[Paper Review] On properness of K-moduli spaces and optimal degenerations of Fano varieties

Harold Blum, Daniel Halpern-Leistner+2 more|arXiv (Cornell University)|Nov 3, 2020

Geometry and complex manifolds47 references10 citations

TL;DR

This paper establishes an algebraic strategy to prove the properness of K-moduli spaces of K-polystable Fano varieties by constructing a $Θ$-stratification on the moduli stack of Fano varieties. It shows that properness follows if the stability threshold $δ(X)$ of every K-unstable Fano variety is computed by a divisorial valuation, using a lexicographic minimization of normalized Futaki invariants to define unique optimal destabilizing test configurations.

ABSTRACT

We establish an algebraic approach to prove the properness of moduli spaces of K-polystable Fano varieties and reduce the problem to a conjecture on destabilizations of K-unstable Fano varieties. Specifically, we prove that if the stability threshold of every K-unstable Fano variety is computed by a divisorial valuation, then such K-moduli spaces are proper. The argument relies on studying certain optimal destabilizing test configurations and constructing a Theta-stratification on the moduli stack of Fano varieties.

Motivation & Objective

• To establish an algebraic approach to proving the properness of K-moduli spaces of K-polystable Fano varieties, avoiding reliance on analytic methods.
• To reduce the properness problem to a conjecture on destabilizations of K-unstable Fano varieties, specifically whether the stability threshold $δ(X)$ is computed by a divisorial valuation.
• To construct a $Θ$-stratification on the moduli stack of Fano varieties using optimal destabilizing test configurations.
• To define a unique, well-ordered destabilization process via lexicographic minimization of normalized Futaki invariants to ensure stratification compatibility with families.
• To generalize the Langton-type algorithm to moduli stacks of Fano varieties by introducing a formal perturbation to resolve non-uniqueness in minimizing test configurations.

Proposed method

• Uses a $Θ$-stratification framework on the moduli stack ${\mathcal{M}}^{Τ\text{-Fano}}_{n,V}$ of $Π$-Fano varieties to stratify unstable points via optimal destabilizing maps.
• Defines optimal destabilizing test configurations by minimizing the bi-valued invariant $\left(\frac{\mathrm{Fut}({\mathcal{X}})}{\|{\mathcal{X}}\|_{\rm m}}, \frac{\mathrm{Fut}({\mathcal{X}})}{\|{\mathcal{X}}\|_{2}}\right)$ under lexicographic order on $\mathbb{R}[\![\epsilon]\!]^2$.
• Introduces a formal perturbation $\bm{\mu}^\prime({\mathcal{X}},{\mathcal{D}}) = \frac{\mathrm{Fut}({\mathcal{X}},{\mathcal{D}})}{\|{\mathcal{X}},{\mathcal{D}}\|_{\rm m} + \epsilon\|{\mathcal{X}},{\mathcal{D}}\|_{2}}$ to resolve non-uniqueness in minimizing the Futaki invariant ratio.
• Applies the theory of $\Theta$-stratifications from [AHLH18] to the moduli stack of Fano varieties, relying on the existence of a unique optimal destabilizing test configuration.
• Reduces the problem of properness to the conjecture that $\delta(X)$ is computed by a divisorial valuation, which ensures the existence of a unique minimizer in the lexicographic order.
• Uses the fact that minimizing $\bm{\mu}^\prime$ is equivalent to minimizing $\bm{\mu}$ among filtrations when the conjecture holds, via formal power series comparison in $\epsilon$.

Experimental results

Research questions

• RQ1 Can the properness of K-moduli spaces of K-polystable Fano varieties be established via an algebraic method, independent of analytic techniques?
• RQ2 Under what conditions does the stability threshold $\delta(X)$ of a K-unstable Fano variety $X$ admit a minimizer among divisorial valuations?
• RQ3 Is there a canonical, unique way to define an optimal destabilizing test configuration for a K-unstable Fano variety, resolving the non-uniqueness issue in minimizing the Futaki invariant ratio?
• RQ4 Can a $\Theta$-stratification be constructed on the moduli stack of Fano varieties using a lexicographic minimization of normalized numerical invariants?
• RQ5 Does the existence of a unique optimal destabilizing test configuration imply the properness of the K-moduli space $M_{n,V}^{\rm Kps}$?

Key findings

• The properness of the K-moduli space $M_{n,V}^{\rm Kps}$ of K-polystable Fano varieties is implied by the conjecture that $\delta(X)$ is computed by a divisorial valuation.
• A unique optimal destabilizing test configuration exists for each K-unstable Fano variety if $\delta(X)$ is computed by a divisorial valuation, resolving non-uniqueness in minimizing the Futaki invariant ratio.
• The lexicographic minimization of the normalized Futaki invariant $\left(\frac{\mathrm{Fut}({\mathcal{X}})}{\|{\mathcal{X}}\|_{\rm m}}, \frac{\mathrm{Fut}({\mathcal{X}})}{\|{\mathcal{X}}\|_{2}}\right)$ yields a well-ordered $\Theta$-stratification on the moduli stack ${\mathcal{M}}^\text{Fano}_{n,V}$.
• The perturbation $\bm{\mu}^\prime({\mathcal{X}},{\mathcal{D}}) = \frac{\mathrm{Fut}({\mathcal{X}},{\mathcal{D}})}{\|{\mathcal{X}},{\mathcal{D}}\|_{\rm m} + \epsilon\|{\mathcal{X}},{\mathcal{D}}\|_{2}}$ defines the same semistability condition as the original invariant and ensures uniqueness of minimizers.
• The construction of the $\Theta$-stratification via this method is compatible with families and satisfies the required properties for Langton-type algorithms to apply.
• Corollary 8.7 confirms that the $\bm{\mu}^\prime$-based stratification coincides with the one in Theorem 7.3 under the conjecture, validating the approach.