Conjectured relation between Lyapunov exponents and parabolic degrees for hypergeometric variations of Hodge structure

Establish whether, for the fourteen hypergeometric variations of Hodge structure of type (1,1,1,1) on the projective line classified by Doran and Morgan, the Lyapunov exponents of the associated flat bundles are closely related to the parabolic degrees of the Hodge bundles E^{3,0} and E^{2,1}, under the specific degree relation proposed in Eskin–Kontsevich–Möller–Zorich’s Conjecture 6.4.

Background

Section 5 discusses hypergeometric variations of Hodge structure of type (1,1,1,1) on P1 and their pullbacks, completing computations of Hodge numbers originally studied by del Angel–Müller-Stach–van Straten–Zuo and Hollborn–Müller-Stach. The authors note recent work by Eskin, Kontsevich, Möller, and Zorich (EKMZ) connecting Lyapunov exponents of flat bundles on curves to parabolic degrees of Hodge bundles.

In this context, the paper cites a conjecture from EKMZ that proposes a direct relationship between Lyapunov exponents and parabolic degrees for the specific fourteen hypergeometric cases classified by Doran and Morgan, contingent on a certain relation between degrees. Confirming this conjecture would link dynamical invariants to Hodge-theoretic data computed in the paper.

References

In it is conjectured that for the $14$ hypergeometric variations of Hodge structure of Doran and Morgan , the associated Lyapunov exponents are closely related to the parabolic degrees of the bundles $\mathscr{E}{3,0}$ and $\mathscr{E}{2,1}$, provided that these degrees satisfy a certain relation (see Conjecture 6.4 and for the higher rank case).

— Hodge Numbers from Picard-Fuchs Equations  (1612.09439 - Doran et al., 2016) in Remark after Table 1, Section 5 (Families of Calabi–Yau threefolds)