Torus and period integrals
$begingroup$
I'm following a course in Riemann surfaces, and I'd like to solve the exercise below.
Let $L$ a lattice in $mathbb{C}$, and let $T:= mathbb{C}/L$ the corresponding torus.
i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the standard coordinate in $mathbb{C}$ and $H_1^{dR}(T)$ the first de Rham cohomology group of $T$).
ii) Chose a canonical basis of $H^1(T)$ and compute the period integrals of $dx$ and $dy$ with respect to that basis (where $H^1(T)$ is the first homology group of $T$).
Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $H^1$?
For example I know that $H^1(T)$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.
Many thanks.
riemann-surfaces
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
add a comment |
$begingroup$
I'm following a course in Riemann surfaces, and I'd like to solve the exercise below.
Let $L$ a lattice in $mathbb{C}$, and let $T:= mathbb{C}/L$ the corresponding torus.
i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the standard coordinate in $mathbb{C}$ and $H_1^{dR}(T)$ the first de Rham cohomology group of $T$).
ii) Chose a canonical basis of $H^1(T)$ and compute the period integrals of $dx$ and $dy$ with respect to that basis (where $H^1(T)$ is the first homology group of $T$).
Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $H^1$?
For example I know that $H^1(T)$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.
Many thanks.
riemann-surfaces
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
add a comment |
$begingroup$
I'm following a course in Riemann surfaces, and I'd like to solve the exercise below.
Let $L$ a lattice in $mathbb{C}$, and let $T:= mathbb{C}/L$ the corresponding torus.
i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the standard coordinate in $mathbb{C}$ and $H_1^{dR}(T)$ the first de Rham cohomology group of $T$).
ii) Chose a canonical basis of $H^1(T)$ and compute the period integrals of $dx$ and $dy$ with respect to that basis (where $H^1(T)$ is the first homology group of $T$).
Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $H^1$?
For example I know that $H^1(T)$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.
Many thanks.
riemann-surfaces
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
$endgroup$
I'm following a course in Riemann surfaces, and I'd like to solve the exercise below.
Let $L$ a lattice in $mathbb{C}$, and let $T:= mathbb{C}/L$ the corresponding torus.
i) Prove that $dx$ and $dy$ span $H_1^{dR}(T)$ (where $x,y$ are the standard coordinate in $mathbb{C}$ and $H_1^{dR}(T)$ the first de Rham cohomology group of $T$).
ii) Chose a canonical basis of $H^1(T)$ and compute the period integrals of $dx$ and $dy$ with respect to that basis (where $H^1(T)$ is the first homology group of $T$).
Our lectures are very abstract so I'm not sure on how should I do the concrete computations. Furthermore I'd like to know what are the general techniques to attack these kind of problems. Especially, for question ii), is there a standard recipe to compute an explicit canonical base of $H^1$?
For example I know that $H^1(T)$ is given by the 1-cycle through the hole and the 1-cycle around the hole, but I can't find a way to write down the concrete integrals.
Many thanks.
riemann-surfaces
riemann-surfaces
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
asked Jan 14 at 15:17
Filippo SneakerheadFilippo Sneakerhead
285
285
add a comment |
add a comment |
1 Answer
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Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
$endgroup$
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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active
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active
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votes
$begingroup$
Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
$endgroup$
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
add a comment |
$begingroup$
Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
$endgroup$
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
add a comment |
$begingroup$
Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
$endgroup$
Remark that for every real $a,b, adx+bdy$ is closed, suppose that this form is exact, if you have $df=adx+bdy$ where $f$ is a differentiable function defined on the torus, then $f$ has an extremum point $u$ and $df_u=0$, this implies that $(adx+dy)_u$ which is equivalent to $a=b=0$. So $dx$ and $dy$ are linearly independent. Since $H^1(T)$ is $2$-dimensional you have your basis.
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
answered Jan 14 at 17:22
Tsemo AristideTsemo Aristide
59.7k11446
59.7k11446
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
add a comment |
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
$begingroup$
Great, thank you, but what about the period matrix and the first homology group?
$endgroup$
– Filippo Sneakerhead
Jan 16 at 14:24
add a comment |
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