Simion Filip

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

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

Families Of K3 surfaces and Lyapunov exponents

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Notes on the multiplicative ergodic theorem

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.

Splitting mixed Hodge structures over affine invariant manifolds

We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich-Zorich cocycle. The number of zero exponents is then as small as possible, given its monodromy.

The algebraic hull of the Kontsevich--Zorich cocycle

Consider a family of K3 surfaces over a hyperbolic curve (i.e., Riemann surface). Their second cohomology groups form a local system, and we show that its top Lyapunov exponent is a rational number. One proof uses the Kuga–Satake construction, which reduces the question to Hodge structures of weight 1. A second proof uses integration by parts. The case of maximal Lyapunov exponent corresponds to modular families coming from the Kummer construction.

Quaternionic covers and monodromy of the Kontsevich-Zorich cocycle in orthogonal groups

For Hoelder cocycles over a Lipschitz base transformation, possibly non-invertible, we show that the subbundles given by the Oseledets Theorem are Hoelder-continuous on compact sets of measure arbitrarily close to 1. The results extend to vector bundle automorphisms, as well as to the Kontsevich-Zorich cocycle over the Teichmueller flow on the moduli space of abelian differentials. Following a recent result of Chaika-Eskin, our results also extend to any given Teichmueller disk.

The Lie algebra of the fundamental group of a surface as a symplectic module

The Oseledets Multiplicative Ergodic theorem is a basic result with numerous applications throughout dynamical systems. These notes provide an introduction to this theorem, as well as subsequent generalizations. They are based on lectures at summer schools in Brazil, France, and Russia.