enumerative geometry - 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

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.

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