[Paper Review] The GIT-stability of Polarised Varieties via discrepancy

Yuji Odaka|arXiv (Cornell University)|Jul 10, 2008

Geometry and complex manifolds10 references15 citations

TL;DR

This paper establishes a fundamental link between GIT stability and singularity theory in algebraic geometry by proving that K-semistable polarized varieties under the $(*)$ conditions must have semi-log-canonical singularities. Using discrepancy theory and Donaldson-Futaki invariants, it generalizes the classical nodal singularity condition in curve moduli to higher dimensions, showing that the stability condition enforces a precise measure of singularities via discrepancy, with stronger results in the Fano case where K-semistability implies log terminal singularities.

ABSTRACT

We prove that various GIT semistabilities of polarized varieties imply semi-log-canonicity.

Motivation & Objective

• To understand the nature of singularities allowed in GIT-stable moduli spaces of polarized varieties.
• To generalize the classical result that stable curves have only nodal singularities to higher-dimensional varieties.
• To establish a precise connection between algebro-geometric stability (K-stability) and birational-geometric invariants (discrepancy).
• To show that K-semistability implies semi-log-canonicity, and in the Fano case, log canonicity.
• To demonstrate that the results are optimal by showing the converse holds in Calabi-Yau and canonical cases.

Proposed method

• Uses the notion of $(*)$-schemes: equidimensional, S2, Gorenstein in codimension 1, and Q-Cartier canonical divisor.
• Applies the theory of discrepancies via log resolutions and normalization to analyze singularities.
• Employs Donaldson-Futaki invariants as algebro-geometric invariants of stability, computing them via test configurations.
• Constructs flag ideals and their blow-ups over $X \times \mathbb{A}^1$ to model degenerations and compute invariants.
• Reduces higher-dimensional cases to lower-dimensional ones via general hyperplane sections to simplify analysis.
• Uses the log minimal model program and the existence of log terminal models to derive contradictions in the non-terminal case.

Experimental results

Research questions

• RQ1 What kind of singularities are allowed in K-semistable polarized varieties satisfying the $(*)$ conditions?
• RQ2 Can the classical nodal singularity result for curves be generalized to higher-dimensional varieties using stability theory?
• RQ3 Is K-semistability sufficient to ensure that singularities are at most semi-log-canonical?
• RQ4 Does K-semistability in the Fano case imply stronger singularity bounds, such as log terminality?
• RQ5 Is the implication from K-semistability to semi-log-canonicity the strongest possible, given known converse results?

Key findings

• K-semistability of a polarized variety $(X,L)$ satisfying $(*)$ implies that $X$ has only semi-log-canonical singularities.
• In the Fano case, K-semistability of $(X, \mathcal{O}_X(-mK_X))$ implies $X$ is log terminal and hence normal.
• Asymptotic (Chow or Hilbert) semistability implies semi-log-canonicity, as it implies K-semistability.
• The $\bar{K}$-semistability condition also implies semi-log-canonicity, confirming its strength relative to K-stability.
• The results are optimal: the converse holds for Calabi-Yau and canonical varieties, showing the bound cannot be improved.
• The discrepancy of the normalization and the conductor divisor play a key role in the proof, with negative discrepancies indicating instability.