Sections of Calabi-Yau threefolds with K3 fibration

Li, Zhiyuan

doi:10.1090/S0002-9947-2014-06002-9

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