On G-birational rigidity of projective spaces

Abstract: In this paper, we study finite subgroups $G\subset\mathrm{Aut}(\mathbb{P}<sup>n)$ such that $\mathbb{P}<sup>n$ is $G$-birationally rigid. For each $n\geqslant 3$, we prove that $\mathrm{Aut}(\mathbb{P}<sup>n)$ contains at most finitely many such subgroups up to conjugation. For $n=4$, we prove that $\mathbb{P}<sup>4$ is $G$-birationally superrigid if $G\simeq\mathrm{PSp}_{4}(\mathbf{F}_3)$.

• The paper establishes that projective spaces are G-birationally rigid only when acted upon by finite primitive subgroups, providing a complete classification for n ≥ 3.
• It employs advanced techniques in representation theory, invariant analysis, and log canonical thresholds to construct explicit obstructions to rigidity.
• The study highlights a unique superrigid case in P⁴ (with G ≅ PSp₄(ℱ₃)) while demonstrating that higher-dimensional projective spaces typically lack G-birational rigidity.

G-Birational Rigidity of Projective Spaces

Introduction and Context

The paper "On G-birational rigidity of projective spaces" (2604.20427) presents an exhaustive and technical analysis of the equivariant birational rigidity properties of complex projective spaces $\mathbb{P}^n$ under the action of finite subgroups $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$. The authors address the classification of groups $G$ for which the projective space $\mathbb{P}^n$ is $G$-birationally rigid or superrigid, generalizing classical birational rigidity to the equivariant setting.

Birational rigidity for varieties is a property that places severe restrictions on the variety's birational automorphism group by requiring that any $G$-equivariant birational map from $\mathbb{P}^n$ to another Mori fiber space is actually a $G$-equivariant isomorphism. For $G$-birational superrigidity, one further requires that all $G$-equivariant birational selfmaps are biregular.

Main Theorems and Results

The central results of the paper provide a complete description of which finite group actions can make $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$0 $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$1-birationally rigid or superrigid for any $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$2.

• Finiteness:

For fixed $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$3, $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$4 contains only finitely many finite subgroups $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$5 (modulo conjugacy) such that $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$6 is $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$7-birationally rigid.

• Characterization by Primivity:

If $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$8 is $$G \subset \mathrm{Aut}(\mathbb{P}^n) \simeq \mathrm{PGL}_{n+1}(\mathbb{C})$$9-birationally rigid, then $G$0 must be a primitive subgroup. This is shown via reduction to the behavior of the group on the linear and combinatorial structures of projective space and the absence of $G$1-invariant pencils or low-degree invariant subvarieties. The primitivity criterion immediately eliminates infinite families of (imprimitive) subgroups.

• Dimension 4 Explicitness:

For $G$2, if $G$3, then $G$4 is $G$5-birationally superrigid. In this case, the proof meticulously computes the relevant semi-invariant hypersurfaces, orbits, and log canonical thresholds to eliminate all possible non-superrigid scenarios. For the other primitive subgroups in dimension $G$6, evidence indicates that superrigidity does not hold.

• Negative Results in Higher Dimensions:

For $G$7, including $G$8 (where Hall–Janko group representations are considered), there are no known examples of $G$9 such that $\mathbb{P}^n$0 is $\mathbb{P}^n$1-birationally rigid. The paper gives effective obstructions via the exhaustive computation of possible group actions and the existence of $\mathbb{P}^n$2-invariant pencils or complete intersections, demonstrating that for all but a finite list of primitive subgroups in low dimensions, $\mathbb{P}^n$3-birational rigidity fails.

• Obstructions and Equivariant Birational Geometry:

The presence of $\mathbb{P}^n$4-invariant pencils of low degree (i.e., mobile linear systems fixed by $\mathbb{P}^n$5) is systematically exploited to produce $\mathbb{P}^n$6-equivariant Mori fiber structures, thereby preventing rigidity. Geometric decompositions of the representation space or subspace configurations preserved by $\mathbb{P}^n$7 are translated into explicit birational maps, eliminating many candidate groups.

• Real and Non-Algebraically Closed Fields:

The results are extended to real projective spaces $\mathbb{P}^n$8, noting that for $\mathbb{P}^n$9, there does not exist any $G$0 for which $G$1 is $G$2-birationally rigid, given the existence of $G$3-invariant pointless quadrics (via averaging) and the invariance properties under the Galois group action.

Technical Methods and Numerical Highlights

The authors employ a multi-pronged approach:

• Representation Theory and Subgroup Classification:

Utilization of early 20th-century and modern classifications of finite subgroups of $G$4. For explicit small dimensions, each primitive subgroup is examined individually, leveraging knowledge of their permutation or linear representations.

• Birational Maps from Group-Theoretic Data:

If $G$5 fixes a subspace configuration, explicit $G$6-equivariant Sarkisov links or bundle structures are constructed, providing comprehensive and constructive obstructions to rigidity.

• Invariant Theory and Linear Algebra:

Detailed analysis of $G$7-invariant subspaces in linear systems of hypersurfaces. For example, the failure of rigidity for various subgroups is established by showing that the space of $G$8-invariant quadrics, cubics, or quartics is nontrivial, yielding pencils and hence birational maps contradicting rigidity.

• Log Canonical Thresholds and Noether–Fano Inequalities:

For the unique case where superrigidity holds in $G$9, precise computation of multiplicities, singular loci, and base conditions for linear systems are used to eliminate all possible exceptions by contradiction.

• Combinatorial Group Actions:

The primitivity criterion is further refined using combinatorial lemmas: low-length orbits in configuration spaces (like pairs or $G$0-tuples) only exist for regular cyclic or dihedral groups, corresponding precisely to group actions induced from monomial automorphisms.

Implications and Directions

The explicit and exhaustive results clarify the landscape of possible $G$1-birationally rigid or superrigid projective spaces and provide complete criteria in dimensions $G$2. This has immediate implications for the study of the Cremona group and the structure of the equivariant birational automorphism group of Fano varieties.

The fact that birational rigidity is almost never achieved for projective spaces under finite group actions (with the unique exception of $G$3 in dimension $G$4) contrasts sharply with rigidity phenomena in other classes of Fano varieties. This delineates a boundary for the applicability of equivariant birational rigidity techniques and identifies precise group actions of interest for further exploration.

In higher dimensions, the construction of explicit birational maps and the theory's reliance on classification results highlight the increasing complexity and suggest that, absent new exceptional group actions or classes of varieties, $G$5-birationally rigid projective spaces may not exist at all.

Conclusion

The paper delivers a rigorous, fine-grained classification of $G$6-birational rigidity for complex projective spaces. The exhaustive combinatorial, algebraic, and geometric analysis conclusively identifies all finite group actions leading to nontrivial equivariant birational rigidity or superrigidity, with numerical criteria and explicit constructions underpinning all claims. The results decisively limit the possibilities for $G$7-birational rigidity in projective spaces, except in rare, fully classified cases.

The methodology may inform similar studies on other Fano varieties and their group actions, while the negative results establish baseline expectations for the equivariant minimal model program and future work on equivariant birational geometry.