# A boundedness theorem for morphisms between threefolds

• Volume: 49, Issue: 2, page 405-415
• ISSN: 0373-0956

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## Abstract

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The main result of this paper is as follows: let $X,Y$ be smooth projective threefolds (over a field of characteristic zero) such that ${b}_{2}\left(X\right)={b}_{2}\left(Y\right)=1$. If $Y$ is not a projective space, then the degree of a morphism $f:X\to Y$ 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 ${c}_{1}\left(Y\right)=0$, 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 ${b}_{2}=1$.

## How to cite

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Amerik, Ekatarina, Rovinsky, Marat, and Van de Ven, Antonius. "A boundedness theorem for morphisms between threefolds." Annales de l'institut Fourier 49.2 (1999): 405-415. <http://eudml.org/doc/75343>.

@article{Amerik1999,
abstract = {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 $f:X\rightarrow Y$ 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 $b_2=1$.},
author = {Amerik, Ekatarina, Rovinsky, Marat, Van de Ven, Antonius},
journal = {Annales de l'institut Fourier},
keywords = {second Betti class; finite morphism; flat variety; Chern class; three-folds; degree of a morphism; Néron-Severi groups},
language = {eng},
number = {2},
pages = {405-415},
publisher = {Association des Annales de l'Institut Fourier},
title = {A boundedness theorem for morphisms between threefolds},
url = {http://eudml.org/doc/75343},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Amerik, Ekatarina
AU - Rovinsky, Marat
AU - Van de Ven, Antonius
TI - A boundedness theorem for morphisms between threefolds
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 2
SP - 405
EP - 415
AB - 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 $f:X\rightarrow Y$ 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 $b_2=1$.
LA - eng
KW - second Betti class; finite morphism; flat variety; Chern class; three-folds; degree of a morphism; Néron-Severi groups
UR - http://eudml.org/doc/75343
ER -

## References

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8. [K] S. KLEIMAN, The transversality of a general translate, Comp. Math., 28 (1974), 287-297. Zbl0288.14014MR50 #13063
9. [KO] S. KOBAYASHI, T. OCHIAI, Meromorphic mappings onto compact complex spaces of general type, Inv. Math., 31 (1975), 7-16. Zbl0331.32020MR53 #5948
10. [Kob] S. KOBAYASHI, Differential geometry of complex vector bundles, Princeton Univ. Press, 1987. Zbl0708.53002MR89e:53100
11. [M] D. MUMFORD, Abelian varieties, Oxford University Press, 1970. Zbl0223.14022MR44 #219
12. [S] C. SCHUHMANN, Mapping threefolds onto three-dimensional quadrics, Math. Ann., 142 (1996), 571-581. Zbl0873.14035MR98a:14053
13. [Y] S. T. YAU, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. USA, 74 (1977), 1798-1799. Zbl0355.32028MR56 #9467

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