[Paper Review] Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces

Ivan Cheltsov, Yanir A. Rubinstein+1 more|arXiv (Cornell University)|Jul 18, 2018

Geometry and complex manifolds39 references4 citations

TL;DR

This paper develops new local intersection estimates to compute basis log canonical thresholds (blct) on logarithmic surfaces, providing a geometric method to assess log canonicity. It proves that asymptotically log del Pezzo surfaces admit Kähler–Einstein edge metrics for all but finitely many families, confirming a key part of a conjecture on K-stability and linking blct to Tian's greatest Ricci lower bound invariant.

ABSTRACT

The purpose of this article is to develop techniques for estimating basis log canonical thresholds on logarithmic surfaces. To that end, we develop new local intersection estimates that imply log canonicity. Our main motivation and application is to show the existence of Kahler-Einstein edge metrics on all but finitely many families of asymptotically log del Pezzo surfaces, partially confirming a conjecture of two of us. In an appendix we show that the basis log canonical threshold of Fujita-Odaka coincides with the greatest lower Ricci bound invariant of Tian.

Motivation & Objective

• To develop intrinsic geometric techniques for estimating basis log canonical thresholds (blct) on logarithmic surfaces.
• To establish a necessary and sufficient condition for K-stability via blct, enabling the existence of Kähler–Einstein edge metrics.
• To resolve a conjecture on asymptotically log del Pezzo surfaces by showing Kähler–Einstein edge metrics exist for all but finitely many families.
• To prove the equivalence between Fujita–Odaka’s basis log canonical threshold and Tian’s greatest Ricci lower bound invariant.

Proposed method

• Introduce new local intersection number criteria to detect log canonicity, replacing reliance on global or toric methods.
• Use vanishing order estimates for basis divisors to bound blct from above and below.
• Apply scaling properties of blct under rational multiples of the anticanonical divisor to analyze stability thresholds.
• Leverage Kähler–Einstein edge (KEE) metric existence theorems and their analytic properties to relate blct to the Kähler–Einstein threshold β(X).
• Use limits of log pairs (X, (1−b)Δm/m, −bKX) as m→∞ to connect discrete and continuous invariants.
• Employ results from K-stability theory, including uniform K-stability and Mabuchi energy bounds, to derive contradictions when blct exceeds 1.

Experimental results

Research questions

• RQ1 Can local intersection estimates be used to compute basis log canonical thresholds on logarithmic surfaces without relying on toric or global geometry?
• RQ2 Does the basis log canonical threshold coincide with Tian’s greatest Ricci lower bound invariant in the log setting?
• RQ3 For which families of asymptotically log del Pezzo surfaces do Kähler–Einstein edge metrics exist?
• RQ4 What is the precise relationship between the Kähler–Einstein threshold β(X) and the basis log canonical threshold blct∞(X,−KX)?

Key findings

• The basis log canonical threshold of Fujita–Odaka coincides with Tian’s greatest Ricci lower bound invariant, establishing a deep link between algebraic and analytic invariants.
• All but finitely many families of asymptotically log del Pezzo surfaces admit Kähler–Einstein edge metrics, confirming a key part of a conjecture by two of the authors.
• The paper proves that blct∞(X,−KX) = β(X), showing that the basis log canonical threshold equals the Kähler–Einstein threshold for log del Pezzo surfaces.
• New local intersection criteria provide a geometric, intrinsic method to test log canonicity, offering a stronger alternative to existing algebraic estimates.
• The method extends beyond toric surfaces and applies to non-toric, non-canonical blow-ups of P², enabling broader applicability to log surfaces.
• The proof strategy uses a contradiction argument via KEE metric existence and stability theory, showing that blct∞(X,−bKX) > 1 leads to a contradiction when b > β(X).