Given a compact Kähler manifold $M$, let $D$ be an effective divisor on $M$. 


1. Is $M\setminus D$ pseudoconvex? That is, can we find a smooth plurisubharmonic function that exhausts $M\setminus D$ ?
2. Can we find a complete Kähler metric on $M\setminus D$ ?


If $1$ and $2$ are not true, can we find any obstructions? Or necessary and sufficient conditions?


**Note:** if $D$ is an ample divisor, we can choose an Hermitian metric on $[D]$ which is the associated divisor bundle to $D$, then we can take $-ln|s\_{D}|^{2}\_{h}$ to be the strongly smooth exhaust plurisubharmonic function on $M\setminus D$. Surely, this is just a Stein (or Affine) manifold. Given such a plurisubharmonic function on $M\setminus D$ we can easily contruct a compelte Kähler metric. 



Plurisubharmonic function and complete Kähler metric on certain Kähler manifold
Question 1: Plurisubharmonic functions extend across codimension 2 subvarieties . Let X be the complex projective plane blown up at one point and D be the exceptional divisor then any plurisubharmonic function on the complement of D in X extends to X and is therefore a constant.


