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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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    $\begingroup$ 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. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 26, 2018 at 22:39
  • $\begingroup$ @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. $\endgroup$
    – user122285
    Commented Mar 26, 2018 at 22:50
  • $\begingroup$ 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. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 27, 2018 at 0:22
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    $\begingroup$ 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... $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 27, 2018 at 0:23
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    $\begingroup$ 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 $X_0$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 27, 2018 at 0:49

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