# Obstructions to locally trivial deformations

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 $$H^2(X, T_X) = 0$$.

• 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? Jul 10, 2019 at 7:23
• 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. Jul 10, 2019 at 8:42
• 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 $\mathcal{Ext}^1$. Jul 10, 2019 at 15:27