Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy

2021/03/03 by Paul Apisa, Apisa, Paul
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2103.02133

openalex publication_date 2021/03/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani and Wright. As a corollary we show that outside of an explicit list of exceptions, if M is a GL(2,R)-invariant subvariety, then the Kontsevich-Zorich cocycle has nonzero Lyapunov exponents in the symplectic orthogonal of the projection of the tangent bundle of M to absolute cohomology.

Citations

• Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle
• Invariant and stationary measures for the SL(2,R) action on Moduli space
• Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture
• Reconstructing orbit closures from their boundaries

Related

• The algebraic hull of the Kontsevich-Zorich cocycle
• Subvarieties of abelian varieties
• Affine Invariant Submanifolds with Completely Degenerate\n Kontsevich-Zorich Spectrum
• Holomorphic symmetric differentials and a birational characterization of Abelian Varieties
• Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular\n Embeddings