Push forward of a Vector bundle is a coherent sheaf?

Question

Let $X$ be a smooth compact Kahler manifold and let $Y\subset X$ be a smooth complex submanifold of complex codimension at least $2$. Let $$j:Y\hookrightarrow X$$ the natural (holomorphic) embedding map. Let $E\rightarrow Y$ a holomorphic vector bundle over $Y$. Are there conditions (on $E$, $j$,$Y$,$X$) under which the push forward sheaf $j_{*}E$ is coherent? Sorry if the question is too standard.

Answer 1

The answer is that $j_*E$ is always coherent under your hypotheses, provided that $Y$ is closed in $X$.

This is a consequence of the the so-called Extension Principle, that can be stated as follows:

Proposition. An analytic sheaf $\mathscr{S}$ on a closed complex subspace $Y$ of a complex space $X$ is $\mathscr{O}_Y$-coherent if and only if the trivial extension of $\mathscr{S}$ to a sheaf on $X$ is $\mathscr{O}_X$-coherent.

In your case $\mathscr{S}=E$, which is a vector bundle, hence a coherent $\mathscr{O}_Y$-sheaf. So the extension $j_*E$ is a coherent $\mathscr{O}_X$-sheaf.

Notice that $X$ and $Y$ are not required to be smooth, and the assumption $\textrm{codim} \, Y \geq 2$ is not necessary.

A reference is Grauert-Remmert, Coherent Analytic Sheaves, page 17 and page 239.