Reinhardt domains determined by their endomorphisms

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.

• The paper demonstrates that isomorphism of endomorphism semigroups forces (anti-)biholomorphic equivalence for pseudoconvex Reinhardt domains in ℂ².
• Methodology relies on conjugating bijections and rigidity arguments to transition set-theoretic invariants into holomorphic or anti-holomorphic maps.
• Results highlight the necessity of pseudoconvexity in establishing a complete complex-analytic invariant via dynamical semigroups.

Reinhardt Domains Determined by Their Endomorphisms

Introduction and Problem Framework

The paper establishes a profound link between the complex-analytic geometry of pseudoconvex Reinhardt domains and the semigroup structure of their holomorphic endomorphisms. The central question is whether an isomorphism between the semigroups $\enmo(D_1)$ and $\enmo(D_2)$ for domains $D_1$ and $D_2$ ensures the (anti-)biholomorphic equivalence of these domains. This investigates to what extent the "dynamical structure" of a complex space, given by its holomorphic self-maps, acts as a complete invariant for the underlying complex manifold.

The authors focus on two principal situations:

1. Pseudoconvex Reinhardt domains in $\mathbb{C}^2$,
2. Stein manifolds containing a properly embedded copy of the punctured complex line $\mathbb{C}_*$ as a retract.

They demonstrate, with strong generality, that in dimension two, pseudoconvex Reinhardt domains are determined up to biholomorphic or anti-biholomorphic equivalence by the isomorphism type of their endomorphism semigroups.

Summary of Main Results

A crucial lemma via Schreier establishes that an isomorphism between endomorphism semigroups arises from conjugation by a set-theoretic bijection $\varphi$ between the base spaces. The analysis therefore centers on characterizing when such a conjugating bijection must in fact be (anti-)biholomorphic.

The two key theorems are:

• Theorem for Pseudoconvex Reinhardt Domains in $\mathbb{C}^2$: If $D_1, D_2$ are pseudoconvex Reinhardt domains in $\mathbb{C}^2$ and $\enmo(D_2)$0 as semigroups, then $\enmo(D_2)$1 and $\enmo(D_2)$2 are (anti-)biholomorphically equivalent.
• Stein Manifolds with a Retract Isomorphic to $\enmo(D_2)$3: For a Stein manifold $\enmo(D_2)$4 that retracts onto a properly embedded copy of $\enmo(D_2)$5, $\enmo(D_2)$6 is determined, up to (anti-)biholomorphic equivalence, by its holomorphic endomorphism semigroup.

The method proceeds via a careful analysis of the structure of the endomorphisms, the properties of conjugating maps, and utilization of complex-analytic and group-theoretic invariants.

Detailed Mechanism and Technical Innovations

Conjugating Mappings and Their Rigidity

At the core is the observation that a conjugating bijection $\enmo(D_2)$7, which satisfies $\enmo(D_2)$8 for all endomorphisms $\enmo(D_2)$9 of $D_1$0, is forced under mild holomorphic-geometric conditions to be (anti-)biholomorphic rather than a more general bijection.

For the punctured plane $D_1$1, the conjugating mappings are shown, through functional equations and rigidity arguments, to be exactly (anti-)biholomorphic automorphisms. The proof involves careful consideration of the structure of holomorphic self-maps of $D_1$2 and properties of exponential function and its conjugations, leading to the conclusion that conjugating maps must be continuous, additive, and ultimately holomorphic or anti-holomorphic.

Structural Analysis of Reinhardt Domains

Pseudoconvex Reinhardt domains in higher dimensions are characterized using their logarithmic image and notions of convexity. The result leverages the fact that such domains are either biholomorphic to bounded domains or have an almost algebraic structure in their group of automorphisms.

For domains not equivalent to bounded ones, reductions show that considerations often localize to subspaces isomorphic to $D_1$3, allowing the use of previous rigidity results. In two dimensions, the full analytic classification of Reinhardt domains due to the irrationality or rationality of lines in their logarithmic image allows the authors to produce a complete description.

The paper provides a classification of holomorphic endomorphisms for the three main types of Reinhardt domains, with explicit normal forms for the endomorphisms and automorphisms. This is essential for tracking the effect of iso-conjugating maps and identifying algebraic invariants preserved under such conjugations.

Algebraic Consequences and Biholomorphic Equivalence

The paper proves that the additivity and multiplicativity properties imposed on the conjugating mapping together with analytic continuation imply holomorphic or anti-holomorphic regularity. This employs functional equations arising from the conjugacy of group actions and exploits the density of certain orbits.

Another key invariant is the set of involutive automorphisms, which differs between certain types of Reinhardt domains, allowing the authors to distinguish classes up to biholomorphic equivalence using only semigroup structure.

Limitations and a Non-Pseudoconvex Counterexample

The requirement of pseudoconvexity is shown to be essential. They exhibit uncountably many pairwise inequivalent non-pseudoconvex Reinhardt domains having trivial endomorphism semigroups (only rotations and constants), highlighting that the pseudoconvexity hypothesis is not merely technical but structurally necessary.

Implications and Future Directions

The results show that for broad classes of complex manifolds—the full family of two-dimensional pseudoconvex Reinhardt domains—the abstract semigroup of holomorphic self-maps is a complete invariant for their (anti-)biholomorphic equivalence class. This elevates holomorphic endomorphism semigroups to a tool of equal power as the holomorphic function ring, extending classical rigidity theorems based on Gelfand duality to a wider class of function spaces.

The extension to higher-dimensional cases is partially achieved under additional conditions (e.g., presence of rational lines in the logarithmic image), and the methods suggest a possible route for attacking the general question for Stein manifolds or even more general domains.

From a dynamical systems perspective, these results connect the structure of dynamical semigroups to analytic geometry, indicating that for many natural classes, the entire complex-analytic structure is recoverable from the "dynamical data" of holomorphic iterations.

Potential future directions involve:

• Completing the classification in higher dimensions without restriction on the rationality of lines in the logarithmic image.
• Characterizing the extent to which weaker semigroup structures (e.g., considering only proper maps or automorphisms) determine the complex structure.
• Extending the framework to other classes of complex manifolds, such as non-Stein or non-holomorphically separable spaces.

Conclusion

This work rigorously establishes that, for pseudoconvex Reinhardt domains in $D_1$4, the semigroup of holomorphic endomorphisms forms a complete invariant for (anti-)biholomorphic equivalence. This deepens the understanding of the interplay between complex-analytic, algebraic, and dynamical structures and suggests broad prospects for further research into the categorical rigidity of complex manifolds under dynamical semigroup invariants.