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

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

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English version:
Russian Mathematical Surveys, 1999, 54:4, 729–752

Bibliographic databases:

Document Type: Article
UDC: 515.1
MSC: Primary 32E10, 32D10, 32J20; Secondary 32F15, 53C23, 57N13, 14J35

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

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• http://mi.mathnet.ru/eng/umn179
• https://doi.org/10.4213/rm179
• http://mi.mathnet.ru/eng/umn/v54/i4/p47

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. Nemirovski S., “Geometric methods in complex analysis”, European Congress of Mathematics, Progress in Mathematics, 202, 2001, 55–64
3. Slapar, M, “Real surfaces in elliptic surfaces”, International Journal of Mathematics, 16:4 (2005), 357
4. Gompf R.E., “Stein Surfaces as Open Subsets of C-2”, J. Symplectic Geom., 3:4, SI (2005), 565–587
5. Joricke, B, “On the continuity principle”, Asian Journal of Mathematics, 11:1 (2007), 167
6. Forstneric, F, “Stein structures and holomorphic mappings”, Mathematische Zeitschrift, 256:3 (2007), 615
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
8. Cerne M., Flores M., “Some Remarks on Hartogs' Extension Lemma”, Proceedings of the American Mathematical Society, 138:10 (2010), 3603–3608
9. Coltoiu M., Joita C., “The Levi Problem in the Blow-Up”, Osaka J Math, 47:4 (2010), 943–947
10. Stefan Nemirovski, “Levi problem and semistable quotients”, Complex Variables and Elliptic Equations, 2011, 1
11. Prezelj J., Slapar M., “The Generalized Oka-Grauert Principle for 1-Convex Manifolds”, Michigan Math J, 60:3 (2011), 495–506
12. Slapar M., “Modeling Complex Points Up to Isotopy”, J. Geom. Anal., 23:4 (2013), 1932–1943
13. Gompf R.E., “Smooth Embeddings with Stein Surface Images”, J. Topol., 6:4 (2013), 915–944
14. Slapar M., “Cancelling Complex Points in Codimension Two”, Bull. Aust. Math. Soc., 88:1 (2013), 64–69
15. Forstneric F., “A Complex Surface Admitting a Strongly Plurisubharmonic Function But No Holomorphic Functions”, J. Geom. Anal., 25:1 (2015), 329–335
16. Stefan Nemirovski, Kyler Siegel, “Rationally convex domains and singular Lagrangian surfaces in
$$\mathbb {C}^2$$
C 2”, Invent. math, 2015
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