Let $X$ be a complex projective variety. If $X$ is smooth, then the first-order infinitesimal deformations are given by $H^1(X, T_X)$ and the obstructions are in $H^2(X, T_X)$.

Now assume that $X$ is singular, but we consider only locally trivial deformations (in the sense of Flenner and Kosarew). Then the statement about $H^1$ remains true, and so does the second one, according to Sernesi's book, Prop. 2.4.6. However, unwinding the proof one finds that in fact a nonsingularity hypothesis is used, namely when applying the Infinitesimal Lifting Property.

Question: Is it true that $H^2(X, T_X)$ is an obstruction space for the locally trivial deformation functor $\mathrm{Def^{lt}}(X)$?

Ideally, an answer would either indicate how to fix the proof of Prop. 2.4.6 above, or give an example of a variety $X$ with $\mathrm{Def^{lt}}(X)$ singular even though $H^2(X, T_X) = 0$.

  • 1
    $\begingroup$ For locally trivial deformations, you only need to specify gluing data. The relevant automorphisms are given by the tangent sheaf, and failure of the cocycle condition is a class in $H^2$. What am I missing? $\endgroup$ Jul 10, 2019 at 7:23
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    $\begingroup$ First you need to extend the automorphisms themselves to one order higher, and this requires the Infinitesimal Lifting Property. E.g. for k[x,y]/(xy), the automorphism given by x \mapsto x+t modulo t^2 does not extend to an automorphism modulo t^3. $\endgroup$
    – pgraf
    Jul 10, 2019 at 8:42
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    $\begingroup$ Good point.The obstruction to deforming $X$ lies in $Ext^2_{\mathcal{O}_X}(\mathbb{L}_X, \mathcal{O}_X)$, where $\mathbb{L}$ is the cotangent complex, and you seek an element of $H^2(X, \mathcal{Ext}^0(\mathbb{L}_X, \mathcal{O}_X))$. I can see why local triviality kills contributions to the obstruction from $\mathcal{Ext}^2$, but that doesn't rule out a contribution involving $\mathcal{Ext}^1$. $\endgroup$ Jul 10, 2019 at 15:27


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