Estimates of the number of rational mappings from a fixed variety to varieties of general type

Bandman, Tanya; Dethloff, Gerd

Annales de l'Institut Fourier, Tome 47 (1997) no. 3, pp. 801-824

First we find effective bounds for the number of dominant rational maps $f:X\to Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$, where $n=\dim X$, $K_X$ is the canonical bundle of $X$ and $A,B$ are some constants, depending only on $n$.

Then we show that for any variety $X$ there exist numbers $c\left(X\right)$ and $C\left(X\right)$ with the following properties:

For any threefold $Y$ of general type the number of dominant rational maps $f:X\to Y$ is bounded above by $c\left(X\right)$.

The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X\to Y$, is bounded above by $C\left(X\right)$.

If, moreover, $X$ is a threefold of general type, we prove that $c\left(X\right)$ and $C\left(X\right)$ only depend on the index $r_{X_c}$ of the canonical model $X_c$ of $X$ and on $K_{X_c}^3$.

Nous démontrons d’abord que le nombre d’applications rationnelles dominantes $f:X\to Y$, entre deux variétés projectives fixes avec fibré canonique ample, peut être majoré par $\{A·K_X^n{\}}^{\{B·K_X^n{\}}^2}$. Ici $n=\dim X$, $K_X$ est le fibré canonique de $X$ et $A,B$ sont quelques constantes, dépendant seulement de $n$.

Ensuite nous démontrons que, pour toute variété $X$, il y a des constantes $c\left(X\right)$ et $C\left(X\right)$ avec les propriétés suivantes  :

Pour toute variété $Y$ de dimension 3 et de type général le nombre d’applications rationnelles dominantes $f:X\to Y$ est majoré par $c\left(X\right)$.

Le nombre de variétés $Y$ de dimension 3 et de type général, modulo équivalence birationnelle, pour lesquelles il existe des applications rationnelles dominantes $f:X\to Y$, est majoré par $C\left(X\right)$.

Si, de plus, $X$ est aussi une variété de dimension 3 et de type général, nous démontrons que $c\left(X\right)$ et $C\left(X\right)$ dépendent seulement de l’index $r_{X_c}$ du modèle canonique $X_c$ de $X$ et de $K_{X_c}^3$.

[Ban1] T. Bandman, Surjective holomorphic mappings of projective manifolds, Siberian Math. Journ., 22 (1982), 204-210. | Zbl | MR

[Ban2] T. Bandman, Topological invariants of a variety and the number of its holomorphic mappings, J. Noguchi (Ed.): Proceedings of the International Symposium Holomorphic Mappings, Diophantine Geometry and Related Topics, RIMS, Kyoto University, 1992, 188-202.

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

[BPV] W. Barth, C. Peters, A. Van De Ven, Compact complex surfaces, Springer Verlag, 1984. | Zbl | MR

[Ben] X. Benveniste, Sur l'anneau canonique de certaine variété de dimension 3, Invent. Math., 73 (1983), 157-164. | Zbl | MR

[BinFle] J. Bingener, H. Flenner, On the fibers of analytic mappings, V. Ancona, A. Silva (Eds.): Complex Analysis and Geometry, Plenum Press, 1993, 45-101. | Zbl | MR

[CatSch] F. Catanese, M. Schneider, Bounds for stable bundles and degrees of Weierstrass schemes, Math. Ann., 293 (1992), 579-594. | Zbl | MR

[DelKat] P. Deligne, N. Katz, Groupes de Monodromie en Géometrie Algébrique, (SGA 7 II), Exp. XVII, LNM 340 (1973), Springer Verlag. | Zbl | MR

[Dem] J.-P. Demailly, A numerical criterion for very ample line bundles, J. Diff. Geom., 37 (1993), 323-374. | Zbl | MR

[DesMen1] M.M. Deschamps and R.L. Menegaux, Applications rationelles séparables dominantes sur une variété de type général, Bull. Soc. Math. France, 106 (1978), 279-287. | Zbl | Numdam | EuDML

