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Fix a finite field $k_p$ of characteristic $p$, as well as a $p$-adic lift of it, $k_0$. In other words $k_0$ is a $p$-adic field and $k_p$ its residue field.

Let $X_p$ be a non-singular variety over the finite field $k_p$.

The usual criterion for existence of lifts of $X_p$ to $k_0$ is the vanishing of obstructions in $H^2(X_p, \mathcal{T}_{X_p})$. When obstructions vanish, the space of lifts is a torsor under $H^1(X_p, \mathcal{T}_{X_p})$.

I am interested in learning about other kinds of existence criteria.

In particular if the $l$-adic cohomology of $X_p$ lifts as a Galois module to characteristic 0, what are additional obstructions to lifting of $X_p$ itself? How do dimensions of deformation spaces of the cohomological Galois representations compare with dimension of $H^1(X_p, \mathcal{T}_{X_p})$?

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    Conjecturally (e.g. on the Tate + semisimplicity conjectures), the $\ell$-adic cohomology of $X_p$ is semisimple, and all simple parts occur inside the cohomology of abelian varieties (see e.g. this post). Since abelian varieties always lift, in particular the Galois action lifts. I don't know if the last statement is also known unconditionally. – R. van Dobben de Bruyn Mar 26 '18 at 22:39
  • @R. van Dobben de Bruyn, thanks! What I am wondering about is: if the Galois representation on the cohomology of $X_p$ does lift, what are further obstructions to lifting $X_p$ to an $X_0$ in char. 0 such that cohomology of $X_0$ is a lift of the cohomology of $X_p$. There is a lot of work on lifting Galois representations, e..g, by Mazur, Boston et al., and I imagine it partly goes towards answering this question. –  Mar 26 '18 at 22:50
  • If $X_0$ is a lift of a smooth proper variety $X_p$ over $k_p$, then the smooth and proper base change theorems give isomorphisms $H^i(X_{\bar k_p},\mathbb Q_\ell) \cong H^i(X_{\bar k_0},\mathbb Q_\ell)$. Thus, lifting the Galois action here only means extending it from $G_{k_p}$ to $G_{k_0}$; the module stays the same. This quite different from the Mazur/Boston story. – R. van Dobben de Bruyn Mar 27 '18 at 00:22
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    Oh and this also trivialises my other comment: the map $G_{k_0} \to G_{k_p}$ is surjective, so any $G_{k_p}$-representation gives rise to a $G_{k_0}$-representation... – R. van Dobben de Bruyn Mar 27 '18 at 00:23
  • @R. van Dobben de Bruyn, sorry for not fully understanding the implications of your comments to answering my question. Perhaps I am being specially obtuse; if so, please bear with me. My question was: Given an $X_p$, a priori we don't know if it'd lift to any $X_0$ (unless it's a curve, K3 or ppav, etc.). Suppose all we know is that its cohomology spaces do lift as Galois representations from $G_{k_p}$ to $G_{k_0}$. Is this sufficient for $X_p$ to lift to an appropriate $X_0$? My instinct would be to say, "Not in general", and then ask what more do I need to ensure a lift exists. –  Mar 27 '18 at 00:43
  • @R. van Dobben de Bruyn, (contd) In other words, does the knowledge of its reduction and etale cohomology suffice to reconstruct a variety over the $p$-adics? –  Mar 27 '18 at 00:45
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    My first comment says that the assumption that the cohomology lifts as a Galois representation is not really an assumption. Since most varieties don't lift, this is not sufficient to conclude that $X_p$ lifts to $X_0$. – R. van Dobben de Bruyn Mar 27 '18 at 00:49

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