# zbMATH — the first resource for mathematics

Hyperbolic and Diophantine analysis. (English) Zbl 0602.14019
The survey is on hyperbolic analysis with the emphasis on its algebraic geometry aspects. It can serve as an excellent introduction to the subject not pretending completeness but reflecting the main ideas including those which appeared at the last decade (after Kobayashi’s survey, 1976). It contains also a number of old and new problems concerning both the hyperbolic analysis itself and its applications to Diophantine equations; among them the author’s (1974) multidimensional versions of Mordell’s conjecture. The central one asserts that a hyperbolic projective variety is Mordellic (i.e. the set of its rational points is finite). The only known case is the Faltings’ theorem in dimension one.
Starting with the notion of the Kobayashi hyperbolicity §§1 and 2 contain the full proofs of certain hyperbolicity criteria - Brody’s one for compact varieties (absence of entire curves) and M. Green’s one for subvarieties of a complex torus (absence of translated subtori). There are given also some variations on these themes.
During the whole article the author adheres to an idea of ”pseudofication”, i.e. a system of relative analogues of the notions and facts under consideration (usually modulo some ”exceptional” sets). The analytic exceptional set Exc(X) of a variety X is defined to be the Zariski closure of the union of all entire curves in X, and an algebraic exceptional set $$Exc_{alg}(X)\subset Exc(X)$$ to be the union of all compact rational and elliptic curves in X. One of the important open questions is whether $$Exc_{alg}(X)=Exc(X)$$ (the algebraic characterization of hyperbolicity).
Sections 3 and 4 are devoted to the Chern-Ricci forms, Ahlfors’ lemma and its multidimensional version working then in the differential geometric conditions for hyperbolicity and measure hyperbolicity.
The problem in §5 are the canonical and ”pseudocanonical” varieties (i.e. respectively the varieties with ample canonical bundle and the varieties of general type). It is established their measure hyperbolicity and discussed the validity of the converse. The Diophantine conjectures 5.7 and 5.8 say: the set of rational points of a pseudocanonical variety X is not Zariski dense; moreover $$X\setminus Exc(X)\neq \emptyset$$ and is Mordellic.
The short §6 is mainly concerned with minimal models. - In §7 the hyperbolicity of a manifold with ample cotangent bundle is proved by the construction of a negatively curved length function (”Finsler metric”). - The Green-Griffiths’ construction of nonpositively curved length functions on jet bundles is given in §8 where also the Green- Griffith’s exceptional set containing Exc(X) is defined.
In the appendix there are some historical comments on the functional analogues of Mordell’s conjecture and its multidimensional versions.
Reviewer: M.Zaidenberg

##### MSC:
 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
Full Text:
##### References:
 [1] James Ax, Some topics in differential algebraic geometry. II. On the zerosof theta functions, Amer. J. Math. 94 (1972), 1205 – 1213. · Zbl 0266.14018 [2] A. BLOCH, Sur les systèmes de fonctions uniformes satisfaisant à l’équation d’une variété algébrique dont l’irrégularité dépasse la dimension, J. de Math. V (1926), 19-66. · JFM 52.0373.04 [3] F. A. Bogomolov, Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041 – 1044 (Russian). [4] E. Bombieri, Canonical models of surfaces of general type, Inst. Hautes Études Sci. Publ. Math. 42 (1973), 171 – 219. · Zbl 0259.14005 [5] R. BRODY, Intrinsic metrics and measures on compact complex manifolds, thesis, Harvard, 1975. [6] Robert Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. 235 (1978), 213 – 219. · Zbl 0416.32013 [7] Robert Brody and Mark Green, A family of smooth hyperbolic hypersurfaces in \?