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Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to A^1$ be a polynomial (or holomorphic) function.

Question: Is it true that the $\lambda \in A^1$ such that the vanishing cycles $\phi_{f-\lambda}(M) = 0$ is a dense open set?

Here are my thoughts:

If $M = O_X$ (or the constant perverse sheaf $A[dim X]$), this is just the fact that the critical values of $f$ are isolated.

In the general case, we can factor $f$ through its graph $X\to X\times A^1$, $x\mapsto (x,f(x))$, reducing to the case where $f$ is the (smooth) projection $t:X\times A^1 \to A^1$. Our sheaf $M$ on $X\times A^1$ has a characteristic variety $\bigcup_\alpha T^*_{S_\alpha}(X\times A^1)$ for a stratification $X\times A^1 = \bigcup_\alpha S_\alpha$. My guess is that $\phi_{t-\lambda}(M) = 0$ when $\{t-\lambda = 0 \}$ is going to be transverse to all the $S_\alpha$ and that this is a generic condition but I'm having trouble making this intuition precise.

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Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. Let $f:X\to \mathbb{C}$A^1$ be a polynomial (or holomorphic) function. Question: Is it true that the $\lambda \in \mathbb{C}$A^1$ such that the vanishing cycles $\phi_{f-\lambda}(M) = 0$ is a dense open set?

Here are my thoughts:

If $M = O_X$ (or the constant perverse sheaf $A[dim X]$), this is just the fact that the critical values of $f$ are isolated.

In the general case, we can factor $f$ through its graph $X\to X\times \mathbb{C}$A^1$, $x\mapsto (x,f(x))$, reducing to the case where$f$is the (smooth) projection $t:X\times \mathbb{C} A^1 \to \mathbb{C}$A^1$. Our sheaf $M$ on $X\times \mathbb{C}$A^1$ has a characteristic variety $\bigcup_\alpha T^*_{S_\alpha}(X\times \mathbb{C})$A^1)$ for a stratification $X\times \mathbb{C} A^1 = \bigcup_\alpha S_\alpha$. My guess is that $\phi_{t-\lambda}(M) = 0$ when $\{t-\lambda \}$ is going to be transverse to all the $S_\alpha$ and that is a generic condition but I'm having trouble making this intuition precise.

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Consider a regular holonomic D-module (or a perverse sheaf) $M$ on a smooth variety $X$. X$. Let $f:X\to \mathbb{C}$ be a polynomial (or holomorphic) function. Question: Is it true that the $\lambda \in \mathbb{C}$ such that the vanishing cycles $\phi_{f-\lambda}(M) = 0$ is a dense open set? Here are my thoughts: If $M = O_X$ (or the constant perverse sheaf $A[dim X]$)X]$), this is just the fact that the critical values of $f$ are isolated.

In the general case, we can factor $f$ through its graph $X\to X\times \mathbb{C}$, mathbb{C}$, $x\mapsto (x,f(x))$, x,f(x))$, reducing to the case where $f$ is the (smooth) projection $t:X\times \mathbb{C} \to \mathbb{C}$. mathbb{C}$. Our sheaf $M$ on $X\times \mathbb{C}$ has a characteristic variety $\bigcup_\alpha T^*{S\alpha}(X\times T^*_{S_\alpha}(X\times \mathbb{C})$ for a stratification $X\times \mathbb{C} = \bigcup_\alpha S_\alpha$. My guess is that $\phi_{t-\lambda}(M) = 0$ when${t-\lambda }$\{t-\lambda \}$ is going to be transverse to all the $S_\alpha$ and that is a generic condition but I'm having trouble making this intuition precise.

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