1st ingredient: cohomology
Choose a finite extension $K'/\mathbf{Q}_p$, with residue field $k'=\mathbf{F}_{p^m}$, so the geometric divisor classes of $X$ are defined over $K'$ and those of $X_p$ over $k'$.
Over characteristic zero we have:
$H:=H^2_{\mathrm{dR}}(X/\mathbf{Q}_p) = F^0 \supset F^1 \supset F^2$, the Hodge filtration
$\operatorname{Pic}(X^{\mathrm{al}})_{\mathbf{Q}} \hookrightarrow F^1_{K'}:=F^1\otimes_{\mathbf{Q}_p}K'$
For $d = 4$, $\dim F^i(X) = 22, 21, 1$.
Over characteristic $p$ we have:
$\operatorname{Pic}(X_p^{\mathrm{al}})_{\mathbf{Q}} \hookrightarrow H^2_{\mathrm{crys}}(X_p\times_{\mathbf{F}_p}k'/W(k'))\otimes_{W(k')}K' \simeq H\otimes_{\mathbf{Q}_p}K'=:H_{K'}=F^0_{K'}\supset F^1_{K'}\supset F^2_{K'}$
SOURCE: O:L1224-1243
SECTION 2.2, The obstruction. Sixteen slides. Five ingredients, two worked calculations, three examples that succeed, stall and fail in turn, and the question those three raise.
Spoken: The Hodge filtration, introduced here rather than in Lecture 1, because this is the first place it does any work.
Spoken: For a K3 surface the graded pieces have dimensions 1, 20, 1, and F^1 is the part of codimension one, cut out by the holomorphic two-form.
TRANSCRIBED FROM: frobenious-dist/nyc-jnts.tex:1224-1239 (the theorem block at 1240-1242 of the same frame is deck slide 6)