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Let $X$ be a smooth projective variety over a field $k$. We know that the deformations of $X$ over $k[t]/(t^2)$ are parametrized by $H^1(X,T_X)$; given such a deformation, we know that it can be extended over $k[t]/(t^3)$ if and only if the corresponding element $\xi \in H^1(X,T_X)$ satisfies $q(\xi )=0$, where $q:H^1(X,T_X)\otimes H^1(X,T_X)\rightarrow H^2(X,T_X)$ is the cup-product deduced from the Lie bracket on $T_X$.

Now consider a closed subvariety $Y\subset X$, say smooth. Again, we know that the deformations of $Y$ inside $X$ over $k[t]/(t^2)$ are parametrized by $H^0(Y, N_{Y/X})$ (the normal bundle of $Y$ in $X$). Given such a deformation, how can we express that it extends to over $k[t]/(t^3)$? What would be the analogue of the map $q$?

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    $\begingroup$ Often you can apply the logic from the first paragraph to the second paragraph by replacing $X$ by the blowing up of $X$ along $Y$. $\endgroup$ Mar 4, 2018 at 23:02
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    $\begingroup$ Are you not essentially asking about a Lie algebroid structure on $N_{Y/X}[-1]=T_{Y/X}$? This has been studied in arxiv.org/abs/1306.5260. $\endgroup$ Mar 5, 2018 at 8:36
  • $\begingroup$ I imagine that you want to first fix a deformation of the ambient $X$ to $k[t]/(t^3)$? If yes, then shouldn't the class of this deformation come into play? $\endgroup$ Mar 5, 2018 at 9:48
  • $\begingroup$ ... oh I see, you want to consider the trivial deformation of $X$. $\endgroup$ Mar 5, 2018 at 9:53
  • $\begingroup$ @Pavel Safronov: Many thanks, the paper you quote looks quite relevant. I will read it. $\endgroup$
    – abx
    Mar 5, 2018 at 10:57

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