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"Weight-monodromy" for open varieties
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 $\geq i+k$. …