Logarithmic abelian varieties, Part I : Complex analytic theory

2We introduce the notions log complex torus and log abelian variety over $\bC$, which are new formulations of degenerations of complex torus and abelian variety over $\bC$, and which have group structures. We compare them with the theory of log Hodge structures. A main result is that the category of the log complex tori (resp.\ log abelian varieties) is equivalent to that of the log Hodge structures (resp.\ fiberwise-polarizable log Hodge structures) of type $(-1,0)+(0,-1)$. The toroidal compactifications of the Siegel spaces are the fine moduli of polarized log abelian varieties with level structure and with the fixed type of local monodromy with respect to the corresponding cone decomposition. In virtue of the fact that log abelian varieties have group structures, we can also show this with a fixed coefficient (rigidified) ring of endomorphisms. The Satake-Baily-Borel compactifications are, in a sense, the coarse moduli. Classical theories of semi-stable degenerations of abelian varieties over $\bC$ can be regarded in our theory as theories of proper models of log abelian varieties.007-11-0