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.
Best Answer
The list of references is way too long. Here are some classical texts containing both the setup and calculations:
1) P. Deligne, Le déterminant de la cohomologie, Current Trends in Arithmetical Algebraic Geometry, Contemp. Math., no. 67, AMS, Providence, 1987.
2) Gerd Faltings, Ching-Li Chai, Degenerations of abelian varieties, Springer-Verlag, 1990.
3) L. Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985).
And here are a couple of more recent papers that deal with the self-intersection and cohomology calculations:
4) Alexis Kouvidakis, Theta line bundles and the determinant of the Hodge bundle, arXiv:alg-geom/9604017.
5) Alexander Polishchuk, Determinant bundles for abelian schemes, arXiv:alg-geom/9703021.