Abstract
We introduce a notion of super-potential for positive closed currents of bidegree (p, p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.
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References
Ahlfors, L., Zur Theorie der Überlagerungsflächen. Acta Math., 65 (1935), 157–194.
Alessandrini, L. & Bassanelli, G., Plurisubharmonic currents and their extension across analytic subsets. Forum Math., 5 (1993), 577–602.
Alexander, H., Projective capacity, in Recent Developments in Several Complex Variables (Proc. Conf. Princeton Univ., Princeton, NJ, 1979), Ann. of Math. Stud., 100, pp. 3–27. Princeton Univ. Press, Princeton, NJ, 1981.
Andersson, M., A generalized Poincaré–Lelong formula. Math. Scand., 101 (2007), 195–218.
Bassanelli, G. & Berteloot, F., Bifurcation currents in holomorphic dynamics on ℙk. J. Reine Angew. Math., 608 (2007), 201–235.
Bedford, E. & Smillie, J., Polynomial diffeomorphisms of C 2. II. Stable manifolds and recurrence. J. Amer. Math. Soc., 4 (1991), 657–679.
Berndtsson, B., Integral formulas on projective space and the Radon transform of Gindikin–Henkin–Polyakov. Publ. Mat., 32 (1988), 7–41.
Blanchard, A., Sur les variétés analytiques complexes. Ann. Sci. É. Norm. Super., 73 (1956), 157–202.
Bost, J.-B., Gillet, H. & Soulé, C., Heights of projective varieties and positive Green forms. J. Amer. Math. Soc., 7:4 (1994), 903–1027.
Briend, J.-Y. & Duval, J., Deux caractérisations de la mesure d’équilibre d’un endomorphisme de P k(C). Publ. Math. Inst. Hautes Études Sci., 93 (2001), 145–159.
Demailly, J.-P., Courants positifs et théorie de l’intersection. Gaz. Math., 53 (1992), 131–159.
— Monge–Ampère operators, Lelong numbers and intersection theory, in Complex Analysis and Geometry, Univ. Ser. Math., pp. 115–193. Plenum, New York, 1993.
— Complex Analytic and Algebraic Geometry. www-fourier.ujf-grenoble.fr/~demailly.
Dinh, T.-C., Decay of correlations for Hénon maps. Acta Math., 195 (2005), 253–264.
— Attracting current and equilibrium measure for attractors on ℙk. J. Geom. Anal., 17 (2007), 227–244.
Dinh, T.-C., Dujardin, R. & Sibony, N., On the dynamics near infinity of some polynomial mappings in ℂ2. Math. Ann., 333 (2005), 703–739.
Dinh, T.-C. & Sibony, N., Dynamique des applications d’allure polynomiale. J. Math. Pures Appl., 82 (2003), 367–423.
— Regularization of currents and entropy. Ann. Sci. É. Norm. Super., 37 (2004), 959–971.
— Green currents for holomorphic automorphisms of compact Kähler manifolds. J. Amer. Math. Soc., 18 (2005), 291–312.
— Une borne supérieure pour l’entropie topologique d’une application rationnelle. Ann. of Math., 161:3 (2005), 1637–1644.
— Distribution des valeurs de transformations méromorphes et applications. Comment. Math. Helv., 81 (2006), 221–258.
— Geometry of currents, intersection theory and dynamics of horizontal-like maps. Ann. Inst. Fourier (Grenoble), 56:2 (2006), 423–457.
— Pull-back currents by holomorphic maps. Manuscripta Math., 123 (2007), 357–371.
— Equidistribution towards the Green current for holomorphic maps. Ann. Sci. Éc. Norm. Supér., 41 (2008), 307–336.
Drasin, D. & Okuyama, Y., Equidistribution and Nevanlinna theory. Bull. Lond. Math. Soc., 39 (2007), 603–613.
Dujardin, R., Hénon-like mappings in ℂ2. Amer. J. Math., 126 (2004), 439–472.
Favre, C. & Jonsson, M., Brolin’s theorem for curves in two complex dimensions. Ann. Inst. Fourier (Grenoble), 53:5 (2003), 1461–1501.
