# Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

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## 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.

## Notes

### Acknowledgments

I am very grateful to my advisor Alex Eskin for suggesting this circle of problems, as well as numerous encouragements and suggestions throughout the work. His advice and help were invaluable at all stages. I have also benefited a lot from conversations with Madhav Nori and Anton Zorich. Giovanni Forni, Julien Grivaux, Pascal Hubert, and Barak Weiss provided useful feedback on the exposition. It was suggested to me by Alex Wright that the methods of this paper could yield the results about real multiplication in Sect. 8.3. I am very grateful to him for that. I am also very grateful to the referee for a thorough reading and detailed feedback, which significantly improved the readability of the paper.

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