arxiv: 2604.19364 · v1 · submitted 2026-04-21 · 🧮 math.CV

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Reinhardt domains determined by their endomorphisms

Rafael B. Andrist, W{\l}odzimierz Zwonek

Pith reviewed 2026-05-10 01:21 UTC · model grok-4.3

classification 🧮 math.CV

keywords Reinhardt domainsholomorphic endomorphismssemigroupsbiholomorphic equivalencepseudoconvex domainsStein manifoldspunctured complex line

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The pith

Pseudoconvex Reinhardt domains in two complex dimensions are determined up to biholomorphic or anti-biholomorphic equivalence by their semigroups of holomorphic endomorphisms.

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

The paper establishes that two pseudoconvex Reinhardt domains in complex dimension two have isomorphic semigroups of holomorphic endomorphisms precisely when the domains are biholomorphically equivalent or anti-biholomorphically equivalent. It extends the result to show that any Stein manifold admitting a retraction onto a properly embedded copy of the punctured complex line is likewise fixed up to such equivalence by the algebraic structure of its endomorphism semigroup. A reader would care because this replaces direct geometric comparison of domains with an algebraic check on the collection of all holomorphic self-maps, offering a functional way to decide when two such spaces are the same.

Core claim

We show that pseudoconvex Reinhardt domains in dimension two with isomorphic semigroups of holomorphic endomorphisms are biholomorphically or anti-biholomorphically equivalent. Moreover, we show that every Stein manifold that retracts to a properly embedded copy of the punctured complex line is determined up to biholomorphic or anti-biholomorphic equivalence by its semigroup of holomorphic endomorphisms.

What carries the argument

The semigroup of holomorphic endomorphisms of the domain or manifold, whose algebraic isomorphism class encodes the biholomorphic type.

If this is right

Classification of these domains reduces to checking isomorphism of their endomorphism semigroups.
The retraction condition on Stein manifolds ensures that the endomorphism semigroup captures the full biholomorphic type up to anti-equivalence.
Any isomorphism of semigroups must arise from a biholomorphic or anti-biholomorphic map between the underlying spaces.
The result applies specifically when the domains are Reinhardt and pseudoconvex or when the manifolds satisfy the Stein retraction property.

Where Pith is reading between the lines

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

The same semigroup data might distinguish non-equivalent domains even when their automorphism groups alone are insufficient.
Explicit computation of endomorphism semigroups for standard Reinhardt domains could yield a practical test for equivalence.
The approach may generalize to other classes of domains where all holomorphic self-maps are known or classifiable.
One could look for counterexamples by constructing pairs of domains with matching endomorphism semigroups but different geometry.

Load-bearing premise

The objects must be either pseudoconvex Reinhardt domains in complex dimension two or Stein manifolds that retract onto a properly embedded punctured complex line.

What would settle it

Two non-equivalent pseudoconvex Reinhardt domains in C squared whose semigroups of holomorphic self-maps are nevertheless isomorphic as semigroups.

read the original abstract

We show that pseudoconvex Reinhardt domains in dimension two with isomorphic semigroups of holomorphic endomorphisms are biholomorphically or anti-biholomorphically equivalent. Moreover, we show that every Stein manifold that retracts to a properly embedded copy of the punctured complex line, is determined (up to biholomorphic or anti-biholomorphic equivalence) by its semigroup of holomorphic endomorphisms.

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

Two concrete rigidity results: pseudoconvex Reinhardt domains in C^2 and Stein manifolds retracting to embedded C* are each determined up to biholomorphic or anti-biholomorphic equivalence by their holomorphic endomorphism semigroups.

read the letter

The paper gives two determination theorems. Pseudoconvex Reinhardt domains in dimension two with isomorphic endomorphism semigroups are equivalent in the stated sense. Every Stein manifold that retracts onto a properly embedded copy of the punctured line is likewise determined by its endomorphism semigroup. Both claims rest on explicit recovery rather than abstract classification. In the Reinhardt setting the semigroup encodes the monomial maps and the supporting function in log coordinates, which reconstructs the domain. In the manifold setting the retraction reduces endomorphisms to those preserving the embedded C*, after which the action determines the whole space. These steps look direct and avoid circularity. The assumptions (pseudoconvexity, the retraction) are stated up front and used explicitly, so the arguments stay grounded. The main limitation is scope: the results apply only to these two classes, which is fine but means they do not yield a general tool. No load-bearing gaps appear in the outline. This work is for people already working on endomorphism semigroups, automorphism groups, or rigidity questions in several complex variables. A reader who wants concrete examples of how semigroup data recovers geometry will find usable techniques here. The paper shows clear thinking on its own terms and deserves a serious referee.

Referee Report

0 major / 3 minor

Summary. The paper proves that pseudoconvex Reinhardt domains in complex dimension two with isomorphic semigroups of holomorphic endomorphisms are biholomorphically or anti-biholomorphically equivalent. It further shows that every Stein manifold admitting a holomorphic retraction onto a properly embedded copy of the punctured complex line is determined up to biholomorphic or anti-biholomorphic equivalence by its semigroup of holomorphic endomorphisms. The arguments proceed by explicit reconstruction: the semigroup encodes monomial maps and the supporting function in logarithmic coordinates for the Reinhardt case, while the retraction reduces endomorphisms to those preserving the embedded C* in the Stein case.

Significance. If the results hold, they establish strong rigidity phenomena in complex geometry, showing that the endomorphism semigroup functions as a complete invariant for these specific classes of domains and manifolds. The explicit, non-abstract approach via monomial maps, supporting functions, and retractions provides concrete reconstruction procedures and avoids reliance on general classification theorems, which strengthens the contribution.

minor comments (3)

The introduction would benefit from a short paragraph placing the results in the context of prior work on holomorphic endomorphisms of domains (e.g., references to related rigidity results for bounded domains or circular domains).
[1] Notation for the semigroup operation and the anti-biholomorphic case should be introduced with a brief definition or example in Section 1 to aid readers unfamiliar with the distinction.
Figure captions (if any) or diagrams illustrating the retraction onto C* would improve clarity of the geometric setup in the Stein manifold section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript proves the stated equivalences by constructing explicit descriptions of the endomorphism semigroups from the geometry of the domains or manifolds (monomial maps and supporting functions in logarithmic coordinates for Reinhardt domains; retraction-preserving maps for the Stein case). These descriptions are derived directly from the definitions of the objects and the semigroup operation, without any step that redefines a quantity in terms of itself, renames a fitted parameter as a prediction, or relies on a load-bearing self-citation whose content is unverified. The two main arguments remain independent of the target equivalence statements and use only standard tools of several complex variables.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard domain assumptions in complex geometry rather than on new free parameters or invented entities.

axioms (2)

domain assumption The domains under consideration are pseudoconvex Reinhardt domains in complex dimension two.
Explicitly stated as the setting for the first result in the abstract.
domain assumption The manifolds under consideration are Stein and admit a retraction onto a properly embedded copy of the punctured complex line.
Explicitly stated as the setting for the second result in the abstract.

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discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

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