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Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ is always pure of weight $i$?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i+1$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ is always pure of some weight?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.

Example: when $X_k$ has ordinary quadratic singularity, then this is true by Picard-Lefschetz formula.

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ is always pure of weight $i$?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_{k^{alg}}, R\Phi \mathbb Q_l)$ is always pure of weight $i$?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.

Let $X$ be a proper scheme over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ is always pure of weight $i$?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.

Let $X$ be a scheme proper and flat over a complete discrete valuation ring $O$ with finite residue field $k$, and choose a prime $l$ not equal to characteristic of $k$. Consider the Galois representation $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ where $R\Phi \mathbb Q_l$ denotes the vanishing cycle, are all of its weights less than $i$? Assume the generic fiber $X_K$ is proper and smooth, do we know that $H^i_{et}(X_s, R\Phi \mathbb Q_l)$ is always pure of weight $i$?

A theorem of Gabber says that $R\Phi \mathbb Q_l$ is perverse (up to shift), maybe this is helpful.