Is there a good reference for the structure of the Chow ring of $\mathcal{A}\_g$, the moduli space of complex principally polarized abelian varieties? More generally, references for the intersection theory, enumerative geometry, and vector bundles on $\mathcal{A}\_g$ would be nice.



Chow Ring of Moduli Space of Abelian Varieties
Van der Geer has written a paper computing what he calls the tautological subring of the chow ring of $\mathcal{A}\_g$. He also computes the tautological ring for a smooth toroidal compactification.


G. van der Geer, Cycles on the Moduli Space of Abelian Varieties, in "Moduli of Curves and Abelian Varieties (The Dutch Intercity Seminar on Moduli)", p. 65-89 (Carel Faber and Eduard Looijenga, editors), Aspects of Mathematics, Vieweg, Wiesbaden 1999.


It is available on the van der Geer's website [here](http://www.science.uva.nl/~geer/publications.html)


Regarding intersection theory, Erdenberger, Grushevsky, and Hulek have been working on this for the toroidal compactifications, mostly for small values of $g$. For example, see the following references.


C. Erdenberger, S. Grushevsky, K. Hulek, Intersection theory of toroidal compactifications of $\mathcal{A}\_4$. Bull. London Math. Soc. 38 (2006), no. 3, 396--400.


C. Erdenberger, S. Grushevsky, K. Hulek, Some intersection numbers of divisors on toroidal compactifications of $\mathcal{A}\_g$. J. Algebraic Geom. 19 (2010), no. 1, 99--132.


S. Grushevsky, Geometry of $\mathcal{A}\_g$ and its compactifications. Algebraic geometry---Seattle 2005. Part 1, 193--234, Proc. Sympos. Pure Math., 80, Part 1, Amer. Math. Soc., Providence, RI, 2009. 