[DesMen2] M.M. Deschamps and R.L. Menegaux, Surfaces de type géneral dominées par une variété fixe, C.R. Acad. Sc. Paris, Ser.A, 288 (1979), 765-767. | Zbl | MR

[DesMen3] M.M. Deschamps and R.L. Menegaux, Surfaces de type géneral dominées par une variété fixe II, C.R. Acad. Sc. Paris, Ser.A, 291 (1980), 587-590. | Zbl

[Det] G. Dethloff, Iitaka-Severi's Conjecture for complex threefolds, Preprint Mathematica Gottingensis, 29-1995, Duke eprint 9505016.

[Elk] R. Elkik, Rationalité des singularités canoniques, Invent. Math., 64 (1981), 1-6. | Zbl | MR | EuDML

[Flen] H. Flenner, Rational singularities, Arch. Math., 36 (1981), 35-44. | Zbl

[Flet] A.R. Fletcher, Contributions to Riemann-Roch on projective threefolds with only canonical singularities and applications, S.J. Bloch (Ed.): Algebraic Geometry, Bowdoin 1985, 221-231. Proc Symp. in Pure Math., vol. 46, 1987. | Zbl

[Fra] M. De Franchis, Un Teorema sulle involuzioni irrazionali., Rend. Circ. Mat. Palermo, 36 (1913), 368. | JFM

[Fuj] T. Fujita, Zariski decomposition and canonical rings of elliptic threefolds, J. Math. Japan, 38 (1986), 20-37. | Zbl | MR

[FulLaz] W. Fulton, R. Lazarsfeld, Positive polynomials for ample vector bundles, Ann. Math., 118 (1983), 35-60. | Zbl | MR

[Gra] H. Grauert, Ein Theorem der analytischen Garbentheorie und die Modulräume komplexer Strukturen, Publ. IHES, 5 (1960), 1-64. | Zbl | Numdam | EuDML

[GriHar] P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley and Sons, 1978. | Zbl | MR

[Gro] A. Grothendieck, EGA III, Publ. IHES, 17 (1963), 1-91. | Numdam

[Han] M. Hanamura, Stability of the pluricanonical maps of threefolds, T. Oda (Ed.): Algebraic Geometry, Sendai 1985, 185-202. Advanced Studies in Pure Math., vol. 10, 1987. | Zbl

[Har] R. Hartshorne, Algebraic Geometry, Springer Verlag, 1977. | Zbl | MR

[Hir] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero I, Ann. Math., 79 (1964), 109-326. | Zbl | MR

[HowSom1] A. Howard, A.J. Sommese, On the orders of the automorphism groupes of certain projective manifolds, J. Hano et al. (Eds.): Manifolds and Lie Groupes. Progress in Math., 14 Birkhäuser 1981, 145-158. | Zbl | MR

[HowSom2] A. Howard and A. Sommese, On the theorem of de Franchis, Annali Scuola Norm. Sup. Pisa, 10 (1983), 429-436. | Zbl | MR | Numdam | EuDML

[Iit] S. Iitaka, Algebraic Geometry, Springer Verlag, 1982. | Zbl | MR

[Kan] E. Kani, Bounds on the number of non-rational subfields of a function field, Inv. Math., 85 (1986), 199-215. | Zbl | MR | EuDML

[Kaw] Y. Kawamata, On the finiteness of generators of pluricanonical ring for a threefold of general type, Amer. J. Math., 106 (1984), 1503-1512. | Zbl | MR

[Kob] S. Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc., 82 (1976), 357-416. | Zbl | MR

[KobOch] S. Kobayashi, T. Ochiai, Meromorphic mappings into complex spaces of general type, Inv. Math., 31 (1975), 7-16. | Zbl | MR | EuDML

