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Fixed mistake by changing "DGA" to "DGLA"
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Braxton Collier
Braxton Collier
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What is the DGADGLA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. A classical result, proven by Kodaira, is that if the cohomology group $H^1(Y,\mathcal{O}(T_{X/Y}))$ vanishes, then the deformation theory of $Y$ as a complex submanifold of $X$ is unobstructed. More precisely, there exists a "maximal family" of deformations of $Y$ in $X$: this consists of a complex manifold $W$ (say with a marked point $w_0\in W$), a complex submanifold $V$ of $W\times X$, such that for each point $w\in W$ the intersection of $V$ with $w\times X$ is a complex compact submanifold of $X$, which in the case $w=w_0$ is equal to $Y$. Furthermore, this family of submanifolds is "maximal" (or "universal") in an appropriate sense. By saying the deformation problem is unobstructed, I mean the following: for any family as described above (maximal or not) there is a canonical injective complex linear map from the tangent space of $W$ at $w_0$ to the space of holomorphic sections of $T_{X/Y}$, i.e. the cohomology group $H^0(Y,\mathcal{O}(T_{X/Y}))$. Given the above assumption that $H^1(Y,\mathcal{O}(T_{X/Y}))=0$, this map is an isomorphism. Roughly speaking, one can view a holomorphic section of the normal bundle of $Y$ as a first order deformation of $Y$ in $X$, and unobstructedness means that each such first order deformation can be extended to an "honest" family of deformations.

On the other hand, Kodaira does not explore what happens if one weakens the condition $H^1(Y,\mathcal{O}(T_{X/Y}))=0$. For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y})$, it seems plausible that it isn't necessary for the entire group $H^1(Y,\mathcal{O}(T_{X/Y}))=0$ to vanish in order for $v$ to extend to an honest family of deformations.

$\textbf{Question 1}$: For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y}))$, is there a way to determine when $v$ extends to an honest family of deformations? In particular, is there a map $T:H^0(Y,\mathcal{O}(T_{X/Y}))\to H^1(Y,\mathcal{O}(T_{X/Y}))$ which can be described in a reasonably explicit way such that $v$ extends if and only if $T(v)=0$?

One possible solution to this problem would be to derive a Maurer-Cartan type of equation. It's my understanding that, on general grounds, one expects the deformation theory of almost any structure to be controlled by a differential graded Lie algebra (DGADGLA), with (perhaps formal) deformations corresponding to solutions of the Maurer-Cartan equation. One candidate for the DGA in the above example would be the space of sections of the Dolbeault complex $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ associated to the holomorphic vector bundle $\Lambda^{\bullet} T_{X/Y}$. This comes equipped with the Dolbeault differential, but no obvious Lie bracket (at least not obvious to me).

$\textbf{Question 2}:$ Can $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ be given the structure of a DGADGLA (i.e. equipped with a bracket compatible with the Dolbeault differential), such that formal deformations of $Y$ as a complex submanifold of $X$ are given by solutions to the Maurer-Cartan equation? If not, is there some other DGADGLA controlling the deformation theory of a complex submanifold which has a "geometric" description (i.e. as the space of sections of some vector bundle)?

What is the DGA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. A classical result, proven by Kodaira, is that if the cohomology group $H^1(Y,\mathcal{O}(T_{X/Y}))$ vanishes, then the deformation theory of $Y$ as a complex submanifold of $X$ is unobstructed. More precisely, there exists a "maximal family" of deformations of $Y$ in $X$: this consists of a complex manifold $W$ (say with a marked point $w_0\in W$), a complex submanifold $V$ of $W\times X$, such that for each point $w\in W$ the intersection of $V$ with $w\times X$ is a complex compact submanifold of $X$, which in the case $w=w_0$ is equal to $Y$. Furthermore, this family of submanifolds is "maximal" (or "universal") in an appropriate sense. By saying the deformation problem is unobstructed, I mean the following: for any family as described above (maximal or not) there is a canonical injective complex linear map from the tangent space of $W$ at $w_0$ to the space of holomorphic sections of $T_{X/Y}$, i.e. the cohomology group $H^0(Y,\mathcal{O}(T_{X/Y}))$. Given the above assumption that $H^1(Y,\mathcal{O}(T_{X/Y}))=0$, this map is an isomorphism. Roughly speaking, one can view a holomorphic section of the normal bundle of $Y$ as a first order deformation of $Y$ in $X$, and unobstructedness means that each such first order deformation can be extended to an "honest" family of deformations.

