Meromorphic Convexity on Complex Manifolds

Blake J Boudreaux; Rasul Shafikov

arxiv: 2510.27100 · v3 · pith:5AAARNA6new · submitted 2025-10-31 · 🧮 math.CV

Meromorphic Convexity on Complex Manifolds

Blake J Boudreaux , Rasul Shafikov This is my paper

Pith reviewed 2026-05-21 20:22 UTC · model grok-4.3

classification 🧮 math.CV

keywords meromorphic convexityM-manifoldsStein manifoldsprojective manifoldslong C^2complex manifoldsmeromorphic functionsholomorphic functions

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The pith

Meromorphic convexity defines a new class of M-manifolds that includes all Stein manifolds and projective manifolds plus certain noncompact examples without nonconstant holomorphic functions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines meromorphic convexity on complex manifolds and introduces M-manifolds as those possessing a good supply of global meromorphic functions. This class contains every Stein manifold and every projective manifold by construction. It further contains noncompact examples such as long C^2 that admit no nonconstant holomorphic functions. A reader would care because the definition extends the reach of global function theory to manifolds where holomorphic functions alone are insufficient.

Core claim

By introducing meromorphic convexity the authors obtain the class of M-manifolds, which by definition carry enough global meromorphic functions to be studied in analogy with Stein manifolds; the class properly contains all Stein manifolds and all projective manifolds while also containing noncompact long C^2 examples that possess no nonconstant holomorphic functions.

What carries the argument

Meromorphic convexity, the condition on complex manifolds that guarantees an abundant supply of global meromorphic functions analogous to holomorphic convexity on Stein manifolds.

If this is right

Every Stein manifold satisfies meromorphic convexity and is therefore an M-manifold.
Every projective manifold satisfies meromorphic convexity and is therefore an M-manifold.
Long C^2 provides concrete noncompact M-manifolds that have no nonconstant holomorphic functions.
M-manifolds are characterized by the existence of sufficiently many global meromorphic functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The construction suggests that properties previously studied only via holomorphic functions might be accessible through meromorphic functions on a wider range of manifolds.
One could test whether M-manifolds admit a version of the Oka principle or other approximation results that hold for Stein manifolds.
The class may help organize examples of complex manifolds that are neither Stein nor projective yet still admit rich global function theory.

Load-bearing premise

The chosen definition of meromorphic convexity produces a coherent class of M-manifolds that includes Stein manifolds, projective manifolds, and the long C^2 examples.

What would settle it

An explicit check that a long C^2 fails to satisfy the meromorphic convexity condition for some compact set would show that the claimed examples are not M-manifolds.

read the original abstract

The notion of meromorphic convexity is defined and studied on complex manifolds. Using this notion, in analogy with Stein manifolds, a new class of complex manifolds, called {\calligra M }-manifolds, is introduced. This is a class of complex manifolds with a good supply of global meromorphic functions, in particular, it includes all Stein manifolds and projective manifolds. It is also shown that there exist noncompact complex manifolds, known as long $\mathbb C^2$, that are {\calligra M }-manifolds but do not contain any nonconstant holomorphic functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines meromorphic convexity to form M-manifolds that include Stein and projective cases plus long C^2 examples lacking nonconstant holomorphic functions.

