Strict positivity of Kähler–Einstein currents

Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $...

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Bibliographic Details
Published inForum of mathematics. Sigma Vol. 12
Main Authors Guedj, Vincent, Guenancia, Henri, Zeriahi, Ahmed
Format Journal Article
LanguageEnglish
Published Cambridge, UK Cambridge University Press 01.01.2024
Subjects
14B05
32Q20
32Q25
32S30
Algebraic and Complex Geometry
Singularities
32Q25
32Q20
32S30
14B05
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Summary:Kähler–Einstein currents, also known as singular Kähler–Einstein metrics, have been introduced and constructed a little over a decade ago. These currents live on mildly singular compact Kähler spaces X and their two defining properties are the following: They are genuine Kähler–Einstein metrics on $X_{\mathrm {reg}}$ , and they admit local bounded potentials near the singularities of X. In this note, we show that these currents dominate a Kähler form near the singular locus, when either X admits a global smoothing, or when X has isolated smoothable singularities. Our results apply to klt pairs and allow us to show that if X is any compact Kähler space of dimension three with log terminal singularities, then any singular Kähler–Einstein metric of nonpositive curvature dominates a Kähler form.
ISSN:2050-5094
2050-5094
DOI:10.1017/fms.2024.54