"Weight-monodromy" for open varieties

Question

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy filtration on the $i$th cohomology groups $H^i_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$ are pure of weight $i+k$.

I was wondering if there is a similar conjecture for open varieties, coming from looking at a good compactification $X\hookrightarrow \overline{X}$. If $X$ is a curve, then I can get my hands on what's happening explicitly, so suppose that $X$ is a curve with compactification $\overline{X}$ and $D$ is the set of missing points.

Then the only interesting group is $H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$ which by excision sits in an exact sequence $$ 0 \rightarrow H^1_\mathrm{et}(\overline{X}_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell) \rightarrow H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell) \rightarrow V \rightarrow 0 $$ where $V$ is a subspace of $H^0_\mathrm{et}(D_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)(-1)$, it is therefore pure of weight $2$. So (pretty tautologically) $V$ turns up in the 2nd graded piece $\mathrm{Gr}_2^W$ of the weight filtration, but with some basic linear algebra I think I've convinced myself that $V$ actually turns up in the 0th graded piece $\mathrm{Gr}_0^M$ of the monodromy filtration on $H^1_\mathrm{et}(X_{\overline{\mathbb{Q}}_p},\mathbb{Q}_\ell)$, and apart from this 'extra' $V$, the graded pieces of the monodromy filtration for $X$ are the same as those for $\overline{X}$.

In other words, $\mathrm{Gr}^M_{-1}$ and $\mathrm{Gr}_1^M$ are pure of weights 0 and 2 respectively, but $\mathrm{Gr}^M_1$ is mixed with weights 1,2. So the naive weight monodromy conjecture fails, but you can still say something - the graded pieces $\mathrm{Gr}_k^M$ are mixed of weights $\geq i+k$.

So I guess this is my question: is there a conjectural 'weight monodromy' for open varieties which says something like the above, e.g. the $k$th graded pieces is mixed with weights in $[i+k,2i+k]$? Maybe this would easily follow from the 'usual' weight monodromy by choosing a good compactification and then just using some linear algebra?

Answer 1

Tony Scholl gave a talk at a conference in Warwick in 2013 on exactly this topic (his talk was called "Remarks on monodromy and weights"). He explained how to formulate a precise version of weight-monodromy for arbitrary varieties over p-adic fields (not necessarily proper or smooth).

The idea is that for any field $K$, and any finite-type $K$-scheme $X$, if we set $H^i(X) = H^i(X_{\overline{K}}, Q_\ell)$ for some $\ell \ne char(K)$, there is a canonical increasing filtration $W_n H^i(X)$ (the "geometric weight filtration") of $H^i(X)$ such that $gr_n^W H^i(X)$ is a subquotient of $H^n(Y)$ for some smooth proper $Y$. (This is due to Deligne and de Jong.)

So it's reasonable to conjecture that if $K$ is a $p$-adic field and $\ell \ne p$, the pieces $gr_n^W H^i$ should be "quasi-pure of weight n" as representations of $G_K$, which means that all eigenvalues of Frobenius are Weil numbers of some weight, and for all $r$ the monodromy operator $N^r$ sends the weight $n + r$ generalized eigenspace isomorphically to the weight $n - r$ one. (Scholl just calls this "pure of weight $n$", but this conflicts with the more restrictive usage of "pure" in your question.)

In Scholl's talk he announced a proof that the usual weight-monodromy conjecture (i.e. for $X$ proper and smooth over $K$) actually implies this more general conjecture. This implication is not obvious, because being quasi-pure is not preserved under passing to subquotients (and Tony didn't say how he gets around this, as far as I can recall).

(It's not totally clear if this answers the question, because one would have to understand how the geometric weight filtration relates to the filtration on $H^i(X)$ given by the Grothendieck abstract-monodromy theorem -- they are definitely not the same, because the latter encodes information about the singularities of the special fibre.)