[Paper Review] Birational superrigidity and slope stability of Fano manifolds

Yuji Odaka, Takuzo Okada|arXiv (Cornell University)|Jul 20, 2011

Geometry and complex manifolds28 references4 citations

TL;DR

This paper establishes a link between birational superrigidity and slope stability in Fano manifolds of Fano index 1 with base point free anticanonical linear systems. By leveraging Seshadri constants and singularity bounds from superrigidity, the authors prove that such Fano manifolds are slope stable, providing a new algebro-geometric criterion for stability that complements K-stability and Yau-Tian-Donaldson conjectures.

ABSTRACT

We show a relation between the birational superrigidity of Fano manifold and its slope stability in the sense of Ross-Thomas.

Motivation & Objective

• To establish a connection between birational superrigidity and slope stability in Fano manifolds, addressing a gap between geometric rigidity and moduli-theoretic stability.
• To show that Fano manifolds with Fano index 1 and base point free anticanonical systems are slope stable, under mild assumptions.
• To generalize results from K-stability to slope stability by exploiting Seshadri constants and singularity control from superrigidity.
• To provide evidence for the broader conjecture that all Fano manifolds of Picard rank 1 are K-semistable.
• To extend the framework to G-equivariant settings, proving G-equivariant slope stability under natural invariance and base point freeness conditions.

Proposed method

• Utilizes Seshadri constants as a tool to bound destabilizing ideal sheaves, linking geometric invariants to stability.
• Applies the main result from [OS12] on Seshadri constants implying K-stability, adapted here to slope stability.
• Employs the notion of log maximal singularity freeness to control singularities of pluri-anticanonical divisors.
• Uses the base point freeness of $|-K_X|$ to deform ideal sheaves while preserving $G$-invariance in the equivariant setting.
• Applies the LCT (log canonical threshold) lower bound from $G$-birationally superrigidity to control singularities in higher codimension.
• Combines divisorial and higher-codimensional analysis via valuations and flag ideals to verify slope stability conditions.

Experimental results

Research questions

• RQ1 Can birational superrigidity imply slope stability in Fano manifolds?
• RQ2 What role do Seshadri constants and singularity bounds play in establishing slope stability?
• RQ3 Does the base point freeness of $|-K_X|$ enable stronger stability conclusions?
• RQ4 Can the results be extended to $G$-equivariant settings with finite group actions?
• RQ5 Is slope stability a universal feature of Fano manifolds of Picard rank 1?

Key findings

• Theorem 1.1 establishes that a birationally superrigid Fano manifold of Fano index 1 with base point free $|-K_X|$ is slope stable.
• Theorem 1.2 strengthens this by proving slope stability under the assumptions of Picard rank 1, Fano index 1, no log maximal singularities, and base point freeness of $|-K_X|$.
• The proof relies on upper bounds for Seshadri constants derived from the mildness of singularities in pluri-anticanonical systems, a consequence of birational superrigidity.
• The key inequality $\mathop{\mathrm{Sesh}}\nolimits(I,(X,-K_X)) \leq \frac{a_i}{c_i} < \frac{(n+1)a_i}{nc_i}$ is derived using deformation techniques and base point freeness.
• A generalization to $G$-equivariant settings is proven: if $X$ is $G$-birationally superrigid and satisfies $G$-invariant base point freeness and divisor bounds, then $X$ is $G$-equivariantly slope stable.
• The paper provides evidence for Conjecture 5.1: every Fano manifold of Picard rank 1 is K-semistable, with supporting results on slope stability along divisors and curves.