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Some questions about Clemens' paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory
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I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the classical deformaion theory developed by Kodaira-Spencer. In particular, I have the following questions:
In the last paragraph of page 11, Clemens write, If for an ideal $mathfrakAsubseteq mathfrakm=t_1,cdots,t_s$, we let
$$Delta_mathfrakA:= SpecfracmathbbC[t]mathfrakAsubseteq Delta.$$
I should add that $Delta$ is a polydisc with coordinates $t_1,cdots,t_s$. My question is that what is the meaning of the inclusion $SpecfracmathbbC[t]mathfrakAsubseteq Delta$? Is this inclusion in the category of schemes or complex sapces? I have the same question for
$$M_mathfrakA:=pi^-1(Delta_mathfrakA),$$
where $pi: Mto Delta$ is a deformation of the central fiber $M_0$.In page 18. What is the meaning of the sentence "$M_mathfrakA/Delta_mathfrakA$is an infinitesimal deformation of compact Kahler manifolds"? What is the definition of extending a family $M_mathfrakA/Delta_mathfrakA$ to a family $M_mathfrakA'/Delta_mathfrakA'$ for some ideal $mathfrakAsupseteq mathfrakA'supseteq mathfrakmmathfrakA$.?
ag.algebraic-geometry complex-geometry deformation-theory
asked 4 hours ago
Wei Xia
1242
add a comment |Â
up vote
1
down vote
favorite
I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the classical deformaion theory developed by Kodaira-Spencer. In particular, I have the following questions:
In the last paragraph of page 11, Clemens write, If for an ideal $mathfrakAsubseteq mathfrakm=t_1,cdots,t_s$, we let
$$Delta_mathfrakA:= SpecfracmathbbC[t]mathfrakAsubseteq Delta.$$
I should add that $Delta$ is a polydisc with coordinates $t_1,cdots,t_s$. My question is that what is the meaning of the inclusion $SpecfracmathbbC[t]mathfrakAsubseteq Delta$? Is this inclusion in the category of schemes or complex sapces? I have the same question for
$$M_mathfrakA:=pi^-1(Delta_mathfrakA),$$
where $pi: Mto Delta$ is a deformation of the central fiber $M_0$.In page 18. What is the meaning of the sentence "$M_mathfrakA/Delta_mathfrakA$is an infinitesimal deformation of compact Kahler manifolds"? What is the definition of extending a family $M_mathfrakA/Delta_mathfrakA$ to a family $M_mathfrakA'/Delta_mathfrakA'$ for some ideal $mathfrakAsupseteq mathfrakA'supseteq mathfrakmmathfrakA$.?
ag.algebraic-geometry complex-geometry deformation-theory
asked 4 hours ago
Wei Xia
1242
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the classical deformaion theory developed by Kodaira-Spencer. In particular, I have the following questions:
In the last paragraph of page 11, Clemens write, If for an ideal $mathfrakAsubseteq mathfrakm=t_1,cdots,t_s$, we let
$$Delta_mathfrakA:= SpecfracmathbbC[t]mathfrakAsubseteq Delta.$$
I should add that $Delta$ is a polydisc with coordinates $t_1,cdots,t_s$. My question is that what is the meaning of the inclusion $SpecfracmathbbC[t]mathfrakAsubseteq Delta$? Is this inclusion in the category of schemes or complex sapces? I have the same question for
$$M_mathfrakA:=pi^-1(Delta_mathfrakA),$$
where $pi: Mto Delta$ is a deformation of the central fiber $M_0$.In page 18. What is the meaning of the sentence "$M_mathfrakA/Delta_mathfrakA$is an infinitesimal deformation of compact Kahler manifolds"? What is the definition of extending a family $M_mathfrakA/Delta_mathfrakA$ to a family $M_mathfrakA'/Delta_mathfrakA'$ for some ideal $mathfrakAsupseteq mathfrakA'supseteq mathfrakmmathfrakA$.?
ag.algebraic-geometry complex-geometry deformation-theory
asked 4 hours ago
Wei Xia
1242
I am reading the paper Cohomology and Obstructions I: Geometry of formal Kuranishi theory written by Herbert Clemens. I have found this paper hard to follow even though I am familiar with the classical deformaion theory developed by Kodaira-Spencer. In particular, I have the following questions:
In the last paragraph of page 11, Clemens write, If for an ideal $mathfrakAsubseteq mathfrakm=t_1,cdots,t_s$, we let
$$Delta_mathfrakA:= SpecfracmathbbC[t]mathfrakAsubseteq Delta.$$
I should add that $Delta$ is a polydisc with coordinates $t_1,cdots,t_s$. My question is that what is the meaning of the inclusion $SpecfracmathbbC[t]mathfrakAsubseteq Delta$? Is this inclusion in the category of schemes or complex sapces? I have the same question for
$$M_mathfrakA:=pi^-1(Delta_mathfrakA),$$
where $pi: Mto Delta$ is a deformation of the central fiber $M_0$.In page 18. What is the meaning of the sentence "$M_mathfrakA/Delta_mathfrakA$is an infinitesimal deformation of compact Kahler manifolds"? What is the definition of extending a family $M_mathfrakA/Delta_mathfrakA$ to a family $M_mathfrakA'/Delta_mathfrakA'$ for some ideal $mathfrakAsupseteq mathfrakA'supseteq mathfrakmmathfrakA$.?