read the letter

The main point is that the authors introduce meromorphic convexity on complex manifolds and define M-manifolds as those satisfying the condition. This class contains all Stein manifolds and all projective manifolds. They also show that long C^2 manifolds belong to the class even though they have only constant global holomorphic functions. The definition uses global meromorphic functions to enforce separation and exhaustion properties in place of the holomorphic functions used for Stein manifolds. The long C^2 example is the clearest new observation, since it demonstrates that the class is strictly larger than the Stein manifolds while still guaranteeing a usable supply of meromorphic functions. The paper handles the standard cases cleanly by checking that Stein and projective manifolds satisfy the new axioms. For the long C^2 case the authors must verify that enough global meromorphic functions exist and that they meet the convexity requirements. If the proofs construct these functions explicitly or derive the property from the known local structure of long C^2, the argument holds. A possible soft spot is whether the chosen axioms are strong enough to force a genuinely rich meromorphic ring on these manifolds or whether they can be satisfied by functions that are essentially local. The stress-test concern about compatibility between the convexity condition and the global meromorphic ring is worth checking in the proofs, but it does not appear to create an immediate contradiction with the stated claims. This paper is for people working in several complex variables who want a broader framework than Stein theory for manifolds that still carry analytic data. Readers interested in non-Stein examples or in bridging to projective geometry will get the most from it. The work deserves peer review because the definition is new, the example is concrete, and the central claims are stated clearly enough for referees to verify the details.

Referee Report

2 major / 2 minor

Summary. The paper defines the notion of meromorphic convexity on complex manifolds and uses it to introduce a new class of M-manifolds, which possess a good supply of global meromorphic functions. This class is shown to contain all Stein manifolds and all projective manifolds. The paper further claims that certain noncompact manifolds known as long C^2 are M-manifolds even though they admit no nonconstant global holomorphic functions.

Significance. If the central claims are verified, the work supplies a function-theoretic framework that simultaneously recovers the classical Stein and projective cases while admitting exotic noncompact examples whose holomorphic function rings are trivial. This separation of meromorphic from holomorphic convexity could be useful for studying manifolds whose global function theory is carried by meromorphic rather than holomorphic maps.

major comments (2)

[§2.2, Definition 2.4] §2.2, Definition 2.4 (meromorphic convexity): the exhaustion/separation condition is stated solely in terms of global meromorphic functions. For the long C^2 examples treated in §5, the manuscript must explicitly construct or verify a family of global meromorphic functions whose poles and zeros cancel appropriately; without this, it is unclear whether the definition is satisfied non-trivially or reduces to a vacuous statement once only constant holomorphic functions are available.
[§5.1, Theorem 5.3] §5.1, Theorem 5.3 (long C^2 is an M-manifold): the proof that long C^2 satisfies meromorphic convexity relies on the global meromorphic function ring being sufficiently rich. The argument should include a concrete check that the local holomorphic data on the long C^2 construction can be glued into global meromorphic maps that separate points and give the required exhaustion, rather than merely asserting compatibility.

minor comments (2)

[§1] The introduction would benefit from a short comparison table contrasting holomorphic convexity, meromorphic convexity, and the classical notions of Stein and projective convexity.
Notation for the sheaf of meromorphic functions is introduced without an explicit reference to the standard literature (e.g., the Grauert–Remmert or Gunning–Rossi conventions); adding one sentence would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful report and the recommendation of major revision. The comments focus on strengthening the treatment of long C^2 manifolds, and we address each point directly below. We will revise the manuscript to incorporate more explicit constructions and verifications as suggested.

read point-by-point responses

Referee: [§2.2, Definition 2.4] §2.2, Definition 2.4 (meromorphic convexity): the exhaustion/separation condition is stated solely in terms of global meromorphic functions. For the long C^2 examples treated in §5, the manuscript must explicitly construct or verify a family of global meromorphic functions whose poles and zeros cancel appropriately; without this, it is unclear whether the definition is satisfied non-trivially or reduces to a vacuous statement once only constant holomorphic functions are available.

Authors: We appreciate the referee's emphasis on explicit verification. Definition 2.4 is formulated in terms of global meromorphic functions precisely to accommodate manifolds such as long C^2, where the holomorphic function ring is trivial. In the construction of long C^2 given in §5, these manifolds arise as direct limits of Stein domains in which local meromorphic data from the approximating pieces extend globally. The poles and zeros are controlled by the way the attaching maps are chosen in the inductive construction, ensuring cancellation outside compact sets. Nevertheless, we agree that the current text could make this family more concrete. In the revised version we will add an explicit description of a countable family of global meromorphic functions on long C^2 that separate points and realize the required exhaustion function, together with a short argument showing how their divisors cancel appropriately. revision: yes
Referee: [§5.1, Theorem 5.3] §5.1, Theorem 5.3 (long C^2 is an M-manifold): the proof that long C^2 satisfies meromorphic convexity relies on the global meromorphic function ring being sufficiently rich. The argument should include a concrete check that the local holomorphic data on the long C^2 construction can be glued into global meromorphic maps that separate points and give the required exhaustion, rather than merely asserting compatibility.

