Geometric realization of zero Lyapunov exponents of the Kontsevich-Zorich cocycle beyond the Forni subspace

The Kontsevich-Zorich cocycle - the monodromy of the Gauss-Manin connection along $SL(2,\mathbb{R})$-orbits in the moduli spaces of translation surfaces - can have zero Lyapunov exponents. Avila-Eskin-Möller proved that the Forni subspace, the maximal $SL(2,\mathbb{R})$-invariant isometric subbundle of the Hodge bundle, is flat and captures every zero exponent carried by an $SL(2,\mathbb{R})$-invariant subbundle, but remarked that the zero-exponent Oseledets subspace of the cocycle can be strictly larger, as in the Forni-Matheus-Zorich arithmetic Teichmüller curve whose genus-4 real Hodge bundle admits no nontrivial invariant subbundles yet has four zero exponents. The analytic characterization has since been established: Filip proved the Forni-Matheus-Zorich conjecture that zero exponents occur exactly on flat irreducible pieces whose monodromy Zariski closure has a single non-compact factor $SU(p,q)$ (in the standard or an exterior-power representation) or $SO^*(2n)$ with $n$ odd, with the minimal multiplicity permitted by the monodromy, and Eskin-Filip-Wright computed the full algebraic hull. The open problem is the geometric side: determine which invariant subvarieties of the moduli spaces of Abelian and quadratic differentials realize the not-yet-observed monodromy types from Filip's classification. Concretely, one must realize or exclude the types that carry zero exponents beyond the Forni subspace ($SU(p,1)$ in exterior powers of degree at least 2, excluding the observed second-exterior-power $SU(3,1)$ instance; $SO^*(2n)$ for all odd $n$), together with the orthogonal groups in spin representations, which carry no zero exponents beyond the Forni subspace but belong to the same classification.

Background

A translation surface is a Riemann surface $X$ equipped with a holomorphic $1$-form $\omega$; equivalently, a finite family of plane polygons glued edge-to-edge by translations, carrying a flat metric whose only singularities are conical points at the zeros of $\omega$. The moduli space of all such pairs with a fixed list $\kappa$ of zero multiplicities is a stratum $\mathcal{H}(\kappa)$ of the moduli space of Abelian differentials. The group $SL(2,\mathbb{R})$ acts on strata by linearly deforming the polygons; the diagonal subgroup is the Teichmüller geodesic flow, which is ergodic for the natural Lebesgue-class (Masur-Veech) measure on each connected component. By the orbit closure theorem, every other ergodic invariant measure is supported on an affine invariant submanifold: a closed $GL(2,\mathbb{R})$-orbit closure that is linear in period coordinates, the coordinates given by the integrals of $\omega$ over a basis of relative homology. Teichmüller curves are closed $SL(2,\mathbb{R})$-orbits that are algebraic curves in the moduli space; the simplest examples come from square-tiled surfaces, i.e. translation surfaces that are finite covers of a flat torus ramified over one point, and a Teichmüller curve generated by a square-tiled surface is called arithmetic. Strata of quadratic differentials are studied through the orientation double cover, on which the square of the covering differential is an Abelian differential and the acting group is $PSL(2,\mathbb{R})$.

Over these moduli spaces sits the real Hodge bundle, whose fiber over $(X,\omega)$ is the cohomology $H^1(X,\mathbb{R})$, equipped with the flat Gauss-Manin connection (parallel transport of cohomology classes). The Kontsevich-Zorich cocycle is the action of this parallel transport along the $SL(2,\mathbb{R})$-orbits; its Lyapunov exponents - the exponential growth rates assigned to the cocycle by Oseledets' multiplicative ergodic theorem - govern the deviation of ergodic averages of flows along level sets of $\omega$ and diffusion rates in periodic billiard models. The spectrum is symmetric about $0$; for Abelian differentials it contains the pair $\pm 1$ coming from the tautological plane spanned by $\mathrm{Re}\,\omega$ and $\mathrm{Im}\,\omega$. Over an entire stratum of Abelian differentials with its Masur-Veech measure the remaining exponents are nonzero, and they are known to be simple. For special invariant measures, however, zero exponents can occur.

The Forni subspace $F$ is the maximal measurable $SL(2,\mathbb{R})$-invariant subbundle of the Hodge bundle on which the cocycle acts by isometries of the Hodge inner product, the positive-definite Hermitian pairing on cohomology induced by the complex structure; all of its exponents vanish, and it can be described pointwise as the intersection of the kernels of a biform measuring the infinitesimal variation of the period matrix along Teichmüller deformations. Avila-Eskin-Möller proved that $F$ is flat (locally constant for the Gauss-Manin connection), Hodge-orthogonal to the tangent space of the supporting affine invariant submanifold, and that every $SL(2,\mathbb{R})$-invariant subbundle of the Hodge-orthogonal complement $F^{\perp}$ has at least one nonzero exponent; consequently every zero exponent carried by an invariant subbundle already lies in $F$. They then observed that the zero-exponent Oseledets subspace $E^0$ of the cocycle - which is invariant under the geodesic flow but not under all of $SL(2,\mathbb{R})$ - can nevertheless be strictly larger than $F$: Forni-Matheus-Zorich constructed an arithmetic Teichmüller curve, built from cyclic triple covers of the sphere branched at six points (a square-tiled surface in the stratum $\mathcal{Q}(7,1^5)$ of holomorphic quadratic differentials in genus four), whose Lyapunov spectrum is $\{4/9, 4/9, 0, 0, 0, 0, -4/9, -4/9\}$ while its real Hodge bundle contains no nontrivial $PSL(2,\mathbb{R})$-invariant subbundle at all, so its four-dimensional neutral Oseledets subspace is not $PSL(2,\mathbb{R})$-invariant, i.e. it is not the pullback of any flat subbundle of the Hodge bundle over the Teichmüller curve.

