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Let $X$ be a smooth projective curve defined over a finite field of char $p$, let $G[1]$ be a truncated Barsotti-Tate grop of level-1. My question is : can $G[1]$ be extended to a truncated Barsotti-Tate group of level-2 (or extended to a Barsotti-Tate) over $X$, if not, what is the obstruction (or obstruction space)? Thank you!

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    $\begingroup$ Yes; see Illusie's paper that surveys Grothendieck's work on the deformation theory of p-divisible groups. Early in there he gives Dieudonne-module arguments of Gabber and Ekedahl to handle the extension problem over any perfect field. $\endgroup$
    – user28172
    Commented May 17, 2013 at 13:22
  • $\begingroup$ I haved see Illusie's paper, it is true even in some affine case ($X$ is affine ), but I think it may has obstruction in case of smooth projective curve. $\endgroup$
    – TOM
    Commented May 18, 2013 at 6:11

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