## Mésures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines.(French)Zbl 0579.32012

Let X be an irreducible n-dimensional Stein space and $$\phi$$ : $$X\to [- \infty,R[$$ a continuous psh (plurisubharmonic) exhaustion function. Each level set $$S(r)=\{x\in X;\quad \phi (x)=r\},$$ $$r<R$$, is shown to carry an intrinsic positive measure $$\mu_ r$$; the measure $$\mu_ r$$ is given by the (2n-1)-form $$(dd^ c\phi)^{n-1}\wedge d^ c\phi$$ when $$\phi$$ is smooth, and otherwise $$\mu_ r$$ is constructed by means of the Monge- Ampère operators introduced by Bedford and Taylor. In this context, a general Lelong-Jensen formula $\mu_ r(V)=\int^{r}_{- \infty}dt\int_{\phi <t}dd^ cV\wedge (dd^ c\phi)^{n- 1}+\int_{\phi <r}V(dd^ c\phi)^ n,$ is proved and used to study the growth and convexity properties of plurisubharmonic or holomorphic functions. If $$(dd^ c\phi)^ n=0$$ on $$\{\phi >r_ 0\}$$, the function $$r\to \mu_ r(V)$$ is shown to be convex and increasing in the interval $$]r_ 0,R[$$. Furthermore, if the volume $$\tau (r)=\int_{\phi <r}(dd^ c\phi)^ n$$ has moderate growth, i.e. if $$\tau (r)=o(r)$$, then bounded holomorphic functions on X are constant; using Siegel’s method, we prove also in that case that the ring of holomorphic functions with $$\phi$$- polynomial growth has a transcendance degree $$\leq n$$. This last result is then applied in order to obtain a necessary and sufficient geometric criterion characterizing affine algebraic manifolds: X is algebraic iff it has finite Monge-Ampère volume and if the Ricci-curvature of the metric $$dd^ c(\exp (\phi))$$ is bounded below by $$-dd^ c\psi,$$ where $$\psi \leq A\phi +B.$$

### MSC:

 32C30 Integration on analytic sets and spaces, currents 32U05 Plurisubharmonic functions and generalizations 32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables 32J10 Algebraic dependence theorems 32J99 Compact analytic spaces
Full Text:

