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

- Is $M\setminus D$ pseudoconvex? That is, can we find a smooth plurisubharmonic function that exhausts $M\setminus D$ ?
- 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.