$$_{3}$$, Duke Math. J. 44 (1977), no. 4, 873 – 874. · Zbl 0383.32009 [8] Shiing-shen Chern, On holomorphic mappings of hermitian manifolds of the same dimension., Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 157 – 170. [9] Alexander Dinghas, Ein \?-dimensionales Analogon des Schwarz-Pickschen Flächensatzes für holomorphe Abbildungen der komplexen Einheitskugel in eine Kähler-Mannigfaltigkeit, Festschr. Gedächtnisfeier K. Weierstrass, Westdeutscher Verlag, Cologne, 1966, pp. 477 – 494 (German). · Zbl 0173.09101 [10] Alexander Dinghas, Über das Schwarzsche Lemma und verwandte Sätze, Israel J. Math. 5 (1967), 157 – 169 (German, with English summary). · Zbl 0166.34001 [11] Igor Dolgachev, Weighted projective varieties, Group actions and vector fields (Vancouver, B.C., 1981) Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 34 – 71. [12] Donald A. Eisenman, Intrinsic measures on complex manifolds and holomorphic mappings, Memoirs of the American Mathematical Society, No. 96, American Mathematical Society, Providence, R.I., 1970. · Zbl 0197.05901 [13] Ian Graham and H. Wu, Some remarks on the intrinsic measures of Eisenman, Trans. Amer. Math. Soc. 288 (1985), no. 2, 625 – 660. · Zbl 0582.32034 [14] Hans Grauert, Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 131 – 149 (German). · Zbl 0161.18401 [15] Hans Grauert and Ulrike Peternell, Hyperbolicity of the complement of plane curves, Manuscripta Math. 50 (1985), 429 – 441. · Zbl 0581.32031 [16] Hans Grauert and Helmut Reckziegel, Hermitesche Metriken und normale Familien holomorpher Abbildungen, Math. Z. 89 (1965), 108 – 125 (German). · Zbl 0135.12503 [17] Mark Lee Green, Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), 43 – 75. · Zbl 0301.32022 [18] Mark L. Green, Holomorphic maps to complex tori, Amer. J. Math. 100 (1978), no. 3, 615 – 620. · Zbl 0384.32007 [19] Mark Green and Phillip Griffiths, Two applications of algebraic geometry to entire holomorphic mappings, The Chern Symposium 1979 (Proc. Internat. Sympos., Berkeley, Calif., 1979), Springer, New York-Berlin, 1980, pp. 41 – 74. · Zbl 0508.32010 [20] Phillip A. Griffiths, Holomorphic mapping into canonical algebraic varieties, Ann. of Math. (2) 93 (1971), 439 – 458. · Zbl 0214.48601 [21] Joe Harris and David Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23 – 88. With an appendix by William Fulton. · Zbl 0506.14016 [22] Robin Hartshorne, Ample vector bundles, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 63 – 94. · Zbl 0173.49003 [23] Shoshichi Kobayashi and Peter Kiernan, Holomorphic mappings into projective space with lacunary hyperplanes, Nagoya Math. J. 50 (1973), 199 – 216. · Zbl 0262.32010 [24] Yujiro Kawamata, Elementary contractions of algebraic 3-folds, Ann. of Math. (2) 119 (1984), no. 1, 95 – 110. , https://doi.org/10.2307/2006964 Yujiro Kawamata, The cone of curves of algebraic varieties, Ann. of Math. (2) 119 (1984), no. 3, 603 – 633. , https://doi.org/10.2307/2007087 János Kollár, The cone theorem. Note to a paper: ”The cone of curves of algebraic varieties” [Ann. of Math. (2) 119 (1984), no. 3, 603 – 633; MR0744865 (86c:14013b)] by Y. Kawamata, Ann. of Math. (2) 120 (1984), no. 1, 1 – 5. · Zbl 0544.14010 [25] Y. Kawamata, Pluricanonical systems on minimal algebraic varieties, Invent. Math. 79 (1985), no. 3, 567 – 588. · Zbl 0593.14010 [26] Shoshichi Kobayashi, Volume elements, holomorphic mappings and Schwarz’s lemma, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., LaJolla, Calif., 1966) Amer. Math. Soc., Providence, R.I., 1968, pp. 253 – 260. [27] Shoshichi Kobayashi, Hyperbolic manifolds and holomorphic mappings, Pure and Applied Mathematics, vol. 2, Marcel Dekker, Inc., New York, 1970. · Zbl 0247.32015 [28] Shoshichi Kobayashi, Intrinsic distances, measures and geometric function