## Sections of Calabi-Yau threefolds with K3 fibration

HTML articles powered by AMS MathViewer

- by Zhiyuan Li PDF
- Trans. Amer. Math. Soc.
**366**(2014), 6313-6328 Request permission

## Abstract:

We study sections of a Calabi-Yau threefold fibered over a curve by K3 surfaces. We show that there exist infinitely many isolated sections on certain K3 fibered Calabi-Yau threefolds and that the group of algebraic $1$-cycles generated by these sections modulo algebraic equivalence is not finitely generated. We also give examples of K$3$ surfaces over the function field $F$ of a complex curve with Zariski dense $F$-rational points, whose geometric model is Calabi-Yau.## References

- Fabio Bardelli,
*Polarized mixed Hodge structures: on irrationality of threefolds via degeneration*, Ann. Mat. Pura Appl. (4)**137**(1984), 287–369 (English, with Italian summary). MR**772264**, DOI 10.1007/BF01789401 - A. Beĭlinson,
*Height pairing between algebraic cycles*, Current trends in arithmetical algebraic geometry (Arcata, Calif., 1985) Contemp. Math., vol. 67, Amer. Math. Soc., Providence, RI, 1987, pp. 1–24. MR**902590**, DOI 10.1090/conm/067/902590 - Spencer Bloch,
*Height pairings for algebraic cycles*, Proceedings of the Luminy conference on algebraic $K$-theory (Luminy, 1983), 1984, pp. 119–145. MR**772054**, DOI 10.1016/0022-4049(84)90032-X - F. A. Bogomolov and Yu. Tschinkel,
*Density of rational points on elliptic $K3$ surfaces*, Asian J. Math.**4**(2000), no. 2, 351–368. MR**1797587**, DOI 10.4310/AJM.2000.v4.n2.a6 - Herbert Clemens,
*Homological equivalence, modulo algebraic equivalence, is not finitely generated*, Inst. Hautes Études Sci. Publ. Math.**58**(1983), 19–38 (1984). MR**720930**, DOI 10.1007/BF02953771 - Herbert Clemens,
*The Néron model for families of intermediate Jacobians acquiring “algebraic” singularities*, Inst. Hautes Études Sci. Publ. Math.**58**(1983), 5–18 (1984). MR**720929**, DOI 10.1007/BF02953770 - Herbert Clemens,
*On the surjectivity of Abel-Jacobi mappings*, Ann. of Math. (2)**117**(1983), no. 1, 71–76. MR**683802**, DOI 10.2307/2006971 - Torsten Ekedahl, Trygve Johnsen, and Dag Einar Sommervoll,
*Isolated rational curves on $K3$-fibered Calabi-Yau threefolds*, Manuscripta Math.**99**(1999), no. 1, 111–133. MR**1697206**, DOI 10.1007/s002290050165 - Mark Green and Phillip Griffiths,
*Algebraic cycles and singularities of normal functions*, Algebraic cycles and motives. Vol. 1, London Math. Soc. Lecture Note Ser., vol. 343, Cambridge Univ. Press, Cambridge, 2007, pp. 206–263. MR**2385303**, DOI 10.1017/CBO9780511721496.006 - Mark Green, Phillip Griffiths, and Matt Kerr,
*Néron models and limits of Abel-Jacobi mappings*, Compos. Math.**146**(2010), no. 2, 288–366. MR**2601630**, DOI 10.1112/S0010437X09004400 - Tom Graber, Joe Harris, and Jason Starr,
*Families of rationally connected varieties*, J. Amer. Math. Soc.**16**(2003), no. 1, 57–67. MR**1937199**, DOI 10.1090/S0894-0347-02-00402-2 - Phillip A. Griffiths,
*Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties*, Amer. J. Math.**90**(1968), 568–626. MR**229641**, DOI 10.2307/2373545 - Tatsuki Hayama,
*Néron models of Green-Griffiths-Kerr and log Néron models*, Publ. Res. Inst. Math. Sci.**47**(2011), no. 3, 803–824. MR**2832807**, DOI 10.2977/PRIMS/52 - Marc Hindry,
*Introduction to abelian varieties and the Mordell-Lang conjecture*, Model theory and algebraic geometry, Lecture Notes in Math., vol. 1696, Springer, Berlin, 1998, pp. 85–100. MR**1678527**, DOI 10.1007/978-3-540-68521-0_{5} - Brendan Hassett and Yuri Tschinkel,
*Potential density of rational points for $K3$ surfaces over function fields*, Amer. J. Math.**130**(2008), no. 5, 1263–1278. MR**2450208**, DOI 10.1353/ajm.0.0023 - Kazuya Kato, Chikara Nakayama, and Sampei Usui,
*Log intermediate Jacobians*, Proc. Japan Acad. Ser. A Math. Sci.**86**(2010), no. 4, 73–78. MR**2657330** - K. Kodaira,
*On stability of compact submanifolds of complex manifolds*, Amer. J. Math.**85**(1963), 79–94. MR**153033**, DOI 10.2307/2373187 - Alan Landman,
*On the Picard-Lefschetz transformation for algebraic manifolds acquiring general singularities*, Trans. Amer. Math. Soc.**181**(1973), 89–126. MR**344248**, DOI 10.1090/S0002-9947-1973-0344248-1 - Masayoshi Nagata,
*On rational surfaces. I. Irreducible curves of arithmetic genus $0$ or $1$*, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math.**32**(1960), 351–370. MR**126443**, DOI 10.1215/kjm/1250776405 - Kapil H. Paranjape,
*Curves on threefolds with trivial canonical bundle*, Proc. Indian Acad. Sci. Math. Sci.**101**(1991), no. 3, 199–213. MR**1143483**, DOI 10.1007/BF02836802 - Morihiko Saito,
*Admissible normal functions*, J. Algebraic Geom.**5**(1996), no. 2, 235–276. MR**1374710** - Claire Voisin,
*Densité du lieu de Noether-Lefschetz pour les sections hyperplanes des variétés de Calabi-Yau de dimension $3$*, Internat. J. Math.**3**(1992), no. 5, 699–715 (French). MR**1189682**, DOI 10.1142/S0129167X92000345 - Claire Voisin,
*The Griffiths group of a general Calabi-Yau threefold is not finitely generated*, Duke Math. J.**102**(2000), no. 1, 151–186. MR**1741781**, DOI 10.1215/S0012-7094-00-10216-5 - Claire Voisin,
*Hodge theory and complex algebraic geometry. I*, Cambridge Studies in Advanced Mathematics, vol. 76, Cambridge University Press, Cambridge, 2002. Translated from the French original by Leila Schneps. MR**1967689**, DOI 10.1017/CBO9780511615344 - Steven Zucker,
*Generalized intermediate Jacobians and the theorem on normal functions*, Invent. Math.**33**(1976), no. 3, 185–222. MR**412186**, DOI 10.1007/BF01404203

## Additional Information

**Zhiyuan Li**- Affiliation: Department of Mathematics, Building 380, Stanford University, 450 Serra Mall, Stanford, California 94305
- Email: zli2@stanford.edu
- Received by editor(s): June 6, 2012
- Received by editor(s) in revised form: October 29, 2012
- Published electronically: June 10, 2014
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**366**(2014), 6313-6328 - MSC (2010): Primary 14-XX
- DOI: https://doi.org/10.1090/S0002-9947-2014-06002-9
- MathSciNet review: 3267011