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Xiaobo Zhuang
Xiaobo Zhuang
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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!

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. 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!

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!

Source Link
Xiaobo Zhuang
Xiaobo Zhuang
  • 503
  • 2
  • 10

Local model of virtual fundamental cycle

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. 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!