Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



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

Authors: Eckart Viehweg and Kang Zuo
Journal: J. Algebraic Geom. 14 (2005), 481-528
Published electronically: February 16, 2005
MathSciNet review: 2129008
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Abstract: Let $\mathcal{M}_{d,n}$ be the moduli stack of hypersurfaces $X \subset\mathbb{P} ^n$ of degree $d\geq n+1$, and let $\mathcal M_{d,n}^{(1)}$ be the sub-stack, parameterizing hypersurfaces obtained as a $d$-fold cyclic covering of $\mathbb{P} ^{n-1}$ ramified over a hypersurface of degree $d$. Iterating this construction, one obtains $\mathcal{M}_{d,n}^{(\nu)}$.

We show that $\mathcal{M}_{d,n}^{(1)}$ is rigid in $\mathcal{M}_{d,n}$, although for $d<2n$ the Griffiths-Yukawa coupling degenerates. However, for all $d\geq n+1$ the sub-stack $\mathcal{M}^{(2)}_{d,n}$ deforms.

We calculate the exact length of the Griffiths-Yukawa coupling over $\mathcal{M}_{d,n}^{(\nu)}$, and we construct a $4$-dimensional family of quintic hypersurfaces $g:\mathcal{Z}\to T$ in $\mathbb{P} ^4$, and a dense set of points $t$ in $T$, such that $g^{-1}(t)$ has complex multiplication.

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Eckart Viehweg
Affiliation: Universität Essen, FB6 Mathematik, 45117 Essen, Germany

Kang Zuo
Affiliation: Universität Mainz, FB17 Mathematik, 55099 Mainz, Germany

Received by editor(s): October 27, 2003
Published electronically: February 16, 2005
Additional Notes: This work has been supported by the Institute of Mathematical Science at the Chinese University of Hong Kong, by the “DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie” and the “DFG-Leibnizprogramm”. The second-named author is supported by grants from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4034/02P) and from the Institute of Mathematical Sciences at the Chinese University of Hong Kong (Program in Algebraic Geometry)

American Mathematical Society