- The paper establishes that the fixed-level equivariant Tian's alpha invariant equals the equivariant global log canonical threshold for all Fano manifolds with compact group actions.
- It employs functional-analytic methods and operator-averaging on Bergman potentials, bypassing traditional toric or combinatorial approaches.
- The results offer new insights into Kähler–Einstein geometry, birational geometry, and equivariant K-stability in complex geometry.
Equivariant Fixed-Level Demailly Identity for Fano Manifolds
Introduction and Problem Statement
This paper addresses the relationship between two prominent invariants associated with Fano manifolds and compact group actions: the fixed-level equivariant Tian's alpha invariant, αk,G, and the fixed-level equivariant global log canonical threshold, $\glct_{k,G}$. Given a Fano manifold X, a line bundle L=−KX, a compact subgroup $G \subset \Aut(X)$, and k∈N such that −kKX is basepoint-free, Jin and Rubinstein introduced these equivariant, fixed-level invariants and proved their equality for toric Fano manifolds, asking whether the equality $\alpha_{k,G} = \glct_{k,G}$ holds in general ["Tian's stabilization problem for toric Fanos" (JR25)]. This question was previously open for arbitrary Fano manifolds and group actions.
The authors of this paper resolve Jin and Rubinstein's question affirmatively for all compact complex manifolds with basepoint-free line bundles and any compact group action. The main technical contribution is a purely functional-analytic proof utilizing equivariant Bergman potentials and operator-theoretic averaging, without reliance on toric or combinatorial structures. A key analytic tool is the semicontinuity of complex singularity exponents by Demailly and Kollár [DK01].
Definitions of Invariants
For a compact complex manifold X and a basepoint-free line bundle L with compact group $\glct_{k,G}$0 acting on $\glct_{k,G}$1 and $\glct_{k,G}$2, the two main invariants are formulated at a fixed integer level $\glct_{k,G}$3:
- Fixed-level equivariant Tian's alpha invariant $\glct_{k,G}$4:
Defined as the supremal $\glct_{k,G}$5 such that, for all $\glct_{k,G}$6-invariant Bergman potentials $\glct_{k,G}$7 from $\glct_{k,G}$8, one has
$\glct_{k,G}$9
Here, X0 is the set of all X1-invariant Bergman potentials at level X2.
- Fixed-level equivariant global log canonical threshold X3:
Defined as
X4
where X5 runs over all nonzero X6-invariant subspaces and X7 measures integrability rates of local sums of squared norms of sections over X8.
These quantities, analytically defined via integrability and algebraically via multiplier ideals, are central in K-stability, the existence of Kähler–Einstein metrics, and birational geometry.
Main Results
The principal theorem establishes that for any compact complex manifold X9, any basepoint-free holomorphic line bundle L=−KX0, any compact group L=−KX1 of automorphisms, and any positive integer L=−KX2, one has
L=−KX3
In the Fano case with L=−KX4, this resolves Jin–Rubinstein's fixed-level equivariant question for all Fano manifolds and group actions.
Crucially, the proof does not utilize the toric structure nor rely on explicit generators or combinatorics of L=−KX5-invariant sections, substantially broadening the scope of previous results.
Proof Outline
The proof can be divided into several key phases:
- Parametrization via Frame Operators:
Bergman potentials are parametrized by positive definite Hermitian "frame operators" on L=−KX6. Operator-averaging (integration over L=−KX7 using Haar measure) ensures invariance and commutation with the L=−KX8-action, reducing the problem to analyzing these commuting operators.
- Upper Bound (L=−KX9):
For any $G \subset \Aut(X)$0, the authors construct a degeneration by approximating the frame operator via a sequence converging to the orthogonal projection onto a worst-case (i.e., minimally integrable) $G \subset \Aut(X)$1-invariant subspace. This leverages semicontinuity and monotonic convergence to show that such $G \subset \Aut(X)$2 is not admissible for $G \subset \Aut(X)$3.
- Uniform Integrability (Key Technical Step):
Establishes a uniform $G \subset \Aut(X)$4 bound for the integrals defining $G \subset \Aut(X)$5 over all possible $G \subset \Aut(X)$6-invariant subspaces, using the effective semicontinuity of complex singularity exponents (Demailly–Kollár [DK01]) and the compactness of the stratification of invariant subspaces inside Grassmannians.
- Lower Bound ($G \subset \Aut(X)$7):
For any $G \subset \Aut(X)$8, an explicit majorization argument controls all $G \subset \Aut(X)$9-invariant potentials from below using their spectral projections, ultimately bounding the integral needed for k∈N0 via the minimal log canonical threshold among the invariant subspaces.
The intersection of upper and lower bounds gives the desired equality.
Implications and Theoretical Significance
The result unifies analytic and algebraic approaches to the study of multiplier ideals and alpha invariants at finite level with equivariance, removing structural constraints such as toric symmetry. The fixed-level, as opposed to asymptotic, nature of the theorem has implications for finite generation problems and the effectivity of K-stability criteria under group actions.
The approach demonstrates the power of operator-theoretic and functional-analytic techniques, such as equivariant averaging and spectral theory, in addressing algebraic-geometric problems. The reliance on only the k∈N1-linearization and basepoint freeness underscores that the result holds in much greater generality than was recognized.
In Kähler–Einstein geometry, these thresholds govern the existence of canonical metrics via the alpha-invariant criterion. Consequently, the theorem provides new tools to analyze equivariant K-stability and log canonical thresholds in moduli problems and in the study of Fano varieties with symmetry.
Future Directions
Several avenues for future research are opened by these results:
- Beyond Basepoint-Freeness:
Investigating whether similar identities or comparison theorems persist when k∈N2 is not basepoint-free, possibly via additional asymptotic or birational techniques.
- Algebro-Geometric Properties:
Exploring the fine behavior of these invariants with respect to variation in families, singular degenerations, and in relation to moduli of Fano varieties with group action.
- Effective Bounds and Computational Aspects:
Utilizing the explicit operator-theoretic framework to obtain effective, computable lower bounds or algorithms for computing k∈N3 and k∈N4 in concrete families.
- Interactions with K-stability:
Applying the fixed-level identity to sharpen the understanding of the equivalence between various notions of stability and integrability invariants in the equivariant setting.
Conclusion
This work rigorously establishes the equality between the fixed-level equivariant Tian's alpha invariant and the fixed-level equivariant global log canonical threshold for arbitrary Fano manifolds and compact group actions. The proof is notable for its generality and purely functional-analytic nature, which departs from combinatorial or toric dependencies. The result has significant implications for Kähler–Einstein geometry, birational geometry, and the study of K-stability with symmetries, and provides a blueprint for further investigations in the equivariant setting.
References:
- Demailly, J.-P., Kollár, J., "Semi-continuity of complex singularity exponents and Kähler–Einstein metrics on Fano orbifolds," Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 4, 525–556. [DK01]
- Jin, C., Rubinstein, Y. A., "Tian’s stabilization problem for toric Fanos," Geom. Topol. 29 (2025), no. 5, 2609–2652. [JR25]
- H. Ju et al., "Automated Conjecture Resolution with Formal Verification," (Ju et al., 4 Apr 2026). [Ju+26]
