Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K\_X +D = 0$ necessary for the existence of Ricci-flat metric?



A necessary condition for existence of Ricci flat metric on pair (X,D)
**No**, the condition $K\_X +D = 0$ is not necessary for the Ricci-flat case, when $D$ is singular. For instance,consider $X=\overline X\setminus D$ and take $\overline X=\mathbb CP^n$ and define the divisor $D:=\{p(z)\prod\_{i=0}^{n-1}z\_i=0\}$, where 


$$f(z)=z\_0^{m-1}z\_n+P(z\_0,...,z\_{n-1})$$ for a homogeneous polynomial $P$ of degree $m\geq 2$. Then $X=(\mathbb C^\*)^n$


After the comment of Jason, this example is for $D$ which is not snc divisor