On the other hand, Kodaira does not explore what happens if one weakens the condition $H^1(Y,\mathcal{O}(T_{X/Y}))=0$. For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y})$, it seems plausible that it isn't necessary for the entire group $H^1(Y,\mathcal{O}(T_{X/Y}))=0$ to vanish in order for $v$ to extend to an honest family of deformations.

$\textbf{Question 1}$: For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y}))$, is there a way to determine when $v$ extends to an honest family of deformations? In particular, is there a map $T:H^0(Y,\mathcal{O}(T_{X/Y}))\to H^1(Y,\mathcal{O}(T_{X/Y}))$ which can be described in a reasonably explicit way such that $v$ extends if and only if $T(v)=0$?

One possible solution to this problem would be to derive a Maurer-Cartan type of equation. It's my understanding that, on general grounds, one expects the deformation theory of almost any structure to be controlled by a differential graded Lie algebra (DGA), with (perhaps formal) deformations corresponding to solutions of the Maurer-Cartan equation. One candidate for the DGA in the above example would be the space of sections of the Dolbeault complex $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ associated to the holomorphic vector bundle $\Lambda^{\bullet} T_{X/Y}$. This comes equipped with the Dolbeault differential, but no obvious Lie bracket (at least not obvious to me).

$\textbf{Question 2}:$ Can $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ be given the structure of a DGA (i.e. equipped with a bracket compatible with the Dolbeault differential), such that formal deformations of $Y$ as a complex submanifold of $X$ are given by solutions to the Maurer-Cartan equation? If not, is there some other DGA controlling the deformation theory of a complex submanifold which has a "geometric" description (i.e. as the space of sections of some vector bundle)?

What is the DGLA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. A classical result, proven by Kodaira, is that if the cohomology group $H^1(Y,\mathcal{O}(T_{X/Y}))$ vanishes, then the deformation theory of $Y$ as a complex submanifold of $X$ is unobstructed. More precisely, there exists a "maximal family" of deformations of $Y$ in $X$: this consists of a complex manifold $W$ (say with a marked point $w_0\in W$), a complex submanifold $V$ of $W\times X$, such that for each point $w\in W$ the intersection of $V$ with $w\times X$ is a complex compact submanifold of $X$, which in the case $w=w_0$ is equal to $Y$. Furthermore, this family of submanifolds is "maximal" (or "universal") in an appropriate sense. By saying the deformation problem is unobstructed, I mean the following: for any family as described above (maximal or not) there is a canonical injective complex linear map from the tangent space of $W$ at $w_0$ to the space of holomorphic sections of $T_{X/Y}$, i.e. the cohomology group $H^0(Y,\mathcal{O}(T_{X/Y}))$. Given the above assumption that $H^1(Y,\mathcal{O}(T_{X/Y}))=0$, this map is an isomorphism. Roughly speaking, one can view a holomorphic section of the normal bundle of $Y$ as a first order deformation of $Y$ in $X$, and unobstructedness means that each such first order deformation can be extended to an "honest" family of deformations.

On the other hand, Kodaira does not explore what happens if one weakens the condition $H^1(Y,\mathcal{O}(T_{X/Y}))=0$. For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y})$, it seems plausible that it isn't necessary for the entire group $H^1(Y,\mathcal{O}(T_{X/Y}))=0$ to vanish in order for $v$ to extend to an honest family of deformations.

$\textbf{Question 1}$: For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y}))$, is there a way to determine when $v$ extends to an honest family of deformations? In particular, is there a map $T:H^0(Y,\mathcal{O}(T_{X/Y}))\to H^1(Y,\mathcal{O}(T_{X/Y}))$ which can be described in a reasonably explicit way such that $v$ extends if and only if $T(v)=0$?

One possible solution to this problem would be to derive a Maurer-Cartan type of equation. It's my understanding that, on general grounds, one expects the deformation theory of almost any structure to be controlled by a differential graded Lie algebra (DGLA), with (perhaps formal) deformations corresponding to solutions of the Maurer-Cartan equation. One candidate for the DGA in the above example would be the space of sections of the Dolbeault complex $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ associated to the holomorphic vector bundle $\Lambda^{\bullet} T_{X/Y}$. This comes equipped with the Dolbeault differential, but no obvious Lie bracket (at least not obvious to me).

