A criterion for being a Teichmüller curve
Résumé
Given a curve in the moduli space of Riemann surfaces, we want to know whether it is a Teichmüller curve. By Deligne semisimplicity theorem the Hodge bundle over the curve decomposes into a direct sum of flat subbundles admitting variations of complex polarized Hodge structures of weight 1. Suppose that the restriction of the canonical pseudo-Hermitian form to one of the blocks of the decomposition has rank $(1,r-1)$. We establish an upper bound for the degree of the corresponding holomorphic line bundle in terms of the (orbifold) Euler characteristic of the curve. Our criterion claims that if the upper bound is attained, the curve is a Teichmüller curve.