Continuity method with movable singularities for classical Monge-Ampère equations

Abstract

On a compact Kähler manifold (X, ω), we study the strong continuity of solutions with prescribed singularities of complex Monge-Ampère equations with convergent integrable Lebesgue densities. Then we address the strong continuity of solutions when the right hand sides are modified to includes all (log-)Kähler Einstein metrics with prescribed singularities. This leads to the closedness of a new continuity method when the densities are modified together with the prescribed singularities setting. For Monge-Ampère equations of Fano type, we also prove an openness result when the singularities decrease. Finally we deduce a strong stability result for (log-)Kähler Einstein metrics on semi-Kähler classes given as modifications of {ω}.

A relative Yau-Tian-Donaldson conjecture and stability thresholds

“…It is possible to extend the result of this paper to the log setting. Namely, replacing X by a log Fano pair (X, ∆), one can convert all the proofs/definitions to such setting getting for instance an algebro-geometric characterization of the existence of ∆-log [ψ]-KE metrics (see [52]). When Aut(X, [ψ]) is not discrete one can replace δ, δ, J NA (•; D) and the uniform Ding and K-stability notions by their G-equivariant versions similarly to [37,41].…”

“…for a smooth, closed and semipositive (1, 1)-form η, where p −1 I = O Y (−D) for an effective divisor D. Moreover the map v → ṽ := (v − u) • p induces a bijection between {v ∈ PSH(X, ω) : v ψ} and PSH(Y, η), which also preserve the total Monge-Ampère mass, i.e. X M A ω (v) = Y M A η (ṽ) [52,Lemma 4.6]. This allows to study the properties of PSH(Y, η) and of the nef class {η} directly over X.…”

“…where dµ = e 2ρ ω up to translate ρ (see [52,Proposition 4.3]). We can assume wlog that dµ is a probability measure: it is the adapted probability measure to h, i.e.…”

“…When ψ is a klt algebraic model type envelope with singularities encoded in (I, c), then by [52,Proposition 4.8] there is a 1-1 correspondence between [ψ]-KE metrics and log KE metrics on the weak Fano pair (Y, ∆) where π : Y → X is given as log-resolution of I while ∆ = cD − K Y /X for D effective divisor such that π −1 I = O Y (−D). The upshot is that "log KE metrics above X" are particular KE metrics with prescribed singularities.…”

“…Proof. As u t ∈ E 1 ψ for any t ∈ R ≥0 and (X, ψ) is klt, there exists ǫ > 0 such that e −ut ∈ L 2(1+ǫ) (X) for any t ∈ R ≥0 (see also [52,Proposition 3.10]). Thus by [26,Proposition 2.2] it follows that e −U is in L 2(1+ǫ) in a neighborhood of (x, t) for any (x, t) ∈ X × D * , i.e.…”