Lyapunov exponents of the Hodge bundle over strata of quadratic differentials with large number of poles

Abstract

We show an upper bound for the sum of positive Lyapunov exponents of any Teichm\"uller curve in strata of quadratic differentials with at least one zero of large multiplicity. As a corollary, it holds for any $SL(2,\mathbb R)$-invariant subspaces defined over $\mathbb Q$ in these strata. This proves Grivaux-Hubert's conjecture about the asymptotics of Lyapunov exponents for strata with a large number of poles in the situation when at least one zero has large multiplicity.