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Uspekhi Mat. Nauk, 1999, Volume 54, Issue 4(328), Pages 47–74 (Mi umn179)  

This article is cited in 17 scientific papers (total in 17 papers)

Complex analysis and differential topology on complex surfaces

S. Yu. Nemirovski

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In the paper, the relationship between the theory of holomorphic functions on two-dimensional complex manifolds and their differential topology is described. The basic fact, established by using the Seiberg–Witten invariants, is that the topological characteristics of embedded real surfaces in Stein surfaces satisfy adjunction-type inequalities. A version of Gromov's $h$-principle for totally real embeddings shows that these topological inequalities are sharp. In some cases, these results can be used to describe the envelopes of holomorphy of embedded real surfaces in a given complex surface. Our examples include real surfaces in $\mathbb C^2$ and $\mathbb{CP}^2$ and in products of $\mathbb{CP}^1$ with non-compact Riemann surfaces. A similar technique can be applied to the study of geometric properties of strictly pseudoconvex domains in dimension two.

DOI: https://doi.org/10.4213/rm179

Full text: PDF file (384 kB)
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English version:
Russian Mathematical Surveys, 1999, 54:4, 729–752

Bibliographic databases:

UDC: 515.1
MSC: Primary 32E10, 32D10, 32J20; Secondary 32F15, 53C23, 57N13, 14J35
Received: 07.05.1999

Citation: S. Yu. Nemirovski, “Complex analysis and differential topology on complex surfaces”, Uspekhi Mat. Nauk, 54:4(328) (1999), 47–74; Russian Math. Surveys, 54:4 (1999), 729–752

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. Yu. Nemirovski, “Topology of Hypersurface Complements in $\mathbb C^n$ and Rationally Convex Hulls”, Proc. Steklov Inst. Math., 235 (2001), 162–172  mathnet  mathscinet  zmath
    2. Nemirovski S., “Geometric methods in complex analysis”, European Congress of Mathematics, Progress in Mathematics, 202, 2001, 55–64  mathscinet  zmath  isi
    3. Slapar, M, “Real surfaces in elliptic surfaces”, International Journal of Mathematics, 16:4 (2005), 357  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Gompf R.E., “Stein Surfaces as Open Subsets of C-2”, J. Symplectic Geom., 3:4, SI (2005), 565–587  crossref  mathscinet  zmath  isi
    5. Joricke, B, “On the continuity principle”, Asian Journal of Mathematics, 11:1 (2007), 167  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Forstneric, F, “Stein structures and holomorphic mappings”, Mathematische Zeitschrift, 256:3 (2007), 615  crossref  mathscinet  zmath  isi  scopus  scopus
    7. Shcherbina, N, “On the set of complex points of a 2-sphere”, Annali Della Scuola Normale Superiore Di Pisa-Classe Di Scienze, 8:1 (2009), 73  mathscinet  zmath  isi
    8. Cerne M., Flores M., “Some Remarks on Hartogs' Extension Lemma”, Proceedings of the American Mathematical Society, 138:10 (2010), 3603–3608  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Coltoiu M., Joita C., “The Levi Problem in the Blow-Up”, Osaka J Math, 47:4 (2010), 943–947  mathscinet  zmath  isi
    10. Stefan Nemirovski, “Levi problem and semistable quotients”, Complex Variables and Elliptic Equations, 2011, 1  crossref  mathscinet  adsnasa  isi  scopus  scopus
    11. Prezelj J., Slapar M., “The Generalized Oka-Grauert Principle for 1-Convex Manifolds”, Michigan Math J, 60:3 (2011), 495–506  crossref  mathscinet  zmath  isi  scopus  scopus
    12. Slapar M., “Modeling Complex Points Up to Isotopy”, J. Geom. Anal., 23:4 (2013), 1932–1943  crossref  mathscinet  zmath  isi  scopus  scopus
    13. Gompf R.E., “Smooth Embeddings with Stein Surface Images”, J. Topol., 6:4 (2013), 915–944  crossref  mathscinet  zmath  isi  scopus  scopus
    14. Slapar M., “Cancelling Complex Points in Codimension Two”, Bull. Aust. Math. Soc., 88:1 (2013), 64–69  crossref  mathscinet  zmath  isi  scopus  scopus
    15. Forstneric F., “A Complex Surface Admitting a Strongly Plurisubharmonic Function But No Holomorphic Functions”, J. Geom. Anal., 25:1 (2015), 329–335  crossref  mathscinet  zmath  isi  scopus  scopus
    16. Stefan Nemirovski, Kyler Siegel, “Rationally convex domains and singular Lagrangian surfaces in
      $$\mathbb {C}^2$$
      C 2”, Invent. math, 2015  crossref  mathscinet  isi  scopus  scopus
    17. Forstneric F., “Stein Manifolds and Holomorphic Mappings: the Homotopy Principle in Complex Analysis, 2Nd Edition”, Stein Manifolds and Holomorphic Mappings: the Homotopy Principle in Complex Analysis, 2Nd Edition, Ergebnisse der Mathematik und Iher Grenzgebiete 3 Folge, 56, Springer-Verlag Berlin, 2017, 1–562  crossref  mathscinet  isi
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