Extra extension properties of equidimensional holomorphic mappings: re\-sults and open questions

S.Ivashkovich

Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time, they have very strong ``extra`` extension properties in compare with mappings in different dimensions. The aim of this paper is to put together the known results on this subject, prove some new ones and formulate open questions in order to give a perspective for the future progress.

1. Andreotti A., Stoll W. Extension of Holomorphic Maps. Annals of Math., 1960, 72, 312-349.

2. Bierstone E., Milman, P.D. Local resolution of singularities. Proc.\ Symp.\ Pure.\ Math., 1991, 52, 42-64.

3. Bogomolov F., Katzarkov L. Symplectic four-manifolds and projective surfaces. Topology and its Applications, 1998, 88, 79-109.

4. Carlson J., Harvey R. A remark on the universal cover of a Moishezon space. Duke Math. J., 1976, 43, 497-500.

5. Chazal, F. Un théorème de prolongement d'applications méromorphes. Math. Ann., 2001, 320, 285-297.

6. Chern S.-S. Differential geometry; its past and its future. Proc.\ International Congress of Mathematicians, Nice, 1970.

7. Gauduchon P. La $1$-forme de torsion d'une vari\'et\'e hermitienne compacte. Math. Ann., 1984, 267, 495-518.

8. Grauert H. Set Theoretic Complex Equivalence Relations. Math. Ann., 1983, 265, 137-148.

9. Grauert H. Meromorphe \"Aquivalenzrelationen. Math. Ann., 1987, 278, 175-183.

10. Griffiths P. Two theorems on extension of holomoprhic mappings. Invent. math., 1971, 14, 27-62.

11. Forstneric F., Slapar M. Stein structures and holomorphic mappings. Math. Z., 2007, 256, 615-646.

12. Hironaka H. Introduction to the theory of infinitely near singular points. Mem.\ Mat.\ Inst.\ Jorge \ Juan, 28, Consejo Superior de Investigaciones Cientificas, Madrid, 1974.

13. Hironaka H. An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures. Ann. of Math. (2), 1962, 75, 190--208.

14. Hirschowitz A. Pseudoconvexit\'e au-dessus d'espaces plus ou moins homogenes. Invent. math., 1974, 26, 303-322.

15. Ilyashenko J. Foliations by analytic curves. Russ. Math. Sbornik, 1972, 88, 558-577.

16. Ivashkovich S. The Hartogs phenomenon for holomorphically convex Kähler manifolds. Math. USSR Izvestija, 1987, 29, № 1, 225-232.

17. Ivashkovich S. The Hartogs type extension theorem for meromorphic maps into compact Kähler manifolds. Invent. math., 1992, 109, 47-54.

18. Ivashkovich S. On convergency properties of meromorphic functions and mappings. In B.~Shabat Memorial vol., Moscow, FASIS, 1997, 145-163 (in russian, english version in math.CV/9804007).

19. Ivashkovich S. Extension properties of meromorphic mappings with values in non-Kähler complex manifolds.Annals of Mathematics, 2004, 160, 795-837.

20. Ivashkovich S. Filling ''holes'' in complex surfaces. Proc. Seminar Complex Anal., Novosibirsk, 1987, p.47.

21. Ivashkovich S. An example concerning extension and separate analyticity properties of meromorphic mappings. Amer. J. Math., 1999, 121, 97-130.

22. Ivashkovich S., Shiffman B. Compact singularities of meromorphic mappings between 3-dimensional manifolds. Math. Res. Letters, 2000, 7, 695-708.

23. Kato M. Examples on an Extension Problem of Holomorphic Maps and a Holomorphic 1-Dimensional Foliation.Tokyo J. Math., 1990, 13, № 1, 139-146.

24. Kato M. A non-Kähler structure on an $\mathbb{S}^2$-bundle over a ruled surface. Unpublished preprint, 1992.

25. L\'arusson F. Compact quotioens of large domains in complex projective space. Annales de l'Institute Fourier, 1998, 48, № 1, 223-246.

26. Mok N. Wong B. Characterization of bounded domains covering Zarisski dense subsets of conpact complex spaces. Amer. J. Math., 1983, 105, 1481-1487.

27. Nemirovski S., Shafikov R. Uniformization of strictly pseudoconvex domains. I, II Izv. Mat., 2005, 69, №6, 1189-1202 and 1203-1210.

28. Ohsawa T. Remark on pseudoconvex domains with analytic complements in compact Kähler manifolds. J. Math. Kyoto Univ., 2007, 47, №1, 115-119.

29. Oka K. Sur les fonctions analytiques de plusieurs variables, IX, Domains finis sans point critique int\'erieur. Japan J. Math., 1953, 23, 97-155.

