[Paper Review] On parametrization, optimization and triviality of test configurations

Yuji Odaka|arXiv (Cornell University)|Jan 3, 2012

Geometry and complex manifolds9 references4 citations

TL;DR

This paper establishes a parametrization of test configurations for polarized varieties via spherical buildings, proving the existence of optimal destabilizing test configurations through results of Kempf and Rousseau. It resolves pathological 'almost trivial' test configurations by redefining K-stability, showing they correspond to a dense subset of the spherical building and are excluded under the amended definition, thus justifying the formalism of Odaka (2011).

ABSTRACT

We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of "optimal" destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give an account of the recent slight amendment to definition of K-stability after Li-Xu, from two other viewpoints: from the one parameter subgroups and from the author's blow up formalism.

Motivation & Objective

• To provide a geometric parametrization of test configurations using spherical buildings, extending classical GIT ideas.
• To establish the existence of an optimal destabilizing test configuration for any unstable polarized variety, analogous to Harder-Narasimhan filtrations.
• To clarify the role of 'almost trivial' test configurations—pathological cases that must be excluded from K-stability—using the spherical building formalism.
• To reconcile the amended definition of K-stability (after Li-Xu) with two perspectives: one-parameter subgroups and the blow-up formalism of Odaka.
• To demonstrate that the set of pathological test configurations forms a dense subset in the spherical building, justifying their removal in stability theory.

Proposed method

• Uses the spherical building Δ(GL(H⁰(X,L))) as a parameter space for test degenerations of exponent 1, leveraging Tits' theory.
• Applies the theorem of Rousseau and Kempf (Tit’s center conjecture) to prove existence of a unique minimizer of the normalized weight function.
• Defines equivalence relations (T-equivalence, U-equivalence) on test configurations to form test degenerations and test classes.
• Introduces the weight function ν on the geometric realization of the spherical building, derived from GIT weights and normalized by a Weyl-invariant norm.
• Identifies the locus of 'almost trivial' test configurations as those corresponding to rational points in S′(T,ℚ), where multiple minimal weights occur.
• Uses lower semicontinuity of the weight function (via Mumford and Kempf) to prove continuity and lower semicontinuity of ν on |Δ(G)|_ℚ.

Experimental results

Research questions

• RQ1 Can test configurations for a polarized variety be uniformly parametrized using a known geometric object like a spherical building?
• RQ2 Does every unstable polarized variety admit a unique optimal destabilizing test configuration, analogous to the Harder-Narasimhan filtration?
• RQ3 What is the geometric and algebraic nature of 'almost trivial' test configurations, and why do they necessitate a modification of the K-stability definition?
• RQ4 How does the spherical building formalism help clarify the relationship between one-parameter subgroups and K-stability?
• RQ5 Can the blow-up formalism of Odaka be justified under the revised definition of K-stability that excludes pathological test configurations?

Key findings

• Test degenerations of exponent 1 for a very ample polarized variety (X,L) are parametrized by the rational points of the spherical building Δ(GL(H⁰(X,L))).
• The existence and uniqueness of an optimal destabilizing test configuration follows from the Kempf-Rousseau theorem applied to the spherical building parameter space.
• The set of 'almost trivial' test configurations corresponds to a dense subset of the spherical building, specifically the image of ∪_T S′(T,ℚ) under the natural map.
• These pathological configurations are excluded in the amended definition of K-stability, which is now consistent with the blow-up formalism of Odaka (2011).
• The normalized Chow weight function ν is continuous and lower semicontinuous on the geometric realization of the spherical building, ensuring the existence of a minimizer.
• The optimal destabilizing test configuration is realized as the point in |Δ(G)|_ℚ with minimal (negative) normalized weight, uniquely determined by the GIT weight function.