Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

• 成果类型: Article
• 署名作者: Filip, Simion
• 署名单位: University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0643-3
• 发表日期: 2016
• 页码: 617-670
• 关键词: interval exchange transformations; veech surfaces; moduli space; teichmuller; DYNAMICS; FLOWS

摘要

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 , 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.