Understanding the obstruction cone of a symmetric obstruction theory

Question

Let $X$ be a scheme over $\mathbb C$ carrying a symmetric perfect obstruction theory $\phi:E\to L_X$, in the sense of Behrend-Fantechi (here $L_X$ is the truncated cotangent complex, in degrees $[-1,0]$). Symmetry implies that the obstruction sheaf is $\textrm{ob}=\Omega_X$. I would like to understand the notion of obstruction cone attached to the obstruction theory.

It should be the image of the intrinsic normal cone $\mathfrak C_X$ under the chain of closed immersions $$\mathfrak C_X\subset \mathfrak N_X\subset \mathcal E=h^1/h^0(E^\vee).$$ The second inclusion is $h^1/h^0(\phi^\vee)$, and is closed by definition of perfect obstruction theory, and the first one is always there (the intrinsic normal sheaf is the abelian hull of the intrinsic normal cone). So we have the obstruction cone $$\mathfrak C\subset \mathcal E.$$

Now, in this paper by Behrend I found another notion of obstruction cone, and I would like to understand why it is the same as the one I just described.

Embedding $X$ in a smooth scheme $M$ (let us assume we can do this), we get a surjection $$p:\Omega_M|_X\twoheadrightarrow \textrm{ob}=\Omega_X.$$

Behrend talks about the cone of curvilinear obstructions $i:\textrm{cv}\hookrightarrow\textrm{ob}$ and says that the obstruction cone is the fiber product of the arrows $p$ and $i$.

Now, there are at least two things I do not understand: first of all, what exactly is the cone of curvilinear obstructions? Apparently it can be identified with the coarse moduli sheaf of $\mathfrak C_X$, but I have no idea how the latter is defined. Secondly, why are the two descriptions of the obstruction cone equivalent? (Probably the second one is the local version of the first, but still I cannot see how they relate to one another).

Thanks!

Answer 1

Behrend introduces precise definitions on page 13 of the linked paper. Locally, $E$ is a quotient, such that we may write $E^\vee=[H \to F]$, hence $\mathcal{E}= [F/H]$. Taking the fibre product of the intrinsic normal cone $\mathfrak{C}_X$ and $F$ over $\mathcal{E}$, we get the (local) obstruction cone $C\subset F$ from your first description. This subcone is $H$-invariant.

The second description approaches this from the side of sheaves: We can take the sheaf associated to $F$ and quotient it by the subsheaf associated to $H$ to get the obstruction sheaf $\mathrm{ob}$. (This is another way of saying that we are taking the $h^1$ of a complex of sheaves on $X$). Similarly, we can take the quotient of the sheaf assocated to $C$ and quotient it by the sheaf associated to $H$. He proves that this gives us a cartesian diagram of sheaves $$\begin{matrix} C & \hookrightarrow & F\\ \downarrow & & \downarrow \\ \mathrm{cv} &\hookrightarrow & \mathrm{ob}.\end{matrix} $$

In the case of a symmetric obstruction theory and a local embedding into a smooth scheme (or DM stack) $M$, we can take $F=\Omega_M |_X$.