Lyapunov spectrum of invariant subbundles of the Hodge bundle

We study the Lyapunov spectrum of the Kontsevich--Zorich cocycle on $SL(2,\mathbb{R})$-invariant subbundles of the Hodge bundle over the support of a $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of Abelian differentials. In particular, we prove formulas for partial sums of Lyapunov exponents in terms of the second fundamental form (or Kodaira--Spencer map) of the Hodge bundle with respect to Gauss--Manin connection and investigate the relations between the central {Oseldets} subbundle and the kernel of the second fundamental form. We illustrate our conclusions in two special cases.