Polarizations Brian

If A is an abelian variety over a field, then to give a projective embedding of A is more or less to give an ample line bundle on A. Over C, such data can be expressed in terms of a positive-definite Riemann form on the homology lattice. Hence, we consider ample line bundles on general abelian varieties to be a “positivity” structure. In a sense that will be explained, just as we view abelian varieties to be a jazzed-up sort of linear algebra datum (as the analytic uniformization makes precise over C), we will interpret the data of ample line bundles as akin to the specification of positive-definite quadratic forms. This will be encoded in the structure to be called a polarization on the abelian variety. It is a fundamental fact that polarized abelian varieties have finite automorphism groups, and this makes them especially well-suited to the formulation of well-behaved moduli problems; we will say nothing more on such aspects of polarizations in these notes. In these notes we provide quite a few references to [Mum] for some further technical details, and the earlier talks (and subsequent write-ups) on analytic and algebraic properties of abelian varieties addressed most such points. We differ slightly from the notes on the analytic theory insofar as our Appel–Humbert normalizations are presented without requiring a choice of √ −1 ∈ C.