[Paper Review] Birational superrigidity and K-stability of Fano complete intersections of index one (with an appendix written jointly with Charlie Stibitz)

Ziquan Zhuang|arXiv (Cornell University)|Feb 23, 2018

Geometry and complex manifolds21 references11 citations

TL;DR

This paper establishes birational superrigidity and K-stability for smooth Fano complete intersections of index one and codimension $ r $ in $ \mathbb{P}^{n+r} $ when $ n \geq 10r $, using a new criterion for K-stability that combines log canonical thresholds with movable boundaries. The key contribution is proving that such varieties are both birationally superrigid and admit Kähler-Einstein metrics, with an application to the complete intersection of a quadric and cubic in $ \mathbb{P}^5 $.

ABSTRACT

We prove that every smooth Fano complete intersection of index $1$ and codimension $r$ in $\mathbb{P}^{n+r}$ is birationally superrigid and K-stable if $n\ge 10r$. We also propose a generalization of Tian's criterion of K-stability and, as an application, prove the K-stability of the complete intersection of a quadric and a cubic in $\mathbb{P}^5$. In the appendix (written jointly with C. Stibitz), we prove the conditional birational superrigidity of Fano complete intersections of higher index in large dimension.

Motivation & Objective

• Address the long-standing conjecture that smooth Fano complete intersections of index one and large dimension are birationally superrigid and K-stable.
• Overcome difficulties in verifying K-stability and birational superrigidity for special members of families by introducing a new, more flexible criterion.
• Generalize Tian’s criterion for K-stability to handle cases where singularities of pairs are hard to control, particularly for non-general members.
• Establish a bridge between birational superrigidity and K-stability by showing that canonical singularities of movable pairs imply K-stability.
• Provide a conditional superrigidity result for higher index Fano complete intersections in large dimensions via an appendix.

Proposed method

• Introduce a new criterion for K-stability based on the log canonicity of pairs $ (X, \frac{1}{n+1}D + \frac{n-1}{n+1}M) $, where $ D \sim_{\mathbb{Q}} -K_X $ and $ M $ is a movable boundary.
• Use the fact that movable boundaries typically have mild singularities, allowing the weighted combination to be more likely to be log canonical than in Tian’s criterion.
• Apply the theory of log canonical thresholds and multiplicity bounds via complete intersection subvarieties to estimate singularities of movable pairs.
• Utilize exponential growth estimates for the number of lattice points in certain polytopes, $ \sigma_{n,\lambda} > c^n $, to control the dimension of subvarieties where singularities may occur.
• Combine the maximal singularity method with dimension bounds on base loci of movable systems to rule out birational maps to Mori fiber spaces.
• Use the Hilbert-Samuel multiplicity and log canonical threshold relations from [dFEM-mult-and-lct] to deduce log canonicity from multiplicity bounds.

Experimental results

Research questions

• RQ1 Can a new criterion for K-stability be formulated that is more effective than Tian’s criterion for Fano varieties with complicated singularities in pairs?
• RQ2 Are smooth Fano complete intersections of index one and codimension $ r $ in $ \mathbb{P}^{n+r} $ birationally superrigid when $ n \geq 10r $?
• RQ3 Does the complete intersection of a quadric and a cubic in $ \mathbb{P}^5 $ admit a Kähler-Einstein metric, despite not being birationally superrigid?
• RQ4 Can the superrigidity of Fano complete intersections of higher index be established conditionally in large dimensions?
• RQ5 Is there a quantitative link between birational superrigidity and K-stability that holds even when the variety is not superrigid?

Key findings

• Theorem 1.2 establishes that every smooth Fano complete intersection of index one, codimension $ r $, and dimension $ n \geq 10r $ in $ \mathbb{P}^{n+r} $ is birationally superrigid.
• Theorem 1.3 proves that such varieties are K-stable, hence admit Kähler-Einstein metrics, under the same dimension condition $ n \geq 10r $.
• An explicit improvement is found for codimension $ r = 2 $, where $ n \geq 12 $ suffices for birational superrigidity and K-stability.
• The complete intersection of a quadric and a cubic in $ \mathbb{P}^5 $ is shown to be K-stable, despite not being birationally superrigid.
• The new criterion in Theorem 1.5 implies both Theorem 1.1 and Theorem 1.4 as special cases and provides a more flexible path to K-stability.
• The appendix proves conditional birational superrigidity for Fano complete intersections of higher index in large dimensions using exponential growth of lattice point counts in polytopes.