Strict positivity of Kähler-Einstein currents

Abstract

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 $3$ with log terminal singularities, then any singular Kähler-Einstein metric of non-positive curvature dominates a Kähler form.

41 pages, v2: expanded section 1.3

Cited by in corpus (1)

• Singular cscK metrics on smoothable varieties