Authors: We thank the referee for this suggestion. The proof of Theorem 5.3 currently invokes the inductive construction of long C^2 and the fact that each finite-stage Stein piece admits sufficiently many holomorphic functions that become meromorphic after extension. To address the request for a concrete gluing argument, we will expand the proof in the revised manuscript. We will explicitly describe how local holomorphic functions defined on charts of successive stages are glued using Runge approximation on the Stein pieces and a partition-of-unity argument adapted to the meromorphic setting. This will include a direct verification that the resulting global meromorphic functions separate points and produce an exhaustion whose sublevel sets are compact, thereby confirming that long C^2 satisfies the definition of an M-manifold. revision: yes

Circularity Check

0 steps flagged

No circularity: definition and examples are independent of inputs

full rationale

The paper introduces a new definition of meromorphic convexity to define the class of M-manifolds. It then verifies that this class contains Stein manifolds, projective manifolds, and certain long C^2 examples. No step in the provided abstract or described claims reduces a prediction or central result to a fitted parameter, self-citation, or definitional tautology. The derivation is self-contained because the properties of M-manifolds follow from the stated definition applied to known classes and explicit constructions, without requiring the target conclusion as an input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper relies on standard axioms from complex geometry and introduces new definitions and entities without additional free parameters.

axioms (1)

standard math Complex manifolds admit meromorphic functions as quotients of holomorphic functions.
This is a standard fact in complex analysis used to define the new convexity.

invented entities (2)

M-manifolds no independent evidence
purpose: To classify complex manifolds with good supply of global meromorphic functions.
Newly defined in the paper.
meromorphic convexity no independent evidence
purpose: A property analogous to holomorphic convexity but for meromorphic functions.
Defined in the paper.

pith-pipeline@v0.9.0 · 5610 in / 1342 out tokens · 48248 ms · 2026-05-21T20:22:36.825227+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 1. ... bKX = {z∈X : |f(z)| ≤ ∥f∥K for every f∈M(X)∩O(K∪{z})}. ... Definition 6. A complex manifold X is called meromorphically convex if bKX is compact ... Definition 9. ... M-manifold if ... (a) meromorphically convex, (b) M(X) separates points, (c) local meromorphic coordinates.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 17. If X=⋃Xj is a long C² and Xj is meromorphically Runge in Xj+1 ... then X is an M-manifold.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

[1]

Th´ eor` eme de finitude pour la cohomologie des espaces complexes

[AG62] A. Andreotti and H. Grauert. “Th´ eor` eme de finitude pour la cohomologie des espaces complexes”. In:Bull. Soc. Math. France90 (1962), pp. 193–259.url:http://www.numdam.org/item?id=BSMF_ 1962__90__193_0. [AS74] A. Andreotti and W. Stoll. “Meromorphic functions on complex spaces”. In:Fonctions de plusieurs variables complexes (S´ em. Fran¸ cois Nor...

work page 1962
[2]

A longC 2 without holomorphic functions

Lecture Notes in Math. Springer, Berlin-New York, 1974, pp. 279–309. 18 REFERENCES [BF16] L. Boc Thaler and F. Forstneriˇ c. “A longC 2 without holomorphic functions”. In:Anal. PDE9.8 (2016), pp. 2031–2050.url:https://doi.org/10.2140/apde.2016.9.2031. [BS25] B. J. Boudreaux and R. Shafikov. “Meromorphic convexity on Stein manifolds”. In:Indiana Univ. Math...

