The problem that we consider in this paper is the following: given a compact Kähler manifold $(X,\omega)$, a compact complex submanifold $V\subset X$, and a closed positive current $T$ on $V$ in the class $[\omega|_{V}]$, can we find a closed positive current $\tilde{T}$ on $X$ in the class $[\omega]$ with $T=\tilde{T}|_{V}$? Extension questions like this have recently generated a great deal of interest thanks to their analytic and geometric applications [4, 7, 9, 15, 17]. The first result in this direction is due to Schumacher [15] who proved that if $[\omega]$ is rational (hence $X$ is projective), then any smooth Kähler metric on $V$ in the class $[\omega|_{V}]$ extends to a smooth Kähler metric on $X$ in the class $[\omega]$ (see also [4, 9, 14, 17]). More recently, Coman-Guedj-Zeriahi proved in [4] that under the same rationality assumption, every closed positive current $T$ on a closed analytic subvariety $V\subset X$ in the class $[\omega|_{V}]$ extends to $X$.

In our main theorem we get rid of rationality/projectivity assumptions in the case of extension of Kähler currents with analytic singularities from a submanifold. More precisely, we prove:

The techniques that have been used in the past to approach this type of extension problems range from Siu’s Stein neighborhood theorem [16], to results of Coltoiu [3] on extending Runge subsets of analytic subsets of $\mathbb{C}^{N}$, to the Ohsawa-Takegoshi $L^{2}$ extension theorem [8]. In this paper we introduce a new constructive extension technique which uses resolution of singularities to obtain estimates which allow us to glue plurisubharmonic functions with analytic singularities near their polar set by a modification of a classical argument of Richberg [13]. The ideas of using resolution of singularities comes from the recent work of Collins-Greenleaf-Pramanik [1] on sharp estimates for singular integral operators, which was motivated by the seminal work of Phong, Stein and Sturm [10, 11, 12]. One advantage of these local techniques is that we can work on general Kähler manifolds with arbitrary Kähler classes, although at present we can only extend Kähler currents with analytic singularities from smooth subvarieties. It would be very interesting to know how far these techniques can be pushed, but we have not undertaken this here.

In our recent work [2], we dealt with a very similar extension problem, and employed a related argument of a more global flavor where we used resolution of singularities from the very beginning and worked on a blowup of $X$. This approach is technically simpler, because it allows us to use directly Richberg’s gluing technique, but it seems not to be strong enough to prove Theorem 1.1, because the extension is achieved only on a blowup of $X$. On the other hand, the proof of Theorem 1.1 can easily be adapted to give another proof of [2, Theorem 3.2], which is the key technical result needed in that paper.