[Paper Review] Semi-continuity of complex singularity exponents and Kähler-Einstein metrics on Fano orbifolds

Jean-Pierre Demailly, Janós Kollár|arXiv (Cornell University)|Oct 22, 1999

Geometry and complex manifolds25 references19 citations

TL;DR

This paper introduces the complex singularity exponent of plurisubharmonic functions and proves its semi-continuity, providing a powerful analytic tool for studying singularities in complex geometry. Using this result, the authors establish a simplified criterion for the existence of Kähler-Einstein metrics on Fano orbifolds, leading to three new examples of rigid Del Pezzo surfaces with quotient singularities that admit such metrics.

ABSTRACT

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and Varchenko, mostly for the study of hypersurface singularities. The plurisubharmonic version is somehow based on a reduction to the algebraic case, but it also takes into account more quantitative informations of great interest for complex analysis and complex differential geometry. We give as an application a new derivation of criteria for the existence of Kähler-Einstein metrics on certain Fano orbifolds, following Nadel's original ideas (but with a drastic simplication in the technique, once the semi-continuity result is taken for granted). In this way, 3 new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are obtained.

Motivation & Objective

• To develop a quantitative analytic framework for measuring singularities of plurisubharmonic functions via complex singularity exponents.
• To establish a general semi-continuity theorem for these exponents, extending known results in algebraic geometry to the analytic setting.
• To apply the semi-continuity result to simplify and re-derive criteria for the existence of Kähler-Einstein metrics on Fano orbifolds.
• To construct new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities using the refined analytic criterion.
• To provide effective, computable conditions for Kähler-Einstein metrics by analyzing curvature and multiplier ideal sheaves in weighted projective spaces.

Proposed method

• Define the complex singularity exponent $ c_K(φ) $ as the supremum of $ c ≥ 0 $ such that $ \exp(-2c\varphi) $ is $ L^1 $-integrable near a compact set $ K $.
• Introduce the Arnold multiplicity $ \lambda_K(\varphi) = c_K(\varphi)^{-1} $ as a dual measure of singularity severity.
• Prove the semi-continuity of $ c_K(\varphi) $ in the $ L^1 $-topology on compact subsets, using $ L^2 $ estimates for $ \bar{\partial} $ and multiplier ideal sheaves.
• Apply the semi-continuity result to reduce the existence problem of Kähler-Einstein metrics on Fano orbifolds to checking curvature positivity and multiplier ideal conditions.
• Use torus actions and degeneration techniques to bound intersection numbers on weighted projective hypersurfaces, enabling effective verification of the Kähler-Einstein criterion.
• Compute the ratio $ \rho_a $ involving weights $ a_i $, degree $ d $, and $ t = d+1 $, and conclude that $ \rho_a < 1 $ implies existence of Kähler-Einstein metric.

Experimental results

Research questions

• RQ1 Can the complex singularity exponent of a plurisubharmonic function be characterized in a way that captures both analytic and algebraic singularity data?
• RQ2 Does the complex singularity exponent exhibit semi-continuity under $ L^1 $-convergence of psh functions, and can this be proven using $ L^2 $ estimates?
• RQ3 Can the semi-continuity of singularity exponents be used to simplify or re-derive criteria for the existence of Kähler-Einstein metrics on Fano orbifolds?
• RQ4 What effective conditions on weighted projective hypersurfaces ensure the existence of Kähler-Einstein metrics, particularly in the case of Del Pezzo surfaces?
• RQ5 Are there new examples of rigid Del Pezzo surfaces with quotient singularities that admit Kähler-Einstein metrics, and how can they be constructed?

Key findings

• The complex singularity exponent $ c_K(\varphi) $ is lower semi-continuous on the space of locally $ L^1 $ plurisubharmonic functions under $ L^1 $-convergence on compact sets.
• The effective version of semi-continuity ensures that $ \exp(-2c\psi) \to \exp(-2c\varphi) $ in $ L^1 $ norm for $ c < c_K(\varphi) $, when $ \psi \to \varphi $ in $ L^1 $.
• The criterion for Kähler-Einstein metrics on Fano orbifolds is simplified to checking that $ (-K_X) \cdot Z > \frac{2}{3}(-K_X)^2 $ and that $ T_X \otimes \mathcal{O}_X(d - a_0 - a_2) $ is nef.
• Three new examples of rigid Kähler-Einstein Del Pezzo surfaces with quotient singularities are constructed: one with weights $ (11,49,69,128) $, $ d=256 $, $ \rho_a \simeq 0.875696 $, and another with $ (13,35,81,128) $, $ d=256 $, $ \rho_a \simeq 0.955311 $.
• A third example is found with weights $ (9,15,17,20) $, $ d=60 $, where improved analysis of the tangent bundle restriction confirms the Kähler-Einstein condition.
• The three examples are rigid as weighted hypersurfaces, meaning no nontrivial deformations exist, and they lead to non-regular Sasakian-Einstein 5-manifolds via [BG00].