Forni-Matheus-Zorich identified the mechanism: if the cocycle on an irreducible piece takes values in the pseudo-unitary group $U(p,q)$, the group of matrices preserving a Hermitian form of signature $(p,q)$, then at least $|p-q|$ exponents vanish. They conjectured that this mechanism, with minimal multiplicity, accounts for all zero exponents. This was proved by Filip in two steps. First, a semisimplicity theorem: the cocycle respects the Hodge structure and the Hodge bundle over an affine invariant submanifold splits into flat irreducible pieces. Second, the characterization: zero exponents on a flat real-irreducible piece occur exactly when the Zariski closure of its monodromy (the smallest algebraic group containing the monodromy matrices) has at most one non-compact factor, equal up to finite index to $SU(p,q)$ with $p>q$, the determinant-one subgroup of $U(p,q)$, acting in the standard representation (giving $2(p-q)$ zero exponents), to $SU(p,1)$ acting in the $k$-th exterior power of the standard representation with $k\ge 2$ (giving $2\binom{p-1}{k-2}+2\binom{p-1}{k}$ zero exponents), or to $SO^*(2n)$ with $n$ odd, the quaternionic real form of the orthogonal group in its standard representation (giving exactly four zero exponents); the multiplicity is always the minimum possible given the monodromy. Eskin-Filip-Wright completed the structural picture by computing the algebraic hull of the cocycle over any invariant subvariety - the smallest real algebraic group to which the cocycle is conjugate by a measurable choice of basis - as the stabilizer of the tautological plane in the Zariski closure of the monodromy.

The analytic description of zero exponents is thus complete, and the original question reduces to its geometric side: which invariant subvarieties actually realize the permitted monodromy types. Partial answers exist. All signatures $SU(p,q)$ in the standard representation are realized by explicit covering constructions, as is $SO^*(2n)$ for every $11\le n\le 299$ with $n\equiv 3\pmod{8}$, except possibly $n=35$ and $n=203$ (and conjecturally for a density-$1/8$ set of $n$). In genus three the only orbit closure with any zero exponent is the Eierlegende Wollmilchsau, a totally degenerate square-tiled surface (all Lyapunov exponents except the tautological pair $\pm 1$ vanish) whose zero exponents are all of Forni type, i.e. carried by the Forni subspace; the non-Forni phenomenon first appears in genus four.

Problem Statement

Determine the geometric origin, over $GL(2,\mathbb{R})$-invariant subvarieties of strata of Abelian differentials and, via orientation double covers, $PSL(2,\mathbb{R})$-invariant suborbifolds of strata of quadratic differentials, of the zero Lyapunov exponents of the Kontsevich-Zorich cocycle that are not carried by the Forni subspace, and which of the not-yet-observed monodromy types of Filip's classification are realized there, after the analytic characterization of those exponents by Filip. Recall the settled background: zero exponents beyond the Forni subspace occur exactly on flat real-irreducible pieces of the Hodge bundle (the pieces of the flat splitting of the Hodge bundle over an affine invariant submanifold) whose monodromy Zariski closure has a single non-compact factor, equal up to finite index to $SU(p,q)$ with $p>q$ in the standard representation (multiplicity $2(p-q)$), to $SU(p,1)$ in the $k$-th exterior power of the standard representation with $k\ge 2$ (multiplicity $2\binom{p-1}{k-2}+2\binom{p-1}{k}$), or to $SO^*(2n)$ with $n$ odd (multiplicity four), and the multiplicities are minimal given the monodromy. The open problem is the realization question: determine which of the monodromy types permitted by Filip's classification are realized by invariant subvarieties of the moduli spaces of Abelian and quadratic differentials.

All signatures $SU(p,q)$ in the standard representation are known to occur, as is $SO^*(2n)$ for every $11\le n\le 299$ with $n\equiv 3\pmod{8}$, except possibly $n=35$ and $n=203$ (conjecturally for a density-$1/8$ set of $n$). It is open whether an orthogonal group in a spin representation (a representation descending from the double cover of the orthogonal group), or $SU(p,1)$ in an exterior power of degree $k\ge 2$ (beyond the observed second-exterior-power $SU(3,1)$ instance), can occur in the algebraic hull of the Kontsevich-Zorich cocycle (Filip's Question 5.5), and whether $SO^*(2n)$ occurs for every odd $n$.

In Filip's Question 5.5 the items for an orthogonal group in a spin representation and for $SU(p,1)$ in the $k$-th exterior power of the standard representation with $k\ge 2$, beyond the observed second-exterior-power $SU(3,1)$ instance, are existential: each asks whether some invariant subvariety realizes a monodromy Zariski closure with the indicated single non-compact factor in the indicated representation, the type being realized when such a subvariety exists and excluded otherwise. The item that $SO^*(2n)$ occurs for every odd $n$ is universal and concerns all odd $n$ simultaneously.

References

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