Zuo, Kang; Viehweg, Eckart:

Complex multiplication, Griffiths-Yukawa couplings, and rigidity for families of hypersurfaces

In: Journal of Algebraic Geometry, Jg. 14 (2005), S. 481 - 528
ISSN: 1056-3911
Zeitschriftenaufsatz / Fach: Mathematik
Let Md,n be the moduli stack of hypersurfaces X ⊂ Pn of degree d ≥
n + 1, and let M(1)
d,n be the sub-stack, parameterizing hypersurfaces
obtained as a d-fold cyclic covering of Pn−1 ramified over a hypersurface
of degree d. Iterating this construction, one obtains M(ν)
We show that M(1)
d,n is rigid in Md,n, although for d < 2n the
Griffiths-Yukawa coupling degenerates. However, for all d ≥ n + 1 the
sub-stack M(2)
d,n deforms.
We calculate the exact length of the Griffiths-Yukawa coupling over
d,n, and we construct a 4-dimensional family of quintic hypersurfaces
g :Z →T in P4, and a dense set of points t in T, such that g−1(t) has
complex multiplication.