### References:

 [1] E. BISHOP . - Conditions for the analyticity of certain sets ; Michigan Math. Jour. 11 ( 1964 ), pp. 289-304. Article | MR 29 #6057 | Zbl 0143.30302 · Zbl 0143.30302 · doi:10.1307/mmj/1028999180 [2] N. BOURBAKI . - Topologie générale , chap. 1 à 4 ; Hermann, Paris, 1971 . Zbl 0249.54001 · Zbl 0249.54001 [3] D. BURNS . - Curvatures of Monge-Ampère foliations and parabolic manifolds ; Ann. of Math. 115 ( 1982 ), pp. 349-373. MR 84a:32031 | Zbl 0507.32011 · Zbl 0507.32011 · doi:10.2307/1971395 [4] E. BEDFORD & B.A. TAYLOR . - The Dirichlet problem for the complex Monge-Ampère equation ; Invent. Math. 37 ( 1976 ), pp. 1-44. MR 56 #3351 | Zbl 0315.31007 · Zbl 0315.31007 · doi:10.1007/BF01418826 [5] E. BEDFORD & B. A. TAYLOR . - A new capacity for plurisubharmonic functions ; Acta Math. 149 ( 1982 ), pp. 1-41. MR 84d:32024 | Zbl 0547.32012 · Zbl 0547.32012 · doi:10.1007/BF02392348 [6] U. CEGRELL . - On the discontinuity of the complex Monge-Ampère operator ; preprint, cf. note Comptes R. Acad. Sc. Paris (mai 1983 ). Zbl 0541.31008 · Zbl 0541.31008 [7] S. S. CHERN , H. I. LEVINE & L. NIRENBERG . - Intrinsic norms on a complex manifold ; Global Analysis (Papers in Honor of K. Kodaira) pp. 119-139, Univ. of Tokyo Press, Tokyo, 1969 . MR 40 #8084 | Zbl 0202.11603 · Zbl 0202.11603 [8] J. P. DEMAILLY . - Différents exemples de fibrés holomorphes non de Stein ; sém. P. Lelong-H. Skoda (Analyse) 1976 / 1977 , Lecture Notes in Math. n^\circ 694, Springer-Verlag 1978 . Zbl 0418.32011 · Zbl 0418.32011 [9] J. P. DEMAILLY . - Un exemple de fibré holomorphe non de Stein à fibre \Bbb C2 ayant pour base le disque ou le plan ; Inv. Math. 48 ( 1978 ), pp. 293-302. MR 81m:32036 | Zbl 0372.32012 · Zbl 0372.32012 · doi:10.1007/BF01390248 [10] J. P. DEMAILLY . - Un exemple de fibré holomorphe non de Stein à fibre \Bbb C2 au-dessus du disque ou du plan ; à paraître au Sém. P. Lelong, P. Dolbeault, H. Skoda (Analyse) 1983 - 1984 . Zbl 0594.32030 · Zbl 0594.32030 [11] J. P. DEMAILLY . - Formules de Jensen en plusieurs variables et applications arithmétiques ; Bull. Soc. Math. France 110 ( 1982 ), pp. 75-102. Numdam | MR 84d:32014 | Zbl 0493.32003 · Zbl 0493.32003 [12] J. P. DEMAILLY . - Sur les nombres de Lelong associés à l’image directe d’un courant positif fermé ; Ann. Inst. Fourier 32, 2 ( 1982 ), pp. 37-66. Numdam | MR 84k:32011 | Zbl 0457.32005 · Zbl 0457.32005 · doi:10.5802/aif.872 [13] J. P. DEMAILLY & H. SKODA . - Relations entre les notions de positivité de P. A. Griffiths et S. Nakano pour les fibrés vectoriels ; Sém. P. Lelong-H. Skoda (Analyse) 1978 / 1979 , Lect. Notes in Math. 822, Springer, 1980 . Zbl 0454.55011 · Zbl 0454.55011 [14] J. DIEUDONNE . - Cours de géométrie algébrique , tome 2 ; Coll. Sup., Presses Univ. de France, 1974 . Zbl 01958374 · Zbl 1092.14500 [15] H. EL MIR . - Sur le prolongement des courants positifs fermés ; Thèse de Doctorat Univ. de Paris VI (nov. 1982), publiée aux Acta Math., vol. 153 ( 1984 ), pp. 1-45 ; cf. aussi Comptes R. Acad. Sc. Paris, série I, t. 294 (1er février 1982 ) pp. 181-184 et t.295 (18 oct. 1982 ) pp. 419-422. · Zbl 0557.32003 [16] J. E. FORNAESS & R. NARASIMHAN . - The Levi problem on complex spaces with singularities ; Math. Ann. 248 ( 1980 ), pp. 47-72. MR 81f:32020 | Zbl 0411.32011 · Zbl 0411.32011 · doi:10.1007/BF01349254 [17] J. E. GOODMAN . - Affine open subsets of algebraic varieties and ample divisors ; Ann. of Math. 89 ( 1969 ), pp. 160-183. MR 39 #4170 | Zbl 0159.50504 · Zbl 0159.50504 · doi:10.2307/1970814 [18] H. GRAUERT . - On Levi’s problem and the imbedding of real analytic manifolds ; Ann. of Math. 68 ( 1958 ), pp. 460-472. MR 20 #5299 | Zbl 0108.07804 · Zbl 0108.07804 · doi:10.2307/1970257 [19] H. HIRONAKA . - Resolution of singularities of an algebraic variety , I-II, Ann. of Math. 79 ( 1964 ), pp. 109-326. MR 33 #7333 | Zbl 0122.38603 · Zbl 0122.38603 · doi:10.2307/1970486 [20] L. HÖRMANDER . - L2 estimates and existence theorems for the \partial -operator ; Acta Math. 113 ( 1965 ), pp. 89-152. Zbl 0158.11002 · Zbl 0158.11002 · doi:10.1007/BF02391775 [21] L. HÖRMANDER . - An introduction to complex analysis in several variables ; 2nd edition, North Holland, vol. 7, 1973 . Zbl 0271.32001 · Zbl 0271.32001 [22] C. O. KISELMAN . - Sur la définition de l’opérateur de Monge-Ampère complexe ; preprint. · Zbl 0562.35021 [23] P. LELONG . - Fonctions plurisousharmoniques et formes différentielles positives ; Gordon and Breach, New-York, et Dunod, Paris, 1969 . Zbl 0195.11603 · Zbl 0195.11603 [24] P. LELONG . - Fonctionnelles analytiques et fonctions entières (n variables) ; Presses de l’Univ. de Montréal, 1968 , Sém. de Math. Supérieures, été 1967 , n^\circ 28. Zbl 0194.38801 · Zbl 0194.38801 [25] P. LELONG . - Fonctions entières (n variables) et fonctions plurisousharmoniques d’ordre fini dans \Bbb Cn ; J. Anal. de Jérusalem 62 ( 1964 ), pp. 365-407. MR 29 #3668 | Zbl 0126.29602 · Zbl 0126.29602 · doi:10.1007/BF02807441 [26] J. MILNOR . - Morse Theory ; Ann. of Math. Studies n^\circ 51, Princeton Univ. Press, 1963 . MR 29 #634 | Zbl 0108.10401 · Zbl 0108.10401 [27] N. MOK . - Courbure bisectionnelle positive et variétés algébriques affines ; C. R. Acad. Sc. Paris, Série I, t. 296 (21 mars 1983 ), pp. 473-476. MR 85a:32015 | Zbl 0579.53043 · Zbl 0579.53043 [28] N. MOK . - An embedding theorem of complete Kähler manifolds of positive bisectional curvature onto affine algebraic varieties ; à paraître au Bull. Soc. Math. France, t. 112 ( 1984 ). Numdam | MR 87a:53103 | Zbl 0536.53062 · Zbl 0536.53062 [29] N. MOK . - Complete non compact Kähler manifolds of positive curvature (survey article) ; to appear in a special volume of the Madison conference on Several complex variables, 1982 . Zbl 0537.53054 · Zbl 0537.53054 [30] N. MOK , Y. T. SIU & S. T. YAU . - The Poincaré-Lelong equation on complete Kähler manifolds , Comp. Math., Vol. 44, fasc. 1-3 ( 1981 ), pp. 183-218. Numdam | MR 84g:32011 | Zbl 0531.32007 · Zbl 0531.32007 [31] S. NAKANO . - Vanishing theorems for weakly 1-complete manifolds II , Publ. R.I.M.S., Kyoto University, 1974 , pp. 101-110. Article | MR 52 #3617 | Zbl 0298.32019 · Zbl 0298.32019 · doi:10.2977/prims/1195192175 [32] R. NARASIMHAN . - Introduction to the theory of analytic spaces ; Lecture Notes in Math. n^\circ 25, 1966 , Springer-Verlag. MR 36 #428 | Zbl 0168.06003 · Zbl 0168.06003 · doi:10.1007/BFb0077071 [33] N. SIBONY . - Quelques problèmes de prolongement de courants en Analyse complexe ; prépublication Univ. de Paris-Sud, Orsay, n^\circ 84 T 15. · Zbl 0578.32023 [34] N. SIBONY & P. M. WONG . - Some remarks on the Casorati-Weierstrass thorem ; Ann. Polon. Math. 39 ( 1981 ), pp. 165-174. MR 82k:32015 | Zbl 0476.32005 · Zbl 0476.32005 [35] C. L. SIEGEL . - Meromorphe Funktionen auf compakten analytischen Mannigfaltigkeiten ; Cöttinger Nachr. ( 1955 ) pp. 71-77. MR 17,530c | Zbl 0064.08201 · Zbl 0064.08201 [36] C. L. SIEGEL . - On meromorphic functions of several variables ; Bull. Calcutta Math. Soc. 50 ( 1958 ) pp. 165-168. MR 21 #2748 | Zbl 0121.06802 · Zbl 0121.06802 [37] Y. T. SIU - S. T. YAU . - Complete Kähler manifolds with non positive curvature of faster than quadratic decay ; Ann. of Math. 105 ( 1977 ), pp. 225-264. MR 55 #10719 | Zbl 0358.32006 · Zbl 0358.32006 · doi:10.2307/1970998 [38] H. SKODA . - Estimations L2 pour l’opérateur \partial et applications arithmétiques ; Sém. P. Lelong (Analyse) 1975 / 1976 , Lecture Notes in Math. n^\circ 538, Springer-Verlag 1977 . Zbl 0363.32004 · Zbl 0363.32004 [39] H. SKODA . - Fibrés holomorphes à base et à fibre de Stein ; Inv. Math. 43 ( 1977 ), pp. 97-107. MR 58 #22657 | Zbl 0365.32018 · Zbl 0365.32018 · doi:10.1007/BF01390000 [40] H. SKODA . - Morphismes surjectifs et fibrés linéaires semi-positifs ; Sém. P. Lelong, H. Skoda (Analyse) 1976 / 1977 , Lecture Notes in Math. n^\circ 694, Springer-Verlag 1978 . Zbl 0396.32009 · Zbl 0396.32009 [41] H. SKODA . - Morphismes surjectifs de fibrés vectoriels semi-positifs ; Ann. Scient. Ec. Norm. Sup., 4e série, t. 11 ( 1978 ), p. 577-611. Numdam | MR 80j:32047 | Zbl 0403.32019 · Zbl 0403.32019 [42] H. SKODA . - Prolongement des courants positifs fermés de masse finie ; Invent. Math. 66 ( 1982 ), pp. 361-376. MR 84k:32020 | Zbl 0488.58002 · Zbl 0488.58002 · doi:10.1007/BF01389217 [43] E. H. SPANIER . - Algebraic topology ; Mc Graw-Hill, 1966 . MR 35 #1007 | Zbl 0145.43303 · Zbl 0145.43303 [44] W. STOLL . - The growth of the area of a transcendental analytic set , I and II, Math. Ann. 156 ( 1964 ), pp. 47-78 and pp. 144-170. MR 29 #3670 | Zbl 0126.09502 · Zbl 0126.09502 · doi:10.1007/BF01359980 [45] W. STOLL . - The characterization of strictly parabolic manifolds ; Ann. Sc. Norm. Sup. Pisa, s. IV, vol. VII, n^\circ 1 ( 1980 ), pp. 87-154. Numdam | MR 81h:32028 | Zbl 0438.32005 · Zbl 0438.32005 [46] P. THIE . - The Lelong number of a point of a complex analytic set ; Math. Ann. 172 ( 1967 ), pp. 269-312. MR 35 #5661 | Zbl 0158.32804 · Zbl 0158.32804 · doi:10.1007/BF01351593
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.