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

Abstract

Let M d,n be the moduli stack of hypersurfaces X ⊂ P n 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 P n−1 ramified over a hypersurface of degree d. Iterating this construction, one obtains M (ν) d,n . We show that M (1) d,n is rigid in M d,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 M (ν)d,n , and we construct a 4-dimensional family of quintic hypersurfaces g : Z → T in P 4 , and a dense set of points t in T , such that g −1 (t) has complex multiplication. Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:26:22 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdfM 5,1 × M 5,1 −→ M 5,4 with image M , such that the CM-points are dense in M . The arguments used in the proof of 0.4 do not extend to the case n ≥ 5 and d = n + 1. They are related to the ones used by B. van Geemen and E. Izadi in [13]. In particular, in [13], quintic surfaces with complex multiplication are studied in Section 6. Acknowledgment This note grew out of discussions started when the first-named author visited the Institute of Mathematical Science and the Department of Mathematics at the Chinese University of Hong Kong. His contributions to the present version were written during two visits to the I.H.E.S., Bures sur Yvette. He would like to thank the members of those three Institutes for their hospitality. Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:26:22 EDT 2015 for download from IP 130.74.92.202.License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf FAMILIES OF HYPERSURFACES 485Shing-Tung Yau, drew our attention to the the work of S. Ferrara and J. Louis [12], an article he was studying to understand similar questions. Bert van Geemen found an error in the first version of Section 8, and he pointed out the relation between our Sections 7 and 8 the Sections 3 and 6 of [13]. Jan Christian Rohde's comments allowed us to lower the number of misprints. We both would like to thank them, and Hélène Esnault for their interest and help. Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:26:22 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:26:22 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Licensed to Univ of Mississippi. Prepared on Thu Jul 2 06:26:22 EDT 2015 for download from IP 130.74.92.202. License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdfSince E 0 + π * d (X ) is a normal ...

Cyclic Coverings, Calabi-Yau Manifolds and Complex Multiplication

“…All new examples C → P n of the preceding remark have a variation V of Hodge structures similar to the examples of J. de Jong and R. Noot [25], and of E. Viehweg and K. Zuo [47], which we call pure (1, n) − V HS. Let Hg(V) denote the generic Hodge group of V and let K denote an arbitrary maximal compact subgroup of Hg ad (V)(R).…”

“…J. de Jong and R. Noot [25] resp., E. Viehweg and K. Zuo [47] have given counterexamples of families with infinitely many CM fibers for g = 4, 6. In our lists here we have counterexamples for g = 5, 7.…”

“…Starting with a family of cyclic covers of P 1 with a dense set of CM fibers, E. Viehweg and K. Zuo [47] have constructed a CMCY family of 3-manifolds. This construction of E. Viehweg and K. Zuo [47] is given by a tower of cyclic coverings, which will be explained in Section 7.3.…”

“…This construction of E. Viehweg and K. Zuo [47] is given by a tower of cyclic coverings, which will be explained in Section 7.3. In Chapter 8 we will give a modified version of a Viehweg-Zuo tower for one of our new examples.…”

“…In [25] J. de Jong and R. Noot gave counterexamples for g = 4 and g = 6 to the conjecture above. In [47] E. Viehweg and K. Zuo gave an additional counterexample for g = 6. The counterexamples are given by families C → P n of cyclic covers of P 1 with infinitely many CM fibers.…”