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.



Push forward of a Vector bundle is a coherent sheaf?
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.


