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

In the present paper, we show that given a compact Kähler manifold (X,ω) with a Kähler metric (not necessarily Hodge 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 gives a partial answer to a question raised by Coman–Guedj–Zeriahi [4].