ag.algebraic-geometry complex-geometry deformation-theory
ag.algebraic-geometry complex-geometry deformation-theory
asked 4 hours ago
Wei Xia
1242
asked 4 hours ago
Wei Xia
1242
asked 4 hours ago
Wei Xia
1242
asked 4 hours ago
Wei Xia
1242
asked 4 hours ago
Wei Xia
1242
1242
add a comment |Â
add a comment |Â
1 Answer
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It is an inclusion of analytic spaces — $Delta $ is not a scheme. If $mathfrakA=(f_1,ldots ,f_p)$, $Delta _mathfrakA$ is the subspace of $Delta $ defined by $f_1=ldots =f_p=0$. I think $mathfrakAneq 0$ is implicitely assumed, so $Delta _mathfrakA$ is a fattening of the origin. Hence $M _mathfrakA$ is what is called an infinitesimal deformation of $M_0$.
If $mathfrakA'subset mathfrakA$, $Delta _mathfrakA$ is a subspace of $Delta _mathfrakA'$, so you look for a family over $Delta _mathfrakA'$ which induces the given family over $Delta _mathfrakA$.
I think your questions would be more appropriate on MSE.
answered 4 hours ago
abx
22.4k34479
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
It is an inclusion of analytic spaces — $Delta $ is not a scheme. If $mathfrakA=(f_1,ldots ,f_p)$, $Delta _mathfrakA$ is the subspace of $Delta $ defined by $f_1=ldots =f_p=0$. I think $mathfrakAneq 0$ is implicitely assumed, so $Delta _mathfrakA$ is a fattening of the origin. Hence $M _mathfrakA$ is what is called an infinitesimal deformation of $M_0$.
If $mathfrakA'subset mathfrakA$, $Delta _mathfrakA$ is a subspace of $Delta _mathfrakA'$, so you look for a family over $Delta _mathfrakA'$ which induces the given family over $Delta _mathfrakA$.
I think your questions would be more appropriate on MSE.
answered 4 hours ago
abx
22.4k34479
add a comment |Â
up vote
3
down vote
It is an inclusion of analytic spaces — $Delta $ is not a scheme. If $mathfrakA=(f_1,ldots ,f_p)$, $Delta _mathfrakA$ is the subspace of $Delta $ defined by $f_1=ldots =f_p=0$. I think $mathfrakAneq 0$ is implicitely assumed, so $Delta _mathfrakA$ is a fattening of the origin. Hence $M _mathfrakA$ is what is called an infinitesimal deformation of $M_0$.
If $mathfrakA'subset mathfrakA$, $Delta _mathfrakA$ is a subspace of $Delta _mathfrakA'$, so you look for a family over $Delta _mathfrakA'$ which induces the given family over $Delta _mathfrakA$.
I think your questions would be more appropriate on MSE.
answered 4 hours ago
abx
22.4k34479
add a comment |Â
up vote
3
down vote
up vote
3
down vote
It is an inclusion of analytic spaces — $Delta $ is not a scheme. If $mathfrakA=(f_1,ldots ,f_p)$, $Delta _mathfrakA$ is the subspace of $Delta $ defined by $f_1=ldots =f_p=0$. I think $mathfrakAneq 0$ is implicitely assumed, so $Delta _mathfrakA$ is a fattening of the origin. Hence $M _mathfrakA$ is what is called an infinitesimal deformation of $M_0$.
If $mathfrakA'subset mathfrakA$, $Delta _mathfrakA$ is a subspace of $Delta _mathfrakA'$, so you look for a family over $Delta _mathfrakA'$ which induces the given family over $Delta _mathfrakA$.
I think your questions would be more appropriate on MSE.
answered 4 hours ago
abx
22.4k34479
It is an inclusion of analytic spaces — $Delta $ is not a scheme. If $mathfrakA=(f_1,ldots ,f_p)$, $Delta _mathfrakA$ is the subspace of $Delta $ defined by $f_1=ldots =f_p=0$. I think $mathfrakAneq 0$ is implicitely assumed, so $Delta _mathfrakA$ is a fattening of the origin. Hence $M _mathfrakA$ is what is called an infinitesimal deformation of $M_0$.
If $mathfrakA'subset mathfrakA$, $Delta _mathfrakA$ is a subspace of $Delta _mathfrakA'$, so you look for a family over $Delta _mathfrakA'$ which induces the given family over $Delta _mathfrakA$.
I think your questions would be more appropriate on MSE.
answered 4 hours ago
abx
22.4k34479
answered 4 hours ago
abx
22.4k34479
answered 4 hours ago
abx
22.4k34479
answered 4 hours ago
abx
22.4k34479
22.4k34479
add a comment |Â
add a comment |Â
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