##
**The algebraic hull of the Kontsevich-Zorich cocycle.**
*(English)*
Zbl 1398.32015

Let \(\Omega_g\) be the space of holomorphic differentials on compact genus-\(g\) Riemann surfaces. Each point \((X, \omega)\) here can be described by an atlas of charts to \(\mathbb C\) (away from the zeros of \(\omega\)) whose transition maps are translations. By integrating \(\omega\) along a basis for homology relative to its singularities, one obtains a description of \((X, \omega)\) as a collection of polygons with parallel sides identified by translation. Hence these objects are known as translation surfaces. There is a natural action on this space by postcomposition with charts, or equivalently, linear action on the defining polygons. To quote from the paper:

“When translation surfaces are described by polygons in the plane, the \(\mathrm{GL}^+(2, \mathbb R)\)-action distorts the polygons linearly. The Kontsevich-Zorich cocycle encodes the procedure of cutting and regluing the polygons to a less distorted shape, and hence carries the mysterious part of the dynamics of the \(\mathrm{GL}^+(2, \mathbb R)\)-action.”

This mysterious cocycle lives in the symplectic group \(\mathrm{Sp}(2g, \mathbb Z)\), and understanding the algebraic hull (essentially the smallest algebraic group the image of the cocycle can be conjugated into) is an important goal of Teichmüller dynamics. In fact, one can compute the algebraic hull associated to each affine \(\mathrm{GL}^+(2, \mathbb R)\)-invariant manifold. The main results of the paper state that one can compute the algebraic hull of the cocycle (even restricted to subbundles \(V\) of the natural bundle where the cocycle lives). Letting \(A_V\) denote this hull, it is shown that \(A_V\) is the stabilizer of the tautological plane in the Zariski closure of the monodromy \(G_V\) if \(V\) contains the tautological plane spanned by the real and imaginary parts of the holomorphic one-form. If \(V\) does not contain the tautological plane, the algebraic hull coincides with the Zariski closure of the monodromy. The monodromy here arises from considering the cocycle along loops.

This main result and further results about monodromy yield applications that say that these groups stabilize if we have sequences of affine invariant manifolds equidistributing in another one, and further yield finiteness results for the number of affine invariant manifolds of rank greater than \(1\) (and degree greater than \(2\)), in particular (for any fixed genus-\(g\)) there are only finitely many Teichmüller curves with trace field of degree greater than \(2\).

“When translation surfaces are described by polygons in the plane, the \(\mathrm{GL}^+(2, \mathbb R)\)-action distorts the polygons linearly. The Kontsevich-Zorich cocycle encodes the procedure of cutting and regluing the polygons to a less distorted shape, and hence carries the mysterious part of the dynamics of the \(\mathrm{GL}^+(2, \mathbb R)\)-action.”

This mysterious cocycle lives in the symplectic group \(\mathrm{Sp}(2g, \mathbb Z)\), and understanding the algebraic hull (essentially the smallest algebraic group the image of the cocycle can be conjugated into) is an important goal of Teichmüller dynamics. In fact, one can compute the algebraic hull associated to each affine \(\mathrm{GL}^+(2, \mathbb R)\)-invariant manifold. The main results of the paper state that one can compute the algebraic hull of the cocycle (even restricted to subbundles \(V\) of the natural bundle where the cocycle lives). Letting \(A_V\) denote this hull, it is shown that \(A_V\) is the stabilizer of the tautological plane in the Zariski closure of the monodromy \(G_V\) if \(V\) contains the tautological plane spanned by the real and imaginary parts of the holomorphic one-form. If \(V\) does not contain the tautological plane, the algebraic hull coincides with the Zariski closure of the monodromy. The monodromy here arises from considering the cocycle along loops.

This main result and further results about monodromy yield applications that say that these groups stabilize if we have sequences of affine invariant manifolds equidistributing in another one, and further yield finiteness results for the number of affine invariant manifolds of rank greater than \(1\) (and degree greater than \(2\)), in particular (for any fixed genus-\(g\)) there are only finitely many Teichmüller curves with trace field of degree greater than \(2\).

Reviewer: Jayadev Athreya (Seattle)

### MSC:

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

14D07 | Variation of Hodge structures (algebro-geometric aspects) |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

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\textit{A. Eskin} et al., Ann. Math. (2) 188, No. 1, 281--313 (2018; Zbl 1398.32015)

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