Bubbling of Kähler-Einstein metrics

Abstract

We prove the finite step termination of bubble trees for singularity formation of polarized Kähler-Einstein metrics in the non-collapsing situation. We also raise several questions and conjectures in connection with algebraic geometry and Riemannian geometry.

Some models for bubbling of (log) Kähler–Einstein metrics

“…In this last section we will first propose a tentative more invariant algebraic picture for detecting the bubbles, slightly reinforcing Sun's conjecture in [38]. Finally, we will formulate a more general framework to study bubbling, by speculating about the existence of what we call multiscale K-moduli compactifications T M K , that is birational modifications of the K-moduli spaces which parameterize algebraic spaces supporting all the possible multiscale bubble limits of non-collapsing KE metrics.…”

“…However, we also point out (Sect. 3.5) that if one instead considers the more general case of log KE metrics (so conical along a divisor) in this dimension, the bubbling picture seems to be more complicated, and related to the jumping phenomena as recently pointed out in [38].…”

“…The claim is that there is the following inductive procedure (terminating in finite steps), mirroring Sun's differential geometric termination of bubbles. Following [38] we call minimal bubble a non-cone rescaled limit whose tangent cone at infinity is equal to the tangent cone at the singularity.…”

“…• The algebraic finite step iterations of the algorithm correspond to the finite number of possible metric bubbles, as described in [38].…”

“…Proof By the established deep general theory of convergence, we know that such metrics GH convergence to a singular KE metric on X 0 whose metric tangent cone is the Calabi's ansatz cone metric on the canonical bundle of the KE Fano on the smooth quadric hypersurface. Moreover, by [38] we know that there should be a rescaled minimal bubble that has such cone as its tangent cone at infinity, and it is a negative weight deformation of it [10]. However, versal deformations of an A 1 singularity are trivial: there is only a smoothing.…”