The following baby version of virtual fundamental cycle is well known:


Let $M\subset V$ be the zero locus of a section $s$ of a vector bandle $E \to V$, in general $s$ is not transversal to the zero section and $M$ does not have the expected dimension, then one uses excess intersection theory to define the virtual fundamental class $[M]^{vir}
$ to be $0\_E^{!}[C\_{M/V}]\in A\_{vdim}(M)$, where $C\_{M/V}\subset E|\_M$ is the normal cone of $M$ in $V$. 


If $V$ is smooth, one can reformulate these into Behrend-Fantachi's language by setting $E^{-1}=E^{\vee}|\_M, E^0=\Omega\_V|\_M$, and get a perfect obstruction theory $E^\* \to L\_{M}$ associated to the model $(V,E,s)$.


My question is whether the reformulation can be reversed:


**Given a perfect obstruction theory $E^\* \to L\_{M}$, can we return to the baby version locally?** I mean whether we can find open covering (in some topology) of $M$, such that for each open subset $U$ in the covering, we can find $U\subset V$,a vector bundle $F \to V$ and $U$ is realized as zero locus of a section $s$. And $[E^\* \to L\_{M}]|\_U$ coincides with the perfect obstruction theory associated to the local model $(V,F,s)$. I will appreciate if you could provide the details.


Thank you!



Local model of virtual fundamental cycle
The answer is yes. Let $[E^{-1} \rightarrow E^0]$ be a perfect obstruction theory on $M$. After localizing in $M$ we can assume that the map $E^0 \rightarrow \Omega\_M$ is induced as $\mathcal{O}\_M \otimes \Omega\_{\mathbf{A}^n} \rightarrow \Omega\_M$ for some map $M \rightarrow \mathbf{A}^n$.(\*) As the map on differentials is surjective the relative differentials vanish and the map is unramified. By EGA IV.18.4.7, we can (at least after localizing further in $M$) factor the map $M \rightarrow \mathbf{A}^n$ as a closed embedding $M \rightarrow V$ followed by an étale map $V \rightarrow \mathbf{A}^n$.


Let $I$ be the ideal of $M$ inside $V$. Then we have an exact sequence


$I/I^2 \rightarrow \mathcal{O}\_M \otimes \Omega\_V \rightarrow \Omega\_M \rightarrow 0$


where the first two terms are $\tau\_{\geq -1} L\_M$. By the definition of an obstruction theory, we are given a map $E^{-1} \rightarrow I/I^2$ inducing a surjection $H^{-1}(E) \rightarrow H^{-1} L\_M$. By the 5-lemma, $E^{-1} \rightarrow I/I^2$ is surjective.


Localizing further in $V$ and $M$, we can assume that the map $E^{-1} \rightarrow I/I^2$ is induced from a map $F \rightarrow I$ of coherent sheaves on $V$ with $F$ locally free. By Nakayama's lemma we can assume, possibly after further localization, that $F \rightarrow I$ is surjective. Then consider the section of the vector bundle $F^\vee$ dual to the composition $F \rightarrow I \subset \mathcal{O}\_V$. The vanishing locus of this section is $M$ and the induced complex $F \rightarrow \mathcal{O}\_M \otimes \Omega\_V$ is the obstruction theory we started with.


(\*) We can represent $\tau\_{\geq -1} L\_M$ as $[ J / J^2 \rightarrow \Omega\_W ]$ for some closed embedding $M \subset W$ and we are given a map of complexes $E^\bullet \rightarrow \tau\_{\geq -1} L\_M$. Replacing $E^\bullet$ with a quasi-isomorphic complex, we can assume that the map $E^0 \rightarrow \Omega\_W$ is surjective. But $E^0$ and $\Omega\_W$ are vector bundles, so after localizing in $M$ we can assume that $\Omega\_W$ is a direct summand of $E^0$. Choosing a basis for the complementary summand, we can identify $E^0 \cong \Omega\_{W \times \mathbf{A}^n}$.