Federer, H., Geometric Measure Theory. Grundlehren der mathematischen Wissenschaften, 153. Springer, Berlin–Heidelberg, 1969.
Fornæss, J.E., Dynamics in Several Complex Variables. CBMS Regional Conference Series in Mathematics, 87. Amer. Math. Soc., Providence, RI, 1996.
Fornæss, J.E. & Sibony, N., Complex dynamics in higher dimensions, in Complex Potential Theory (Montreal, PQ, 1993), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 439, pp. 131–186. Kluwer Acad. Publ., Dordrecht, 1994.
— Complex dynamics in higher dimension. II, in Modern Methods in Complex Analysis (Princeton, NJ, 1992), Ann. of Math. Stud., 137, pp. 135–182. Princeton Univ. Press, Princeton, NJ, 1995.
— Oka’s inequality for currents and applications. Math. Ann., 301 (1995), 399–419.
Friedland, S. & Milnor, J., Dynamical properties of plane polynomial automorphisms. Ergodic Theory Dynam. Systems, 9 (1989), 67–99.
Gelfand, I. M., Kapranov, M. M. & Zelevinsky, A. V., Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston, MA, 1994.
Gillet, H. & Soulé, C., Arithmetic intersection theory. Publ. Math. Inst. Hautes Études Sci., 72 (1990), 93–174.
Gromov, M., Convex sets and Kähler manifolds, in Advances in Differential Geometry and Topology, pp. 1–38. World Sci. Publ., Teaneck, NJ, 1990.
Harvey, R. & Polking, J., Extending analytic objects. Comm. Pure Appl. Math., 28:6 (1975), 701–727.
Hörmander, L., An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, 7. North-Holland, Amsterdam, 1990.
Lelong, P., Fonctions plurisousharmoniques et formes différentielles positives. Gordon & Breach, Paris, 1968.
Méo, M., Image inverse d’un courant positif fermé par une application analytique surjective. C. R. Acad. Sci. Paris Sér. I Math., 322:12 (1996), 1141–1144.
Nguyên, V. A., Algebraic degrees for iterates of meromorphic self-maps of ℙk. Publ. Mat., 50 (2006), 457–473.
Polyakov, P. L. & Henkin, G. M., Homotopy formulas for the \( \overline \partial \)-operator on CP n and the Radon–Penrose transform. Izv. Akad. Nauk SSSR Ser. Mat., 50 (1986), 566–597, 639 (Russian); English translation in Math. USSR–Izv., 50 (1986), 555–587.
de Rham, G., Differentiable Manifolds. Grundlehren der Mathematischen Wissenschaften, 266. Springer, Berlin–Heidelberg, 1984.
Sibony, N., Dynamique des applications rationnelles de P k, in Dynamique et géométrie complexes (Lyon, 1997), Panor. Synthèses, 8, pp. ix–x, xi–xii, 97–185. Soc. Math. France, Paris, 1999.
Sibony, N. & Wong, P. M., Some results on global analytic sets, in Séminaire Pierre Lelong–Henri Skoda (Analyse), Années 1978/79, Lecture Notes in Math., 822, pp. 221–237. Springer, Berlin–Heidelberg, 1980.
Sion, M., On general minimax theorems. Pacific J. Math., 8 (1958), 171–176.
Siu, Y.T., Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Invent. Math., 27 (1974), 53–156.
de Thélin, H., Sur les exposants de Lyapounov des applications méromorphes. Invent. Math., 172 (2008), 89–116.
Triebel, H., Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, 18. North-Holland, Amsterdam, 1978.
Vigny, G., Dynamics semi-conjugated to a subshift for some polynomial mappings in ℂ2. Publ. Mat., 51 (2007), 201–222.
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Dinh, TC., Sibony, N. Super-potentials of positive closed currents, intersection theory and dynamics. Acta Math 203, 1–82 (2009). https://doi.org/10.1007/s11511-009-0038-7
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DOI: https://doi.org/10.1007/s11511-009-0038-7
Key words
- super-potential
- structural disc of currents
- intersection theory
- pull-back operator
- complex dynamics
- regular polynomial automorphism
- algebraically p-stable maps