Purity of vanishing cycle for proper scheme over DVR with smooth generic fiber

Question

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.

Answer 1

Here's a typical example of what can go wrong in high dimension:

Let $\mathcal E$ be an elliptic curve degenerating to a nodal cubic. Let $X = \mathcal E^n$.

Then $H^* ( X_{\eta}, \mathbb Q_\ell ) = H^* (E_{\eta}, \mathbb Q_\ell)^{\otimes n}$ where the most interesting piece $H^1( E_{\eta}, \mathbb Q_\ell)$ is a two-dimensional representation with weights $0$ and $2$. Similarly $H^* ( X_{s}, \mathbb Q_\ell ) = H^* (E_{s}, \mathbb Q_\ell)^{\otimes n}$ where $H^1( E_{s}, \mathbb Q_\ell)$ is one-dimensional and has weight zero.

The specialization map $H^*( X_s, \mathbb Q_\ell) \to H^*(X_\eta, \mathbb Q_\ell)$ will be the $n$th tensor power of the specialization map for $\mathcal E$ and in particular will be injective. So the cohomology of vanishing cycles, which is the mapping cone of this map, will just be the cokernel.

Using this, you can see that the weights do all kinds of crazy things and that the weights in degree $n$ go all the way up to $2n$, not just $n+1$.