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Matthieu Romagny
Matthieu Romagny
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If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$. Now if I take a deformation $A_{\epsilon}/k[\epsilon]$, does there still existsexist a canonical lift of this deformation to a deformation of (the generic fiber of) $\hat{A}$?

By the Kodaira-Spencer mapping deformations are essentially encoded by differentials of $A$, so the question boils down to whether differentials on $A$ liftslift canonically to differentials on $\hat{A}$. By Katz, Serre-Tate local moduli, Section 3, differentials on any lift $\tilde{A}$ correspond to points in $T_p(A^\vee)(k)$, and his main theorem 3.7.1 describedescribes the compatibility of this identification with the Kodaira-Spencer map. Is there a way to use this to lift differentials canonically on the canonical lift?

If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$. Now if I take a deformation $A_{\epsilon}/k[\epsilon]$, does there still exists a canonical lift of this deformation to a deformation of (the generic fiber of) $\hat{A}$?

By the Kodaira-Spencer mapping deformations are essentially encoded by differentials of $A$, so the question boils down to whether differentials on $A$ lifts canonically to differentials on $\hat{A}$. By Katz, Serre-Tate local moduli, Section 3, differentials on any lift $\tilde{A}$ correspond to points in $T_p(A^\vee)(k)$, and his main theorem 3.7.1 describe the compatibility of this identification with the Kodaira-Spencer map. Is there a way to use this to lift differentials canonically on the canonical lift?

If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$. Now if I take a deformation $A_{\epsilon}/k[\epsilon]$, does there still exist a canonical lift of this deformation to a deformation of (the generic fiber of) $\hat{A}$?

By the Kodaira-Spencer mapping deformations are essentially encoded by differentials of $A$, so the question boils down to whether differentials on $A$ lift canonically to differentials on $\hat{A}$. By Katz, Serre-Tate local moduli, Section 3, differentials on any lift $\tilde{A}$ correspond to points in $T_p(A^\vee)(k)$, and his main theorem 3.7.1 describes the compatibility of this identification with the Kodaira-Spencer map. Is there a way to use this to lift differentials canonically on the canonical lift?

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YCor
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RandomMathUser
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Canonical lift of the deformation of an ordinary abelian variety

If $A/k$ is a principally polarised ordinary abelian variety ($k$ a perfect field of characteristic $p$, we may assume it is finite for simplicity), we have a canonical lift $\hat{A}/W(k)$. Now if I take a deformation $A_{\epsilon}/k[\epsilon]$, does there still exists a canonical lift of this deformation to a deformation of (the generic fiber of) $\hat{A}$?

By the Kodaira-Spencer mapping deformations are essentially encoded by differentials of $A$, so the question boils down to whether differentials on $A$ lifts canonically to differentials on $\hat{A}$. By Katz, Serre-Tate local moduli, Section 3, differentials on any lift $\tilde{A}$ correspond to points in $T_p(A^\vee)(k)$, and his main theorem 3.7.1 describe the compatibility of this identification with the Kodaira-Spencer map. Is there a way to use this to lift differentials canonically on the canonical lift?