Geodesic
On defining functions and cores for unbounded domains. III
Sbornik. Mathematics, Tome 212 (2021) no. 6, pp. 859-885 Cet article a éte moissonné depuis la source Math-Net.Ru

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We extend the authors' results on existence of global defining functions to a number of different settings. In particular, we relax the assumption on strict pseudoconvexity of the domain to strict $q$-pseudoconvexity and we consider more general situations, where the ambient space is an almost complex manifold or a complex space. We also investigate to what extent the assumption on smoothness of the boundary of the domains under consideration is necessary in our results. Bibliography: 27 titles.
Keywords: strictly pseudoconvex domains, plurisubharmonic defining functions
Mots-clés : core of a domain. 
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T. Harz; N. Shcherbina; G. Tomassini. On defining functions and cores for unbounded domains. III. Sbornik. Mathematics, Tome 212 (2021) no. 6, pp. 859-885. http://geodesic.mathdoc.fr/item/SM_2021_212_6_a4/

[1] A. Andreotti, H. Grauert, “Théorèmes de finitude pour la cohomologie des espaces complexes”, Bull. Soc. Math. France, 90 (1962), 193–259 | DOI | MR | Zbl

[2] J.-P. Demailly, Complex analytic and differential geometry, 2012 http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf

[3] K. Diederich, A. Sukhov, “Plurisubharmonic exhaustion functions and almost complex Stein structures”, Michigan Math. J., 56:2 (2008), 331–355 | DOI | MR | Zbl

[4] K. Diederich, J. E. Fornaess, “Smoothing $q$-convex functions and vanishing theorems”, Invent. Math., 82:2 (1985), 291–305 | DOI | MR | Zbl

[5] G. Fischer, Complex analytic geometry, Lecture Notes in Math., 538, Springer-Verlag, Berlin–New York, 1976, vii+201 pp. | DOI | MR | Zbl

[6] J. E. Fornaess, R. Narasimhan, “The Levi problem on complex spaces with singularities”, Math. Ann., 248:1 (1980), 47–72 | DOI | MR | Zbl

[7] J. E. Fornæss, B. Stensønes, Lectures on counterexamples in several complex variables, Math. Notes, 33, Princeton Univ. Press, Princeton, NJ; Univ. of Tokyo Press, Tokyo, 1987, viii+248 pp. | MR | Zbl

[8] M. Fraboni, T. Napier, “Strong $q$-convexity in uniform neighborhoods of subvarieties in coverings of complex spaces”, Math. Z., 265:3 (2010), 653–685 | DOI | MR | Zbl

[9] H. Grauert, “Über Modifikationen und exzeptionelle analytische Mengen”, Math. Ann., 146 (1962), 331–368 | DOI | MR | Zbl

[10] H. Grauert, O. Riemenschneider, “Kählersche Mannigfaltigkeiten mit hyper-$q$-konvexem Rand”, Problems in analysis, Lectures sympos. in honor of S. Bochner (Princeton Univ., Princeton, NJ, 1969), Princeton Univ. Press, Princeton, NJ, 1970, 61–79 | MR | Zbl

[11] F. R. Harvey, H. B. Lawson, “Potential theory on almost complex manifolds”, Ann. Inst. Fourier (Grenoble), 65:1 (2015), 171–210 | DOI | MR | Zbl

[12] F. R. Harvey, H. B. Lawson, Jr., S. Pliś, “Smooth approximation of plurisubharmonic functions on almost complex manifolds”, Math. Ann., 366:3-4 (2016), 929–940 | DOI | MR | Zbl

[13] T. Harz, N. Shcherbina, G. Tomassini, “On defining functions and cores for unbounded domains. I”, Math. Z., 286:3-4 (2017), 987–1002 | DOI | MR | Zbl

[14] T. Harz, N. Shcherbina, G. Tomassini, “On defining functions and cores for unbounded domains. II”, J. Geom. Anal., 30:3 (2020), 2293–2325 | DOI | MR | Zbl

[15] G. M. Henkin, J. Leiterer, Theory of functions on complex manifolds, Monogr. Math., 79, Birkhäuser Verlag, Basel, 1984, 226 pp. | DOI | MR | Zbl

[16] G. M. Henkin, J. Leiterer, Andreotti–Grauert theory by integral formulas, Progr. Math., 74, Birkhäuser Boston, Inc., Boston, MA, 1988, 270 pp. | DOI | MR | Zbl

[17] S. Ivashkovich, J.-P. Rosay, “Schwarz-type lemmas for solutions of $\overline{\partial}$-inequalities and complete hyperbolicity of almost complex manifolds”, Ann. Inst. Fourier (Grenoble), 54:7 (2004), 2387–2435 | DOI | MR | Zbl

[18] J. Morrow, H. Rossi, “Some theorems of algebraicity for complex spaces”, J. Math. Soc. Japan, 27:2 (1975), 167–183 | DOI | MR | Zbl

[19] R. Narasimhan, “The Levi problem for complex spaces”, Math. Ann., 142 (1960/61), 355–365 | DOI | MR | Zbl

[20] R. Narasimhan, “The Levi problem for complex spaces. II”, Math. Ann., 146 (1962), 195–216 | DOI | MR | Zbl

[21] A. Nijenhuis, W. B. Woolf, “Some integration problems in almost-complex and complex manifolds”, Ann. of Math. (2), 77:3 (1963), 424–489 | DOI | MR | Zbl

[22] S. Pliś, “The Monge–Ampère equation on almost complex manifolds”, Math. Z., 276:3-4 (2014), 969–983 | DOI | MR | Zbl

[23] S. Pliś, “Monge–Ampère operator on four dimensional almost complex manifolds”, J. Geom. Anal., 26:4 (2016), 2503–2518 | DOI | MR | Zbl

[24] R. Richberg, “Stetige streng pseudokonvexe Funktionen”, Math. Ann., 175 (1968), 257–286 | DOI | MR | Zbl

[25] P. A. N. Smith, “Smoothing plurisubharmonic functions on complex spaces”, Math. Ann., 273:3 (1986), 397–413 | DOI | MR | Zbl

[26] G. Tomassini, “Sur les algèbres $A^0(\overline D)$ et $A^\infty(\overline D)$ d'un domaine pseudoconvexe non borné”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10:2 (1983), 243–256 | MR | Zbl

[27] V. Vâjâitu, “On the equivalence of the definitions of $q$-convex spaces”, Arch. Math. (Basel), 61:6 (1993), 567–575 | DOI | MR | Zbl