[Kol1] J. Kollár, Towards moduli of singular varieties, Compos. Math., 56 (1985), 369-398. | Zbl | MR | Numdam | EuDML

[Kol2] J. Kollár, Higher direct images of dualizing sheaves I, Annals of Math., 123, 11-42. | Zbl | MR

[KolMor] J. Kollár and S. Mori, Classification of three-dimensional flips, J. of the AMS, 5 (1992), 533-703. | Zbl | MR

[Luo1] T. Luo, Global 2-forms on regular threefolds of general type, Duke Math. J., 71 (1993), 859-869. | Zbl | MR

[Luo2] T. Luo, Plurigenera of regular threefolds, Math. Z., 217 (1994), 37-46. | Zbl | MR | EuDML

[Mae1] K. Maehara, Families of varieties dominated by a variety, Proc. Japan Acad., Ser. A, 55 (1979), 146-151. | Zbl | MR

[Mae2] K. Maehara, A finiteness property of variety of general type, Math. Ann., 262 (1983), 101-123. | Zbl | MR | EuDML

[Mae3] K. Maehara, Diophantine problem of algebraic varieties and Hodge theory, J. Noguchi (Ed.): Proceedings of the International Symposium Holomorphic Mappings, Diophantine Geometry and Related Topics, 167-187. RIMS, Kyoto University, 1992.

[MatMum] T. Matsusaka, D. Mumford, Two fundamental theorems on deformations of polarized varieties, Amer. J. Math., 86 (1964), 668-684. | Zbl | MR

[Mil] J. Milnor, On the Betti numbers of real projective varieties, Proc. AMS, 15 (1964), 275-280. | Zbl | MR

[Mor] S. Mori, Flip theorem and the existence of minimal models for threefolds, J. AMS, 1 (1988), 117-253. | Zbl | MR

[Nog] J. Noguchi, Meromorphic mappings into compact hyperbolic complex spaces and geometric diophantine problems, Inte. J. Math., 3 (1992), 277-289, 677. | Zbl | MR

[Rei1] M. Reid, Canonical threefolds, A. Beauville (Ed.): Algebraic Geometry, Angers 1979, 273-310, Sijthoff and Noordhoff, 1980. | Zbl

[Rei2] M. Reid, Young person's guide to canonical singularities, S.J. Bloch (Ed.): Algebraic Geometry, Bowdoin 1985, 345-414. Proc Symp. in Pure Math., 46, 1987. | Zbl

[Sam] P. Samuel, Complements a un article de Hans Grauert sur la conjecture de Mordell, Publ. Math. IHES, 29 (1966), 311-318. | Zbl | MR | Numdam | EuDML

[Suz] M. Suzuki, Moduli spaces of holomorphic mappings into hyperbolically embedded complex spaces and hyperbolic fibre spaces, J. Noguchi (Ed.): Proceedings of the International Symposium Holomorphic Mappings, Diophantine Geometry and Related Topics, 157-166 RIMS, Kyoto University 1992.

[Sza] E. Szabo, Bounding the automorphisms groups, Math. Ann., 304 (1996), 801-811. | Zbl | MR | EuDML

[Tsa1] I.H. Tsai, Dominating the varieties of general type, to appear: J. reine angew. Math. (1996), 29 pages. | Zbl | EuDML

[Tsa2] I.H. Tsai, Dominant maps and dominated surfaces of general type, Preprint (1996), 44 pages.

[Tsa3] I.H. Tsai, Chow varieties and finiteness theorems for dominant maps, Preprint (1996), 33 pages. | Zbl

[Uen] K. Ueno, Classification theory of algebraic varieties and compact complex spaces, LNM 439 (1975), Springer Verlag. | Zbl | MR

[ZaiLin] M.G. Zaidenberg and V.Ya. Lin, Finiteness theorems for holomorphic maps, Several Complex Variables III, Encyclopaedia Math. Sciences, vol., 9, Springer Verlag, 1989, 113-172. | Zbl