$\textbf{Question 2}:$ Can $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ be given the structure of a DGLA (i.e. equipped with a bracket compatible with the Dolbeault differential), such that formal deformations of $Y$ as a complex submanifold of $X$ are given by solutions to the Maurer-Cartan equation? If not, is there some other DGLA controlling the deformation theory of a complex submanifold which has a "geometric" description (i.e. as the space of sections of some vector bundle)?

Source Link
Braxton Collier
Braxton Collier
  • 361
  • 1
  • 8

What is the DGA controlling the deformation theory of a complex submanifold?

Let $X$ be a complex manifold, $Y\hookrightarrow X$ a complex compact submanifold. Let $T_{X/Y}$ denote the normal bundle of $Y$ in $X$, and $\mathcal{O}(T_{X/Y})$ its sheaf of holomorphic sections. A classical result, proven by Kodaira, is that if the cohomology group $H^1(Y,\mathcal{O}(T_{X/Y}))$ vanishes, then the deformation theory of $Y$ as a complex submanifold of $X$ is unobstructed. More precisely, there exists a "maximal family" of deformations of $Y$ in $X$: this consists of a complex manifold $W$ (say with a marked point $w_0\in W$), a complex submanifold $V$ of $W\times X$, such that for each point $w\in W$ the intersection of $V$ with $w\times X$ is a complex compact submanifold of $X$, which in the case $w=w_0$ is equal to $Y$. Furthermore, this family of submanifolds is "maximal" (or "universal") in an appropriate sense. By saying the deformation problem is unobstructed, I mean the following: for any family as described above (maximal or not) there is a canonical injective complex linear map from the tangent space of $W$ at $w_0$ to the space of holomorphic sections of $T_{X/Y}$, i.e. the cohomology group $H^0(Y,\mathcal{O}(T_{X/Y}))$. Given the above assumption that $H^1(Y,\mathcal{O}(T_{X/Y}))=0$, this map is an isomorphism. Roughly speaking, one can view a holomorphic section of the normal bundle of $Y$ as a first order deformation of $Y$ in $X$, and unobstructedness means that each such first order deformation can be extended to an "honest" family of deformations.

On the other hand, Kodaira does not explore what happens if one weakens the condition $H^1(Y,\mathcal{O}(T_{X/Y}))=0$. For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y})$, it seems plausible that it isn't necessary for the entire group $H^1(Y,\mathcal{O}(T_{X/Y}))=0$ to vanish in order for $v$ to extend to an honest family of deformations.

$\textbf{Question 1}$: For a given first order deformation $v\in H^0(Y,\mathcal{O}(T_{X/Y}))$, is there a way to determine when $v$ extends to an honest family of deformations? In particular, is there a map $T:H^0(Y,\mathcal{O}(T_{X/Y}))\to H^1(Y,\mathcal{O}(T_{X/Y}))$ which can be described in a reasonably explicit way such that $v$ extends if and only if $T(v)=0$?

One possible solution to this problem would be to derive a Maurer-Cartan type of equation. It's my understanding that, on general grounds, one expects the deformation theory of almost any structure to be controlled by a differential graded Lie algebra (DGA), with (perhaps formal) deformations corresponding to solutions of the Maurer-Cartan equation. One candidate for the DGA in the above example would be the space of sections of the Dolbeault complex $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ associated to the holomorphic vector bundle $\Lambda^{\bullet} T_{X/Y}$. This comes equipped with the Dolbeault differential, but no obvious Lie bracket (at least not obvious to me).

$\textbf{Question 2}:$ Can $\Omega^{\bullet}(Y;\Lambda^{\bullet}T_{X/Y})$ be given the structure of a DGA (i.e. equipped with a bracket compatible with the Dolbeault differential), such that formal deformations of $Y$ as a complex submanifold of $X$ are given by solutions to the Maurer-Cartan equation? If not, is there some other DGA controlling the deformation theory of a complex submanifold which has a "geometric" description (i.e. as the space of sections of some vector bundle)?