A boundedness theorem for morphisms between threefolds
Annales de l'Institut Fourier, Volume 49 (1999) no. 2, p. 405-415
The main result of this paper is as follows: let X,Y be smooth projective threefolds (over a field of characteristic zero) such that b 2 (X)=b 2 (Y)=1. If Y is not a projective space, then the degree of a morphism f:XY is bounded in terms of discrete invariants of X and Y. Moreover, suppose that X and Y are smooth projective n-dimensional with cyclic Néron-Severi groups. If c 1 (Y)=0, then the degree of f is bounded iff Y is not a flat variety. In particular, to prove our main theorem we show the non-existence of a flat 3-dimensional projective variety with b 2 =1.
Le résultat principal de cet article est le théorème suivant : soient X,Y des variétés lisses projectives complexes de dimension trois telles que b 2 (X)=b 2 (Y)=1. Si Y n’est pas l’espace projectif, alors le degré d’un morphisme f:XY est borné (par des invariants discrets de X et de Y). En plus, supposons X,Y lisses projectives de dimension quelconque et telles que leurs groupes de Néron-Severi soient cycliques. Si c 1 (Y)=0, nous montrons que le degré de f est borné si et seulement si Y n’est pas une variété plate. Une partie de la preuve du théorème principal revient donc à montrer la non-existence d’une variété projective plate de dimension trois avec b 2 =1.
@article{AIF_1999__49_2_405_0,
     author = {Amerik, Ekatarina and Rovinsky, Marat and Van De Ven, Antonius},
     title = {A boundedness theorem for morphisms between threefolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {2},
     year = {1999},
     pages = {405-415},
     doi = {10.5802/aif.1679},
     mrnumber = {2000f:14056},
     zbl = {0923.14008},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_1999__49_2_405_0}
}
Amerik, Ekatarina; Rovinsky, Marat; Van De Ven, Antonius. A boundedness theorem for morphisms between threefolds. Annales de l'Institut Fourier, Volume 49 (1999) no. 2, pp. 405-415. doi : 10.5802/aif.1679. https://aif.centre-mersenne.org/item/AIF_1999__49_2_405_0/

[A] E. Amerik, Maps onto certain Fano threefolds, Documenta Mathematica, 2 (1997), 195-211, http://www.mathematik.uni-bielefeld.de/documenta. | MR 98h:14049 | Zbl 0922.14007

[A1] E. Amerik, On a problem of Noether-Lefschetz type, Compositio Mathematica, 112 (1998), 255-271. | MR 99f:14059 | Zbl 0929.14003

[B] K.S. Brown, Cohomology of groups, Springer, 1982. | MR 83k:20002 | Zbl 0584.20036

[BD] T. Bandman, G. Dethloff, Estimates of the number of rational mappings from a fixed variety to varieties of general type, Ann. Inst. Fourier, 47-3 (1997), 801-824. | Numdam | MR 98h:14016 | Zbl 0868.14008

[BM] T. Bandman, D. Markushevich, On the number of rational maps between varieties of general type, J. Math. Sci. Tokyo, 1 (1994), 423-433. | MR 96c:14012 | Zbl 0824.14009

[D] I. Dolgachev, Weighted projective spaces, in: J.B. Carrell (ed.), Group actions and vector fields, Lecture Notes in Math., 956, Springer, 1982. | MR 85g:14060 | Zbl 0516.14014

[I] V. A. Iskovskih, Fano 3-folds I, II, Math. USSR Izv., 11 (1977), 485-52, and 12 (1978), 469-506. | Zbl 0382.14013

[K] S. Kleiman, The transversality of a general translate, Comp. Math., 28 (1974), 287-297. | Numdam | MR 50 #13063 | Zbl 0288.14014

[KO] S. Kobayashi, T. Ochiai, Meromorphic mappings onto compact complex spaces of general type, Inv. Math., 31 (1975), 7-16. | MR 53 #5948 | Zbl 0331.32020

[Kob] S. Kobayashi, Differential geometry of complex vector bundles, Princeton Univ. Press, 1987. | MR 89e:53100 | Zbl 0708.53002

[M] D. Mumford, Abelian varieties, Oxford University Press, 1970. | MR 44 #219 | Zbl 0223.14022

[S] C. Schuhmann, Mapping threefolds onto three-dimensional quadrics, Math. Ann., 142 (1996), 571-581. | MR 98a:14053 | Zbl 0873.14035

[Y] S. T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA, 74 (1977), 1798-1799. | MR 56 #9467 | Zbl 0355.32028