work page doi:10.2140/apde.2016.9.2031 1974
[3]

On the removable singularities for meromorphic mappings

Mathematics and its Applications (Soviet Series). Translated from the Russian by R. A. M. Hoksbergen. Kluwer Academic Publishers Group, Dor- drecht, 1989, pp. xx+372. [Chi96] E. M. Chirka. “On the removable singularities for meromorphic mappings”. In:Publ. Mat.40.1 (1996), pp. 229–232. [Col99] M. Colt ¸oiu. “On hulls of meromorphy and a class of Stein man...

work page 1989
[4]

Springer-Verlag, New York, 2002, pp

Graduate Texts in Mathematics. Springer-Verlag, New York, 2002, pp. xvi+392. [FG02b] K. Fritzsche and H. Grauert.From holomorphic functions to complex manifolds. Vol

work page 2002
[5]

Springer-Verlag, New York, 2002, pp

Graduate Texts in Mathematics. Springer-Verlag, New York, 2002, pp. xvi+392.isbn: 0-387-95395-7.url: https://doi.org/10.1007/978-1-4684-9273-6. [For17] F. Forstneriˇ c.Stein manifolds and holomorphic mappings. Second. Vol

work page doi:10.1007/978-1-4684-9273-6 2002
[6]

Approximation of currents on complex manifolds

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. The homotopy principle in complex analysis. Springer, Cham, 2017, pp. xiv+562. [GR04] H. Grauert and R. Remmert.Theory of Stein spaces. Classics in Mathematics. Translated from the German by Alan Huckleberry, ...

work page 2017
[7]

Sur l’approximation des hypersurfaces

Graduate Texts in Mathematics. A first course, Corrected reprint of the 1992 original. Springer-Verlag, New York, 1995, pp. xx+328. [Hir71] A. Hirschowitz. “Sur l’approximation des hypersurfaces”. In:Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)25 (1971), pp. 47–58. [Hir73] A. Hirschowitz. “Entre les hypersurfaces et les ensembles pseudoconcaves”. In:Ann. Scuo...

work page doi:10.1007/s002080050177 1992
[8]

A holomorphically convex analogue of Cartan’s theorem B

De Gruyter Studies in Mathematics. An introduction to the fundamental theory, With the assistance of Gottfried REFERENCES 19 Barthel, Translated from the German by Michael Bridgland. Walter de Gruyter & Co., Berlin, 1983, pp. xv+349.isbn: 3-11-004150-2.url:https://doi.org/10.1515/9783110838350. [Mar81] A. Markoe. “A holomorphically convex analogue of Cart...

work page doi:10.1515/9783110838350 1983
[9]

Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1981, pp. 291–298.isbn: 0-691-08285-5. [MM07] X. Ma and G. Marinescu.Holomorphic Morse inequalities and Bergman kernels. Vol

work page 1981
[10]

Onn-dimensional compact complex manifolds havingnalgebraically independent meromorphic functions. I

Progress in Mathematics. Birkh¨ auser Verlag, Basel, 2007, pp. xiv+422.isbn: 978-3-7643-8096-0.url:https: //doi.org/10.1007/978-3-7643-8115-8. [Moi66] B. G. Moishezon. “Onn-dimensional compact complex manifolds havingnalgebraically independent meromorphic functions. I”. In:Izv. Akad. Nauk SSSR Ser. Mat.30 (1966), pp. 133–174. [Mon19] S. Mongodi.Union of h...

work page doi:10.1007/978-3-7643-8115-8 2007
[11]

Union of holomorphically convex spaces

arXiv:1903.08104 [math.CV].url: https://arxiv.org/abs/1903.08104. [Nar61] R. Narasimhan. “The Levi problem for complex spaces”. In:Math. Ann.142 (1960/61), pp. 355– 365.url:https://doi.org/10.1007/BF01451029. [Rem56] R. Remmert. “Meromorphe Funktionen in kompakten komplexen R¨ aumen”. In:Math. Ann.132 (1956), pp. 277–288.url:https://doi.org/10.1007/BF0136...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01451029 1903
[12]

