On the extensions of Kähler currents on compact Kähler manifolds

Abstract

Let $(X,\omega )$ be a compact Kähler manifold with a Kähler form $\omega$ of complex dimension $n$, and $V\subset X$ is a compact complex submanifold of positive dimension $k<n$. Suppose that $V$ can be embedded in $X$ as a zero section of a holomorphic vector bundle or rank $n-k$ over $V$. Let $\phi$ be a strictly $\omega {|}_V$-psh function on $V$. In this paper, we prove that there is a strictly $\omega$-psh function $\Phi$ on $X$, such that $\Phi {|}_V=\phi$. This result gives a partial answer to an open problem raised by Collins-Tosatti and Dinew-Guedj-Zeriahi, for the case of Kähler currents. We also discuss possible extensions of Kähler currents in a big class.

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