On the Albanese variety of the moduli space of polarized K3 surfaces (Q580463)

It follows from the global Torelli theorem that the moduli space of polarized K3 surfaces of degree 2k has the form \(D_ k/\Gamma _ k\), where \(D_ k\) is a disjoint union of two copies of a 19-dimensional bounded symmetric domain of \(type\quad IV\) and \(\Gamma _ k\) is an arithmetic subgroup acting properly discontinuously on \(D_ k\). We prove that if \([\Gamma _ k,\Gamma _ k]\) denotes the commutator subgroup of \(\Gamma _ k\), then \(\Gamma _ k/[\Gamma _ k,\Gamma _ k]\) is finite. As corollaries, the Albanese variety of \(D_ k/\Gamma _ k\) is trivial and \(H_ 1(D_ 1/\Gamma _ k,{\mathbb{Q}})=0\). In case of curves, Mumford gave a similar result by using the theory of Teichmüller modular groups. In our case, by means of the theory of symmetric bilinear forms, we can give a generator of \(\Gamma _ k\), and discuss its conjugate classes. This method is also applicable to the moduli space of polarized abelian surfaces.