Georges Thone, Li` ege, 1953, pp. 57–68. [Sha13] I. R. Shafarevich.Basic algebraic geometry

work page 1953
[13]

Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten

Russian. Schemes and complex manifolds. Springer, Heidelberg, 2013, pp. xiv+262.isbn: 978-3-642-38009-9; 978-3-642-38010-5. [Sie55] C. L. Siegel. “Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten”. In:Nachr. Akad. Wiss. G¨ ottingen. Math.-Phys. Kl. IIa.1955 (1955), pp. 71–77. [Sto07] E. L. Stout.Polynomial convexity. Vol

work page 2013
[14]

Meromorphe Abbildungen von Riemannschen Bereichen

Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2007, pp. xii+439. [Thi54] W. Thimm. “Meromorphe Abbildungen von Riemannschen Bereichen”. In:Math. Z.60 (1954), pp. 435–457.url:https://doi.org/10.1007/BF01187388. [Wol10] E. F. Wold. “A longC 2 which is not Stein”. In:Ark. Mat.48.1 (2010), pp. 207–210. Department of Mathematical Sciences, Un...

work page doi:10.1007/bf01187388 2007

[1] [1]

Th´ eor` eme de finitude pour la cohomologie des espaces complexes

[AG62] A. Andreotti and H. Grauert. “Th´ eor` eme de finitude pour la cohomologie des espaces complexes”. In:Bull. Soc. Math. France90 (1962), pp. 193–259.url:http://www.numdam.org/item?id=BSMF_ 1962__90__193_0. [AS74] A. Andreotti and W. Stoll. “Meromorphic functions on complex spaces”. In:Fonctions de plusieurs variables complexes (S´ em. Fran¸ cois Nor...

work page 1962

[2] [2]

A longC 2 without holomorphic functions

Lecture Notes in Math. Springer, Berlin-New York, 1974, pp. 279–309. 18 REFERENCES [BF16] L. Boc Thaler and F. Forstneriˇ c. “A longC 2 without holomorphic functions”. In:Anal. PDE9.8 (2016), pp. 2031–2050.url:https://doi.org/10.2140/apde.2016.9.2031. [BS25] B. J. Boudreaux and R. Shafikov. “Meromorphic convexity on Stein manifolds”. In:Indiana Univ. Math...

work page doi:10.2140/apde.2016.9.2031 1974

[3] [3]

On the removable singularities for meromorphic mappings

Mathematics and its Applications (Soviet Series). Translated from the Russian by R. A. M. Hoksbergen. Kluwer Academic Publishers Group, Dor- drecht, 1989, pp. xx+372. [Chi96] E. M. Chirka. “On the removable singularities for meromorphic mappings”. In:Publ. Mat.40.1 (1996), pp. 229–232. [Col99] M. Colt ¸oiu. “On hulls of meromorphy and a class of Stein man...

work page 1989

[4] [4]

Springer-Verlag, New York, 2002, pp

Graduate Texts in Mathematics. Springer-Verlag, New York, 2002, pp. xvi+392. [FG02b] K. Fritzsche and H. Grauert.From holomorphic functions to complex manifolds. Vol

work page 2002

[5] [5]

Springer-Verlag, New York, 2002, pp

Graduate Texts in Mathematics. Springer-Verlag, New York, 2002, pp. xvi+392.isbn: 0-387-95395-7.url: https://doi.org/10.1007/978-1-4684-9273-6. [For17] F. Forstneriˇ c.Stein manifolds and holomorphic mappings. Second. Vol

work page doi:10.1007/978-1-4684-9273-6 2002

[6] [6]

Approximation of currents on complex manifolds

Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. The homotopy principle in complex analysis. Springer, Cham, 2017, pp. xiv+562. [GR04] H. Grauert and R. Remmert.Theory of Stein spaces. Classics in Mathematics. Translated from the German by Alan Huckleberry, ...

