An extension theorem for Kähler currents with analytic singularities

Abstract

Abstract. We prove an extension theorem for Kähler currents with analytic singularities in a Kähler class on a complex submanifold of a compact Kähler manifold.

Transcendental Morse inequality and generalised Okounkov bodies

“…When α is any ample class, there is a very similar theorem which has appeared in the proof of Proposition 4.13. However, the proof there relies on the difficult extension theorem in [CT14]. Here we give a simple and direct proof when X is a complex surface.…”

“…We choose t > 0 small enough such that ω − tY 1 is still a Kähler class. By the main theorem of [CT14], any Kähler current T ∈ (ω − tY 1 )| Y 1 with analytic singularities can be extended to a Kähler current T ∈ ω − tY 1 , thus we have…”

“…Here we give a simple and direct proof when X is a complex surface. Anyway, the idea of proof here is borrowed from [CT14].…”

Nakamaye’s theorem on complex manifolds

“…We conclude this article by noting that a refinement of the extension and gluing techniques that we just presented allowed Collins and the author [15] to prove the following extension theorem for Kähler currents: It is expected that this extension result should hold more generally when V is an analytic subvariety and T is just a closed positive current in the class [ω| V ] (in which case T should extend to a global closed positive current). This was proved by Coman-Guedj-Zeriahi [18] when X is projective and [ω] is a rational class, using rather different techniques.…”

On the extension of Kähler currents on compact Kähler manifolds: holomorphic retraction case

“…• When ω is a Kähler metric and ϕ is a quasi-psh function on V, which has analytic singularities, such that ω| V + √ −1∂ ∂ϕ > ǫω| V for some ǫ > 0, there is a quasi-psh function Φ on X, such that Φ| V = ϕ and ω + √ −1∂ ∂Φ > ǫ ′ ω on X by Collins-Tosatti [CT14]. • When ω is a Kähler metric and ϕ is a quasi-psh function with arbitrary singularity on V, such that ω| V + √ −1∂ ∂ϕ > ǫω| V for some ǫ > 0.…”