Boundary distance functions and q-convexity of pseudoconvex domains of general order in Kähler manifolds

Kazuko MATSUMOTO

doi:10.2969/jmsj/04810085

Kazuko MATSUMOTO

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JOURNAL FREE ACCESS

1996 Volume 48 Issue 1 Pages 85-107

DOI https://doi.org/10.2969/jmsj/04810085

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Published: 1996 Received: March 18, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-259.
2) W. Barth, Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projectiven Raum, Math. Ann., 187 (1970), 150-162.
3) L. Bungart, Piecewise smooth approximations to q-plurisubharmonic functions, Pacific J. Math., 142 (1990), 227-244.
4) K. Diederich and J. E. Fornaess, Smoothing q-convex functions and vanishing theorems, Invent. Math., 82 (1985), 291-305.
5) M. G. Eastwood and G. V. Suria, Cohomologically complete and pseudoconvex domains, Comment. Math. Helv., 55 (1980), 413-426.
6) G. Elencwajg, Pseudo-convexité locale dans les variétés kählériennes, Ann. Inst. Fourier (Grenoble), 25 (1975), 295-314.
7) T. Frankel, Manifolds with positive curvature, Pacific J. Math., 11 (1961), 165-174.
8) O. Fujita, Domaines pseudoconvexes d'ordre général et fonctions pseudoconvexes d'ordre général, J. Math. Kyoto Univ., 30 (1990), 637-649.
9) O. Fujita, On the equivalence of the q-plurisubharmonic functions and the pseudoconvex functions of general order, preprint.
10) S. I. Goldberg and S. Kobayashi, Holomorphic bisectional curvature, J. Differential Geom., 1 (1967), 225-233.
11) R. E. Greene and H. Wu, On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs, Abh. Math. Sem. Univ. Hamburg, 47 (1978), 171-185.
12) L. R. Hunt and J. J. Murray, q-plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J., 25 (1978), 299-316.
13) K. Matsumoto, Pseudoconvex domains of general order in Stein manifolds, Mem. Fac. Sci. Kyushu Univ. Ser. A, 43 (1989), 67-76.
14) K. Matsumoto, A note on the differentiability of the distance function to regular submanifolds of Riemannian manifolds, Nihonkai Math. J., 3 (1992), 81-85.
15) K. Matsumoto, Pseudoconvex domains of general order and q-convex domains in the complex projective space, J. Math. Kyoto Univ., 33 (1993), 685-695.
16) M. Peternell, Continuous q-convex exhaustion functions, Invent. Math., 85 (1986), 249-262.
17) M. Peternell, q-completeness of subsets in complex projective space, Math. Z., 195 (1987), 443-450.
18) O. Riemenschneider, Über den Flächeninhalt analytischer Mengen and die Erzeugung k-pseudokonvexer Gebiete, Invent. Math., 2 (1967), 307-331.
19) M. Schneider, Über eine Vermutung von Hartshorne, Math. Ann., 201 (1973), 221-229.
20) W. Schwarz, Local q-completeness of complements of smooth CR-submanifolds, Math. Z., 210 (1992), 529-553.
21) Z. Slodkowski, The Bremermann-Dirichlet problem for q-plurisubharmonic functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 303-326.
22) Z. Slodkowski, Local maximum property and q-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl., 115 (1986), 105-130.
23) G. V. Suria, q-pseudoconvex and q-complete domains, Compositio Math., 53 (1984), 105-111.
24) O. Suzuki, Pseudoconvex domains on a Kähler manifold with positive holomorphic bisectional curvature, Publ. Res. Inst. Math. Sci., 12 (1976), 191-214.
25) M. Tadokoro, Sur les ensembles pseudoconcaves généraux, J. Math. Soc. Japan, 17 (1965), 281-290.
26) A. Takeuchi, Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan, 16 (1964), 159-181.
27) A. Takeuchi, Domaines pseudoconvexes sur les variétés kählériennes, J. Math. Kyoto Univ., 6 (1967), 323-357.

