A higher dimensional version of generalized elliptic curves.
Abstract: I will discuss a higher dimensional version of Deligne and
Rapoport's notion of "generalized elliptic curve" and explain how one
can use it to obtain canonical (arithmetic) compactifications of
moduli spaces for abelian varieties with polarization (not
necessarily principal) and level structure. These moduli spaces are
obtained as solutions to certain moduli problems. In the case of
$A_g$ (the moduli space of principally polarized abelian varieties)
the compactification is equal to the normalization of the main
component of Alexeev's canonical compactification of $A_g$.