify that both $Ann(B^{\mathbb{R}})$ and $E^0$ are $SL(2,\mathbb{R})$-invariant. As a consequence, we derive that $E^0 = Ann(B^{\mathbb{R}})$ (see Theorem 7). In other words, the neutral Oseledets bundle $E^0$ (responsible for the zero exponents of the Kontsevich-Zorich cocycle) has a natural geometric explanation: it is the annihilator $Ann(B^{\mathbb{R}})$ of the second fundamental form.  Some nice consequences of this fact are that the Kontsevich-Zorich cocycle acts by isometries along $E^0$, and $E^0$ is continuous (and actually real-analytic) in the case of square-tiled cyclic covers: indeed, this is so because the second fundamental form $B^{\mathbb{R}}$ is a continuous (actually, real-analytic) object. However, as we announce  in Appendix B (leaving the details for a forthcoming paper), the neutral Oseledets bundle $E^0$ doesn’t coincide with $Ann(B^{\mathbb{R}})$ in general! In fact, based on some constructions of C. McMullen, we exhibit an example where $E^0$ is not $SO(2,\mathbb{R})$-invariant (and hence $E^0 \neq Ann(B^{\mathbb{R}})$) despite the fact that $E^0$ and $Ann(B^{\mathbb{R}})$ have the same rank! In any event, even though the annihilator of the second fundamental form is not responsible for the zero exponents in this example, the mechanism for the existence of $E^0$ is not very complicated: essentially, we are dealing with a cocycle of matrices preserving an indefinite non-degenerate Hermitian form (i.e., we have a cocycle of $U(p,q)$ matrices where $(p,q)$ is the signature of the invariant Hermitian form), and a simple (linear-algebra) argument allows to prove the existence of zero exponents in this context. Also, let me point out that in this example the Kontsevich-Zorich cocycle also acts by isometries, but the conceptual reason behind this is different from the square-tiled cyclic covers case! Of course, we will come back to this issue later in this blog (most likely when the promised forthcoming paper comes out).

Closing this post, let me make two points. The first one is that, besides the “applications” given in the Appendices to this survey article, we feel that this point of view (of using the second fundamental form to understand the Kontsevich-Zorich cocycle) might be helpful in other contexts (and that’s what motivated us to write down this survey). For instance, Alex Eskin communicated to us that the discussion in our survey is useful when trying to derive certain semisimplicity statements of the “algebraic hulls” (in the sense of Zimmer) of the Kontsevich-Zorich cocycle (needed in his work with Maryam Mirzakhani on classification of $SL(2,\mathbb{R})$-invariant measures in moduli spaces). The second point is that, while in the case of square-tiled cyclic covers we showed that the neutral Oseledets bundle $E^0$ is continuous by comparison with the annihilator of the second fundamental form, the same kind of reasoning can’t be applied to the second class of examples (in Appendix B of our article), and so it is natural to ask about the regularity of $E^0$ in this situation. Here, in a work (in progress) by A. Avila, J.-C. Yoccoz and myself, we are able to show (among other things) that $E^0$ is not continuous at all, so that it is only measurable at best. In particular, this gives an example of a symplectic cocycle whose neutral Oseledets bundle is not continuous (in contrast with “most” examples in the literature where zero Lyapunov exponents “usually” are associated to continuous subbundles).  Evidently, I also plan to come more on this work in progress in due time, but for now I think that’s all I have to say on zero exponents of the Kontsevich-Zorich cocycle!