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

Abstract

In the present paper, we show that given a compact Kähler manifold (X, ω) with a Kähler metric ω, and a complex submanifold V ⊂ X of positive dimension, if V has a holomorphic retraction structure in X, then any quasi-plurisubharmonic function ϕ on V such that ω| V + √ −1∂ ∂ϕ ≥ εω| V with ε > 0 can be extended to a quasi-plurisubharmonic function Φ on X, such that ω + √ −1∂ ∂Φ ≥ ε ′ ω for some ε ′ > 0. This is an improvement of results in [WZ20]. Examples satisfying the assumption that there exists a holomorphic retraction structure contain product manifolds, thus contains many compact Kähler manifolds which are not necessarily projective.

On the extension of quasiplurisubharmonic functions

“…A further result in this direction was recently obtained in [WZ20,NWZ21]. Assuming that there exists a holomorphic retraction U → X on a neighborhood U ⊂ V of X, it is proved in [NWZ21] that any strictly ω| X -psh function on X extends to a strictly ω-psh function on V . These results all assume that X is smooth.…”