Right : [1] A. Andreotti and H. Grauert, Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193-259.
[2] W. Barth, Der Abstand von einer algebraischen Mannigfaltigkeit im komplex-projectiven Raum, Math. Ann., 187 (1970), 150-162.
[3] L. Bungart, Piecewise smooth approximations to q-plurisubharmonic functions, Pacific J. Math., 142 (1990), 227-244.
[4] K. Diederich and J. E. Fornaess, Smoothing q-convex functions and vanishing theorems, Invent. Math., 82 (1985), 291-305.
[5] M. G. Eastwood and G. V. Suria, Cohomologically complete and pseudoconvex domains, Comment. Math. Helv., 55 (1980), 413-426.
[6] G. Elencwajg, Pseudo-convexité locale dans les variétés kählériennes, Ann. Inst. Fourier (Grenoble), 25 (1975), 295-314.
[7] T. Frankel, Manifolds with positive curvature, Pacific J. Math., 11 (1961), 165-174.
[8] O. Fujita, Domaines pseudoconvexes d'ordre général et fonctions pseudoconvexes d'ordre général, J. Math. Kyoto Univ., 30 (1990), 637-649.
[9] O. Fujita, On the equivalence of the q-plurisubharmonic functions and the pseudoconvex functions of general order, preprint.
[10] S. I. Goldberg and S. Kobayashi, Holomorphic bisectional curvature, J. Differential Geom., 1 (1967), 225-233.
[11] R. E. Greene and H. Wu, On Kähler manifolds of positive bisectional curvature and a theorem of Hartogs, Abh. Math. Sem. Univ. Hamburg, 47 (1978), 171-185.
[12] L. R. Hunt and J. J. Murray, q-plurisubharmonic functions and a generalized Dirichlet problem, Michigan Math. J., 25 (1978), 299-316.
[13] K. Matsumoto, Pseudoconvex domains of general order in Stein manifolds, Mem. Fac. Sci. Kyushu Univ. Ser. A, 43 (1989), 67-76.
[14] K. Matsumoto, A note on the differentiability of the distance function to regular submanifolds of Riemannian manifolds, Nihonkai Math. J., 3 (1992), 81-85.
[15] K. Matsumoto, Pseudoconvex domains of general order and q-convex domains in the complex projective space, J. Math. Kyoto Univ., 33 (1993), 685-695.
[16] M. Peternell, Continuous q-convex exhaustion functions, Invent. Math., 85 (1986), 249-262.
[17] M. Peternell, q-completeness of subsets in complex projective space, Math. Z., 195 (1987), 443-450.
[18] O. Riemenschneider, Über den Flächeninhalt analytischer Mengen und die Erzeugung k-pseudokonvexer Gebiete, Invent. Math., 2 (1967), 307-331.
[19] M. Schneider, Über eine Vermutung von Hartshorne, Math. Ann., 201 (1973), 221-229.
[20] W. Schwarz, Local q-completeness of complements of smooth CR-submanifolds, Math. Z., 210 (1992), 529-553.
[21] Z. Slodkowski, The Bremermann-Dirichlet problem for q-plurisubharmonic functions, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 303-326.
[22] Z. Slodkowski, Local maximum property and q-plurisubharmonic functions in uniform algebras, J. Math. Anal. Appl., 115 (1986), 105-130.
[23] G. V. Suria, q-pseudoconvex and q-complete domains, Compositio Math., 53 (1984), 105-111.
[24] O. Suzuki, Pseudoconvex domains on a Kähler manifold with positive holomorphic bisectional curvature, Publ. Res. Inst. Math. Sci., 12 (1976), 191-214.
[25] M. Tadokoro, Sur les ensembles pseudoconcaves généraux, J. Math. Soc. Japan, 17 (1965), 281-290.
[26] A. Takeuchi, Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan, 16 (1964), 159-181.
[27] A. Takeuchi, Domaines pseudoconvexes sur les variétés kählériennes, J. Math. Kyoto Univ., 6 (1967), 323-357.

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