(I have asked the question
[The commutativity of minimal extension $\cdots$](https://mathoverflow.net/questions/301134/the-commutativity-of-minimal-extension-and-direct-image-by-blowing-down) and I simplify this question to the next simple question:)


Let $X$ be a rational variety over $\mathbb{C}$, $\phi : \hat{X} \rightarrow X$ be the blow-up of one point $\{p\}$, and $M$ be a simple (holonomic) $D\_{\hat{X}}$-module.
Then


*Is it true that the direct image $\int\_{\phi}M (=\phi\_+M)$ is also simple (holonomic) $D\_X$-module ?*



Is direct image of simple $D$-module is also simple?
No. I’ll provide an example for what WillSawin suggested. Let $X$ be $\Bbb C^2$. Let $M$ be the irreducible holonomic module on $\hat X$ supported on the special fiber whose restriction to the special fiber is the structure sheaf. Then the direct image of $M$ is just the de Rham complex of $\Bbb P^1$, or rather the direct image of this complex via the (exact) functor $H^0i\_+$, where $i\colon \{0\}\to X$ is inclusion. In particular, $\phi\_+M$ has cohomology in more than one degree. 


