English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respectthe Hodge structure. In particular, we establish a version of Delignesemisimplicity in this context. This implies that invariant subbundles mustvary polynomially on affine manifolds. All results apply to tensor powers ofthe cocycle and this implies that the measurable and real-analytic algebraichulls coincide.  We also prove that affine manifolds parametrize Jacobians with non-trivialendomorphisms. Typically a factor has real multiplication.  The tools involve curvature properties of the Hodge bundles and estimatesfrom random walks. In the appendix, we explain how methods from ergodic theoryimply some of the global consequences of Schmid's work on variations of Hodgestructures. We also derive the Kontsevich-Forni formula using differentialgeometry.

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle