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Regarding wells' p.156 conjugate liner 1-form
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I am kind of confused with the conjugate linear 1-form, because it seems that to any conjugate linear 1-form $x$ and any $vin E$, $x(v)=0$, in this sense the conjugation doesn't make sense.
I am considering an example: Let $X$ be a complex manifold of dim $n$ and $E$ be the tangent space of $X$ at $a$: $T_a^1,0X=<...,fracpartialpartial x_j-ifracpartialpartial y_j,...>=<...,fracpartialpartial z_j,...>$, where $fracpartialpartial z_j=1/2(fracpartialpartial x_j-ifracpartialpartial y_j)$,
Now with this notation, we have $F$ above should be $(T_a^*X_0)_mathbbC=(T_a^*X)^1,0oplus (T_a^*X)^0,1=<...,dz_j,...>oplus <...,dbarz_j,...>$, s.t. $dz_i(fracpartialpartial z_j)=delta_ij$ and $dbarz_i(fracpartialpartial z_j)=0$. And $wedge^1,0F=(T_a^*X)^1,0$, $wedge^0,1F=(T_a^*X)^0,1$.
But by $dbarz_i(fracpartialpartial z_j)=0$, to any $vin E=T_a^1,0X=<...,fracpartialpartial z_j,...>$, we have $dbarz_i(v)=0$. But $dbarz_i$ is the conjugate of $dz_i$, and $dz_i(v)$ might not have to be $0$, thus there is a contradiction. Where did I go wrong?
More generally, let $E=<d_1,..d_n>=<x_1+iy_1,...,x_n+iy_n>$, then $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then the $wedge^1,0F$ as the set of complex linear 1-forms should be $<...,1/2(x_j^*-iy_j^*),...>$, then what's the conjugate? It seems that it has a similar problem.
differential-geometry differential-forms complex-geometry
asked Jul 31 at 2:44
Danny
777212
add a comment |Â
up vote
2
down vote
favorite
I am kind of confused with the conjugate linear 1-form, because it seems that to any conjugate linear 1-form $x$ and any $vin E$, $x(v)=0$, in this sense the conjugation doesn't make sense.
I am considering an example: Let $X$ be a complex manifold of dim $n$ and $E$ be the tangent space of $X$ at $a$: $T_a^1,0X=<...,fracpartialpartial x_j-ifracpartialpartial y_j,...>=<...,fracpartialpartial z_j,...>$, where $fracpartialpartial z_j=1/2(fracpartialpartial x_j-ifracpartialpartial y_j)$,
Now with this notation, we have $F$ above should be $(T_a^*X_0)_mathbbC=(T_a^*X)^1,0oplus (T_a^*X)^0,1=<...,dz_j,...>oplus <...,dbarz_j,...>$, s.t. $dz_i(fracpartialpartial z_j)=delta_ij$ and $dbarz_i(fracpartialpartial z_j)=0$. And $wedge^1,0F=(T_a^*X)^1,0$, $wedge^0,1F=(T_a^*X)^0,1$.
But by $dbarz_i(fracpartialpartial z_j)=0$, to any $vin E=T_a^1,0X=<...,fracpartialpartial z_j,...>$, we have $dbarz_i(v)=0$. But $dbarz_i$ is the conjugate of $dz_i$, and $dz_i(v)$ might not have to be $0$, thus there is a contradiction. Where did I go wrong?
More generally, let $E=<d_1,..d_n>=<x_1+iy_1,...,x_n+iy_n>$, then $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then the $wedge^1,0F$ as the set of complex linear 1-forms should be $<...,1/2(x_j^*-iy_j^*),...>$, then what's the conjugate? It seems that it has a similar problem.
differential-geometry differential-forms complex-geometry
asked Jul 31 at 2:44
Danny
777212
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am kind of confused with the conjugate linear 1-form, because it seems that to any conjugate linear 1-form $x$ and any $vin E$, $x(v)=0$, in this sense the conjugation doesn't make sense.
I am considering an example: Let $X$ be a complex manifold of dim $n$ and $E$ be the tangent space of $X$ at $a$: $T_a^1,0X=<...,fracpartialpartial x_j-ifracpartialpartial y_j,...>=<...,fracpartialpartial z_j,...>$, where $fracpartialpartial z_j=1/2(fracpartialpartial x_j-ifracpartialpartial y_j)$,
Now with this notation, we have $F$ above should be $(T_a^*X_0)_mathbbC=(T_a^*X)^1,0oplus (T_a^*X)^0,1=<...,dz_j,...>oplus <...,dbarz_j,...>$, s.t. $dz_i(fracpartialpartial z_j)=delta_ij$ and $dbarz_i(fracpartialpartial z_j)=0$. And $wedge^1,0F=(T_a^*X)^1,0$, $wedge^0,1F=(T_a^*X)^0,1$.
But by $dbarz_i(fracpartialpartial z_j)=0$, to any $vin E=T_a^1,0X=<...,fracpartialpartial z_j,...>$, we have $dbarz_i(v)=0$. But $dbarz_i$ is the conjugate of $dz_i$, and $dz_i(v)$ might not have to be $0$, thus there is a contradiction. Where did I go wrong?
More generally, let $E=<d_1,..d_n>=<x_1+iy_1,...,x_n+iy_n>$, then $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then the $wedge^1,0F$ as the set of complex linear 1-forms should be $<...,1/2(x_j^*-iy_j^*),...>$, then what's the conjugate? It seems that it has a similar problem.
differential-geometry differential-forms complex-geometry
asked Jul 31 at 2:44
Danny
777212
I am kind of confused with the conjugate linear 1-form, because it seems that to any conjugate linear 1-form $x$ and any $vin E$, $x(v)=0$, in this sense the conjugation doesn't make sense.
I am considering an example: Let $X$ be a complex manifold of dim $n$ and $E$ be the tangent space of $X$ at $a$: $T_a^1,0X=<...,fracpartialpartial x_j-ifracpartialpartial y_j,...>=<...,fracpartialpartial z_j,...>$, where $fracpartialpartial z_j=1/2(fracpartialpartial x_j-ifracpartialpartial y_j)$,
Now with this notation, we have $F$ above should be $(T_a^*X_0)_mathbbC=(T_a^*X)^1,0oplus (T_a^*X)^0,1=<...,dz_j,...>oplus <...,dbarz_j,...>$, s.t. $dz_i(fracpartialpartial z_j)=delta_ij$ and $dbarz_i(fracpartialpartial z_j)=0$. And $wedge^1,0F=(T_a^*X)^1,0$, $wedge^0,1F=(T_a^*X)^0,1$.
But by $dbarz_i(fracpartialpartial z_j)=0$, to any $vin E=T_a^1,0X=<...,fracpartialpartial z_j,...>$, we have $dbarz_i(v)=0$. But $dbarz_i$ is the conjugate of $dz_i$, and $dz_i(v)$ might not have to be $0$, thus there is a contradiction. Where did I go wrong?
More generally, let $E=<d_1,..d_n>=<x_1+iy_1,...,x_n+iy_n>$, then $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then the $wedge^1,0F$ as the set of complex linear 1-forms should be $<...,1/2(x_j^*-iy_j^*),...>$, then what's the conjugate? It seems that it has a similar problem.
differential-geometry differential-forms complex-geometry
asked Jul 31 at 2:44
Danny
777212
edited Jul 31 at 4:07
asked Jul 31 at 2:44
Danny
777212
asked Jul 31 at 2:44
Danny
777212
asked Jul 31 at 2:44
Danny
777212
777212
add a comment |Â
add a comment |Â
1 Answer
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The confusion is at the line $E = T^1,0X$. It is true that there is an (complex isomorphism between $(E, J)$ and $(T^1,0X, sqrt-1)$ given by
$$ vin Emapsto v - sqrt-1 Jv in T^1,0X,$$
but if you think of $E cong Eotimes 1$ in $Eotimes mathbb C$, then $Ecap T^1,0X = 0$.
answered Jul 31 at 12:21
John Ma
37.5k93669
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
 |Â
show 2 more comments
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
The confusion is at the line $E = T^1,0X$. It is true that there is an (complex isomorphism between $(E, J)$ and $(T^1,0X, sqrt-1)$ given by
$$ vin Emapsto v - sqrt-1 Jv in T^1,0X,$$
but if you think of $E cong Eotimes 1$ in $Eotimes mathbb C$, then $Ecap T^1,0X = 0$.
answered Jul 31 at 12:21
John Ma
37.5k93669
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
 |Â
show 2 more comments
up vote
1
down vote
accepted
The confusion is at the line $E = T^1,0X$. It is true that there is an (complex isomorphism between $(E, J)$ and $(T^1,0X, sqrt-1)$ given by
$$ vin Emapsto v - sqrt-1 Jv in T^1,0X,$$
but if you think of $E cong Eotimes 1$ in $Eotimes mathbb C$, then $Ecap T^1,0X = 0$.
answered Jul 31 at 12:21
John Ma
37.5k93669
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
 |Â
show 2 more comments
up vote
1
down vote
accepted
up vote
1
down vote
accepted
The confusion is at the line $E = T^1,0X$. It is true that there is an (complex isomorphism between $(E, J)$ and $(T^1,0X, sqrt-1)$ given by
$$ vin Emapsto v - sqrt-1 Jv in T^1,0X,$$
but if you think of $E cong Eotimes 1$ in $Eotimes mathbb C$, then $Ecap T^1,0X = 0$.
answered Jul 31 at 12:21
John Ma
37.5k93669
The confusion is at the line $E = T^1,0X$. It is true that there is an (complex isomorphism between $(E, J)$ and $(T^1,0X, sqrt-1)$ given by
$$ vin Emapsto v - sqrt-1 Jv in T^1,0X,$$
but if you think of $E cong Eotimes 1$ in $Eotimes mathbb C$, then $Ecap T^1,0X = 0$.
answered Jul 31 at 12:21
John Ma
37.5k93669
answered Jul 31 at 12:21
John Ma
37.5k93669
answered Jul 31 at 12:21
John Ma
37.5k93669
answered Jul 31 at 12:21
John Ma
37.5k93669
37.5k93669
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
 |Â
show 2 more comments
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
I feel a little confused, here $E$ is a complex space, but it seems you see $E$ as a real space.
– Danny
Jul 31 at 12:59
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Also, for my last part, the general case, the basis of the conjugate should be $ ...,1/2(x_j^*+iy_j^*,...$, then still we have $1/2(x_j^*+iy_j^*)(x_k+iy_k)=0$, thus the conjugate element acts trivially on $E$.
– Danny
Jul 31 at 13:04
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Yes I am thinking of $E$ as $2n$-dimensional real space and this is confusing. $E$ is complex so it has a complex structure $J$. We do not call it $i$, since that will cause confusion when you consider $Eotimes mathbb C$. So your $x_j + i y_j$ is actually not in $E$ but only in $Eotimes mathbb C$.
– John Ma
Jul 31 at 13:13
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
Now if $C^n=E=<d_1,..d_n>cong <x_1, y_1,...,x_n,y_n>=R^2n$ and $F=<x_1^*,y_1^*,...,x_n^*,y_n^*>oplus i<x_1^*,y_1^*,...,x_n^*,y_n^*>$, then how do elements of $F$ act on $E$? Can you give an example?
– Danny
Jul 31 at 13:22
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
For example $(x_1^* + i y_1^*) (x_1) = x_1^* (x_1) + i y_1^*(x_1) = 1 + icdot 0 = 1$.
– John Ma
Jul 31 at 13:35
 |Â
show 2 more comments
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