Remarks on $L^2$-jet extension and extension of singular Hermitian metric with semipositive curvature

Abstract

We give a new variant of L 2 -extension theorem for the jets of holomorphic sections and discuss the relation between the extension problem of singular Hermitian metrics with semipositive curvature.

Kähler currents and null loci

“…The main technical ingredient in the proof of our main theorem is a technical extension-type result for Kähler currents with analytic singularities in a nef and big class which are defined on submanifold of X (Theorem 3.2). The problem of extending positive currents from subvarieties has recently generated a great deal of interest in both analytic and geometric applications [16,29,42,61]. At the moment, the best results available are due to Coman-Guedj-Zeriahi [16], who proved that every positive current in a Hodge class on a (possibly singular) subvariety V ⊂ CP N is the restriction of a globally defined, positive current.…”

Open problems in pluripotential theory

“…A result of Matsumura [Mat13] shows that, conversely, such an extension property forces the associated line bundle to be ample. An alternative approach to the case of Hodge forms has been proposed by Hisamoto [His12]. It requires a better understanding of the Ohsawa-Takegoshi extension theorem with prescription of jets of high order, a problem of independent interest.…”

An extension theorem for Kähler currents with analytic singularities

“…The problem that we consider in this paper is the following: given a compact Kähler manifold (X, ω), a compact complex submanifold V ⊂ X, and a closed positive current T on V in the class [ω| V ], can we find a closed positive current T on X in the class [ω] with T = 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 [ω] is rational (hence X is projective), then any smooth Kähler metric on V in the class [ω| V ] extends to a smooth Kähler metric on X in the class [ω] (see also [4,9,14,17]).…”