For the purposes of this question, let the Hodge bundle $\lambda$ be the bundle on a fibration of abelian varieties $X\to B$ with fiber over $b\in B$ the space of 1-forms on $X_b$, or the pullback to $B$ along the zero section of the sheaf of relative differentials. The most interesting examples are when $B$ is $M_g$ or $A_g$, and the fibrations are the Jacobian fibration or the universal family of abelian varieties.

Is there a good reference for the properties of this bundle? And for the determinant line bundle on these spaces? Things like self-intersection numbers, cohomology (in particular global sections) are particularly of interest.