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

TL;DR

This paper studies the rigidity and deformation properties of certain hypersurface families in projective space, analyzing Griffiths-Yukawa couplings and complex multiplication phenomena.

Contribution

It introduces new results on the rigidity of hypersurface moduli sub-stacks and computes Griffiths-Yukawa coupling lengths for these families.

Findings

M(d,n;1) is rigid in M(d,n) despite degenerating Griffiths-Yukawa coupling for d<2n

M(d,n;2) deforms for all d>n

Constructed a 4D family of quintic hypersurfaces with dense CM points

Abstract

Let M(d,n) be the moduli stack of hypersurfaces of degree d > n in the complex projective n-space, and let M(d,n;1) be the sub-stack, parameterizing hypersurfaces obtained as a d fold cyclic covering of the projective n-1 space, ramified over a hypersurface of degree d. Iterating this construction, one obtains M(d,n;r). We show that M(d,n;1) is rigid in M(d,n), although the Griffiths-Yukawa coupling degenerates for d<2n. On the other hand, for all d>n the sub-stack M(d,n;2) deforms. We calculate the exact length of the Griffiths-Yukawa coupling over M(d,n;r), and we construct a 4-dimensional family of quintic hypersurfaces, and a dense set of points in the base, where the fibres have complex multiplication.