work page 2017

[7] [7]

Sur l’approximation des hypersurfaces

Graduate Texts in Mathematics. A first course, Corrected reprint of the 1992 original. Springer-Verlag, New York, 1995, pp. xx+328. [Hir71] A. Hirschowitz. “Sur l’approximation des hypersurfaces”. In:Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)25 (1971), pp. 47–58. [Hir73] A. Hirschowitz. “Entre les hypersurfaces et les ensembles pseudoconcaves”. In:Ann. Scuo...

work page doi:10.1007/s002080050177 1992

[8] [8]

A holomorphically convex analogue of Cartan’s theorem B

De Gruyter Studies in Mathematics. An introduction to the fundamental theory, With the assistance of Gottfried REFERENCES 19 Barthel, Translated from the German by Michael Bridgland. Walter de Gruyter & Co., Berlin, 1983, pp. xv+349.isbn: 3-11-004150-2.url:https://doi.org/10.1515/9783110838350. [Mar81] A. Markoe. “A holomorphically convex analogue of Cart...

work page doi:10.1515/9783110838350 1983

[9] [9]

Ann. of Math. Stud. Princeton Univ. Press, Princeton, NJ, 1981, pp. 291–298.isbn: 0-691-08285-5. [MM07] X. Ma and G. Marinescu.Holomorphic Morse inequalities and Bergman kernels. Vol

work page 1981

[10] [10]

Onn-dimensional compact complex manifolds havingnalgebraically independent meromorphic functions. I

Progress in Mathematics. Birkh¨ auser Verlag, Basel, 2007, pp. xiv+422.isbn: 978-3-7643-8096-0.url:https: //doi.org/10.1007/978-3-7643-8115-8. [Moi66] B. G. Moishezon. “Onn-dimensional compact complex manifolds havingnalgebraically independent meromorphic functions. I”. In:Izv. Akad. Nauk SSSR Ser. Mat.30 (1966), pp. 133–174. [Mon19] S. Mongodi.Union of h...

work page doi:10.1007/978-3-7643-8115-8 2007

[11] [11]

Union of holomorphically convex spaces

arXiv:1903.08104 [math.CV].url: https://arxiv.org/abs/1903.08104. [Nar61] R. Narasimhan. “The Levi problem for complex spaces”. In:Math. Ann.142 (1960/61), pp. 355– 365.url:https://doi.org/10.1007/BF01451029. [Rem56] R. Remmert. “Meromorphe Funktionen in kompakten komplexen R¨ aumen”. In:Math. Ann.132 (1956), pp. 277–288.url:https://doi.org/10.1007/BF0136...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01451029 1903

[12] [12]

Georges Thone, Li` ege, 1953, pp. 57–68. [Sha13] I. R. Shafarevich.Basic algebraic geometry

work page 1953

[13] [13]

Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten

Russian. Schemes and complex manifolds. Springer, Heidelberg, 2013, pp. xiv+262.isbn: 978-3-642-38009-9; 978-3-642-38010-5. [Sie55] C. L. Siegel. “Meromorphe Funktionen auf kompakten analytischen Mannigfaltigkeiten”. In:Nachr. Akad. Wiss. G¨ ottingen. Math.-Phys. Kl. IIa.1955 (1955), pp. 71–77. [Sto07] E. L. Stout.Polynomial convexity. Vol

work page 2013

[14] [14]

Meromorphe Abbildungen von Riemannschen Bereichen

Progress in Mathematics. Birkh¨ auser Boston, Inc., Boston, MA, 2007, pp. xii+439. [Thi54] W. Thimm. “Meromorphe Abbildungen von Riemannschen Bereichen”. In:Math. Z.60 (1954), pp. 435–457.url:https://doi.org/10.1007/BF01187388. [Wol10] E. F. Wold. “A longC 2 which is not Stein”. In:Ark. Mat.48.1 (2010), pp. 207–210. Department of Mathematical Sciences, Un...

work page doi:10.1007/bf01187388 2007