theory, Bull. Amer. Math. Soc. 82 (1976), no. 3, 357 – 416. · Zbl 0346.32031 [29] Shoshichi Kobayashi, Negative vector bundles and complex Finsler structures, Nagoya Math. J. 57 (1975), 153 – 166. · Zbl 0326.32016 [30] Shoshichi Kobayashi and Takushiro Ochiai, Mappings into compact manifolds with negative first Chern class, J. Math. Soc. Japan 23 (1971), 137 – 148. · Zbl 0203.39101 [31] Shoshichi Kobayashi and Takushiro Ochiai, Meromorphic mappings onto compact complex spaces of general type, Invent. Math. 31 (1975), no. 1, 7 – 16. · Zbl 0331.32020 [32] Serge Lang, Integral points on curves, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 27 – 43. · Zbl 0112.13402 [33] Serge Lang, Higher dimensional diophantine problems, Bull. Amer. Math. Soc. 80 (1974), 779 – 787. · Zbl 0298.14014 [34] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013 [35] Serge Lang, Introduction to transcendental numbers, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. · Zbl 0144.04101 [36] Ju. I. Manin, Rational points on algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395 – 1440 (Russian). · Zbl 0166.16901 [37] Shigefumi Mori, Threefolds whose canonical bundles are not numerically effective, Ann. of Math. (2) 116 (1982), no. 1, 133 – 176. · Zbl 0557.14021 [38] Shigefumi Mori and Shigeru Mukai, The uniruledness of the moduli space of curves of genus 11, Algebraic geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math., vol. 1016, Springer, Berlin, 1983, pp. 334 – 353. · Zbl 0557.14015 [39] Oscar Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560 – 615. · Zbl 0124.37001 [40] Junjiro Noguchi, A higher-dimensional analogue of Mordell’s conjecture over function fields, Math. Ann. 258 (1981/82), no. 2, 207 – 212. · Zbl 0459.14002 [41] Junjiro Noguchi, Logarithmic jet spaces and extensions of de Franchis’ theorem, Contributions to several complex variables, Aspects Math., E9, Friedr. Vieweg, Braunschweig, 1986, pp. 227 – 249. · Zbl 0598.32021 [42] Takushiro Ochiai, On holomorphic curves in algebraic varieties with ample irregularity, Invent. Math. 43 (1977), no. 1, 83 – 96. · Zbl 0374.32006 [43] H. RECKZIEGEL, Hyperbolische Raüme und normale Familien holomorpher Abbildungen, Dissertation, Göttingen, 1967. [44] Dieter Riebesehl, Hyperbolische komplexe Räume und die Vermutung von Mordell, Math. Ann. 257 (1981), no. 1, 99 – 110 (German). · Zbl 0451.32018 [45] H. L. Royden, Remarks on the Kobayashi metric, Several complex variables, II (Proc. Internat. Conf., Univ. Maryland, College Park, Md., 1970) Springer, Berlin, 1971, pp. 125 – 137. Lecture Notes in Math., Vol. 185. [46] Ira H. Shavel, A class of algebraic surfaces of general type constructed from quaternion algebras, Pacific J. Math. 76 (1978), no. 1, 221 – 245. · Zbl 0422.14022 [47] C. L. SIEGEL, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. Phys. Math. Kl. (1929), 41-69. · JFM 56.0180.05 [48] H. P. F. Swinnerton-Dyer, A solution of \?$$^{5}$$+\?$$^{5}$$+\?$$^{5}$$=\?$$^{5}$$+\?$$^{5}$$+\?$$^{5}$$, Proc. Cambridge Philos. Soc. 48 (1952), 516 – 518. · Zbl 0046.26602 [49] Yung-Sheng Tai, On the Kodaira dimension of the moduli space of abelian varieties, Invent. Math. 68 (1982), no. 3, 425 – 439. · Zbl 0508.14038 [50] P. VOJTA, Springer Lecture Notes, to appear; see also his thesis, Harvard, 1984. [51] Jürgen Wolfart, Taylorentwicklungen automorpher Formen und ein Transzendenzproblem aus der Uniformisierungstheorie, Abh. Math. Sem. Univ. Hamburg 54 (1984), 25 – 33 (German, with English summary). · Zbl 0556.10020 [52] J. WOLFART and G. WUSTHOLZ, Der Üherlagerungsradius algehraischer Kurven und die Werte der Betafunktion an rationalen Stellen (to appear). [53] Shing Tung Yau, Intrinsic measures of compact complex manifolds, Math. Ann. 212 (1975), 317 – 329. · Zbl 0313.32031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.