Recently Eskin-Kontsevich-Moller-Zorich prove my conjecture that the sum of the top k Lyapunov exponents is always greater or equal to the degree of any rank k holomorphic subbundle(They generalize the original context from Teichm(u)ller curves to any local system over a curve with non-expanding cusp monodromies).Furthermore,they conjecture that equality of the sum of Lyapunov exponents and the degree is related to the monodromy group being a thin subgroup of its Zariski closure.I will introduce some backgrounds on those conjectures and some applications to Teichm(u)ller dynamics and Calabi-Yau type families.