30. Siegel C. Analytic functions of several complex variables. Institute for Advanced Atudy, Princeton, NJ, 1950.

31. Stein K. Topics on holomorphic correspondences. Rocky Mountain J. Math., 1972, 2, 443-463.

32. Takeuchi A. Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif. J. Math. Soc. Japan, 1964, 16, 159-181.

	Extra extension properties of equidimensional holomorphic mappings: results and open questions
Author	S.Ivashkovich ivachkov@math.univ-lille1.fr U.F.R. de Mathématiques, Université de Lille-1, 59655 Villeneuve d'Ascq, France, IAPMM Acad. Sci. Ukraine, 79601 Lviv, Naukova 3b, Ukraine
Abstract	Holomorphic (nondegenerate) mappings between complex manifolds of the same dimension are of special interest. For example, they appear as coverings of complex manifolds. At the same time, they have very strong ``extra`` extension properties in compare with mappings in different dimensions. The aim of this paper is to put together the known results on this subject, prove some new ones and formulate open questions in order to give a perspective for the future progress.
Keywords	extra extension property, equidimensional mapping, complex manifold
DOI	doi:10.30970/ms.30.2.198-213
Reference	1. Andreotti A., Stoll W. Extension of Holomorphic Maps. Annals of Math., 1960, 72, 312-349. 2. Bierstone E., Milman, P.D. Local resolution of singularities. Proc.\ Symp.\ Pure.\ Math., 1991, 52, 42-64. 3. Bogomolov F., Katzarkov L. Symplectic four-manifolds and projective surfaces. Topology and its Applications, 1998, 88, 79-109. 4. Carlson J., Harvey R. A remark on the universal cover of a Moishezon space. Duke Math. J., 1976, 43, 497-500. 5. Chazal, F. Un théorème de prolongement d'applications méromorphes. Math. Ann., 2001, 320, 285-297. 6. Chern S.-S. Differential geometry; its past and its future. Proc.\ International Congress of Mathematicians, Nice, 1970. 7. Gauduchon P. La $1$-forme de torsion d'une vari\'et\'e hermitienne compacte. Math. Ann., 1984, 267, 495-518. 8. Grauert H. Set Theoretic Complex Equivalence Relations. Math. Ann., 1983, 265, 137-148. 9. Grauert H. Meromorphe \"Aquivalenzrelationen. Math. Ann., 1987, 278, 175-183. 10. Griffiths P. Two theorems on extension of holomoprhic mappings. Invent. math., 1971, 14, 27-62. 11. Forstneric F., Slapar M. Stein structures and holomorphic mappings. Math. Z., 2007, 256, 615-646. 12. Hironaka H. Introduction to the theory of infinitely near singular points. Mem.\ Mat.\ Inst.\ Jorge \ Juan, 28, Consejo Superior de Investigaciones Cientificas, Madrid, 1974. 13. Hironaka H. An example of a non-Kählerian complex-analytic deformation of Kählerian complex structures. Ann. of Math. (2), 1962, 75, 190--208. 14. Hirschowitz A. Pseudoconvexit\'e au-dessus d'espaces plus ou moins homogenes. Invent. math., 1974, 26, 303-322. 15. Ilyashenko J. Foliations by analytic curves. Russ. Math. Sbornik, 1972, 88, 558-577. 16. Ivashkovich S. The Hartogs phenomenon for holomorphically convex Kähler manifolds. Math. USSR Izvestija, 1987, 29, № 1, 225-232. 17. Ivashkovich S. The Hartogs type extension theorem for meromorphic maps into compact Kähler manifolds. Invent. math., 1992, 109, 47-54. 18. Ivashkovich S. On convergency properties of meromorphic functions and mappings. In B.~Shabat Memorial vol., Moscow, FASIS, 1997, 145-163 (in russian, english version in math.CV/9804007). 19. Ivashkovich S. Extension properties of meromorphic mappings with values in non-Kähler complex manifolds.Annals of Mathematics, 2004, 160, 795-837. 20. Ivashkovich S. Filling ''holes'' in complex surfaces. Proc. Seminar Complex Anal., Novosibirsk, 1987, p.47. 21. Ivashkovich S. An example concerning extension and separate analyticity properties of meromorphic mappings. Amer. J. Math., 1999, 121, 97-130. 22. Ivashkovich S., Shiffman B. Compact singularities of meromorphic mappings between 3-dimensional manifolds. Math. Res. Letters, 2000, 7, 695-708. 23. Kato M. Examples on an Extension Problem of Holomorphic Maps and a Holomorphic 1-Dimensional Foliation.Tokyo J. Math., 1990, 13, № 1, 139-146. 24. Kato M. A non-Kähler structure on an $\mathbb{S}^2$-bundle over a ruled surface. Unpublished preprint, 1992. 25. L\'arusson F. Compact quotioens of large domains in complex projective space. Annales de l'Institute Fourier, 1998, 48, № 1, 223-246. 26. Mok N. Wong B. Characterization of bounded domains covering Zarisski dense subsets of conpact complex spaces. Amer. J. Math., 1983, 105, 1481-1487. 27. Nemirovski S., Shafikov R. Uniformization of strictly pseudoconvex domains. I, II Izv. Mat., 2005, 69, №6, 1189-1202 and 1203-1210. 28. Ohsawa T. Remark on pseudoconvex domains with analytic complements in compact Kähler manifolds. J. Math. Kyoto Univ., 2007, 47, №1, 115-119. 29. Oka K. Sur les fonctions analytiques de plusieurs variables, IX, Domains finis sans point critique int\'erieur. Japan J. Math., 1953, 23, 97-155. 30. Siegel C. Analytic functions of several complex variables. Institute for Advanced Atudy, Princeton, NJ, 1950. 31. Stein K. Topics on holomorphic correspondences. Rocky Mountain J. Math., 1972, 2, 443-463. 32. Takeuchi A. Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif. J. Math. Soc. Japan, 1964, 16, 159-181.
Pages	198-213
Volume	30
Issue	2
Year	2008
Journal	Matematychni Studii
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Table of content of issue	html

Extra extension properties of equidimensional holomorphic mappings: results and open questions