(I have asked the question The commutativity of minimal extension $\cdots$ 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, 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 ?

(I have asked the question The commutativity of minimal extension $\cdots$ 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 ?

(I have asked the question The commutativity of minimal extension $\cdots$ and I deduce 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, 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 ?

(I have asked the question The commutativity of minimal extension $\cdots$ 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, 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 ?