Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata
Abstract
We study the Kontsevich--Zorich cocycle, its fixed tensor constructions, and their finite direct sums over products of strata of Abelian differentials. The acting groups are closed subgroups G⊂(SL2ℝ)n that, up to factor permutations and coordinatewise conjugation by rotations, are products of diagonal SL2ℝ blocks and a finite central group. For ergodic invariant probability measures, we prove Deligne-type semisimplicity: the bundles admit Hodge-compatible decompositions into G-invariant pieces Vi⊗AiWi, where Ai are division algebras, Vi carry pure variations of Hodge structure, and the multiplicity spaces Wi carry finite direct sums of pure Hodge structures. Every invariant subbundle arises from Ai-submodules of the Wi. If block marginal measures have smooth local supports and are locally equivalent to smooth volume, Hodge-orthogonal projections onto invariant subbundles are locally polynomial in period coordinates on the normalized support. The proof obtains a local product-of-quadrics model and joint polynomiality from polynomiality along stable and unstable leaves. On the cone obtained by scaling each diagonal block together, projection coefficients admit homogeneous degree-zero expressions with integral powers of reciprocal block areas. For the full product group (SL2ℝ)n, single-stratum measure classification supplies the support hypothesis, yielding rigidity for every ergodic invariant probability measure. Under the transported support hypothesis, arbitrary coordinatewise SL2ℝ-conjugates also admit polynomial equivariant projectors, though not necessarily Hodge-orthogonal. Genus-two examples show that nonrotational conjugation can destroy Deligne-torus invariance, and independent factor scalings can require square roots of area functions.
Related
- The Faraday Form of Conformal Products: Weyl Curvature and Four-Dimensional Einstein Rigidity
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- Rationality of Kontsevich moduli of stable maps, and their Galois descents
- Shintani Cocycle Decomposition of Topological Polylogarithms for Totally Real Fields
- Frobenius forms and semisimplicity of cyclotomic quiver Hecke superalgebras
BibTeX
@misc{baron2026semisimplicity,
title = {{Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata}},
author = {Polina Baron},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21247},
eprint = {2610.21247},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21247},
}
RIS
TY - UNPB
TI - Semisimplicity and Rigidity of the Kontsevich--Zorich Cocycle for Products of Strata
AU - Polina Baron
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21247
UR - https://arxiv.org/abs/2610.21247
AB - We study the Kontsevich--Zorich cocycle, its fixed tensor constructions, and their finite direct sums over products of strata of Abelian differentials. The acting groups are closed subgroups G⊂(SL_2ℝ)^n that, up to factor permutations and coordinatewise conjugation by rotations, are products of diagonal SL_2ℝ blocks and a finite central group. For ergodic invariant probability measures, we prove Deligne-type semisimplicity: the bundles admit Hodge-compatible decompositions into G-invariant pieces V_i⊗_A_iW_i, where A_i are division algebras, V_i carry pure variations of Hodge structure, and the multiplicity spaces W_i carry finite direct sums of pure Hodge structures. Every invariant subbundle arises from A_i-submodules of the W_i. If block marginal measures have smooth local supports and are locally equivalent to smooth volume, Hodge-orthogonal projections onto invariant subbundles are locally polynomial in period coordinates on the normalized support. The proof obtains a local product-of-quadrics model and joint polynomiality from polynomiality along stable and unstable leaves. On the cone obtained by scaling each diagonal block together, projection coefficients admit homogeneous degree-zero expressions with integral powers of reciprocal block areas. For the full product group (SL_2ℝ)^n, single-stratum measure classification supplies the support hypothesis, yielding rigidity for every ergodic invariant probability measure. Under the transported support hypothesis, arbitrary coordinatewise SL_2ℝ-conjugates also admit polynomial equivariant projectors, though not necessarily Hodge-orthogonal. Genus-two examples show that nonrotational conjugation can destroy Deligne-torus invariance, and independent factor scalings can require square roots of area functions.
ER -