Lefschetz operator
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Let $omega=sum_{i=1}^n dx_iwedge dy_iinbigwedge^2mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $kleq n$, the Lefschetz operator
$L^{n-k}:bigwedge^kmathbb{R}^{2n}to
bigwedge^{2n-k}mathbb{R}^{2n}$ defined by $$
L^{n-k}alpha=alphawedgeunderbrace{omegawedgeldotswedgeomega}_{n-k}=alphawedgeomega^{n-k}
$$ is an isomorphism.
This follows from Proposition 1.2.30 in Huybrechts' Complex Geometry. The proof is slick and it uses representation theory.
Question: What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
proof by brute force. A proof by brute force is not difficult but ugly
and perhaps in some reference I can find an elegant presentation of
such a proof.
This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except the one by Huybrechts.
linear-algebra sg.symplectic-geometry
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
add a comment |
up vote
5
down vote
favorite
Let $omega=sum_{i=1}^n dx_iwedge dy_iinbigwedge^2mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $kleq n$, the Lefschetz operator
$L^{n-k}:bigwedge^kmathbb{R}^{2n}to
bigwedge^{2n-k}mathbb{R}^{2n}$ defined by $$
L^{n-k}alpha=alphawedgeunderbrace{omegawedgeldotswedgeomega}_{n-k}=alphawedgeomega^{n-k}
$$ is an isomorphism.
This follows from Proposition 1.2.30 in Huybrechts' Complex Geometry. The proof is slick and it uses representation theory.
Question: What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
proof by brute force. A proof by brute force is not difficult but ugly
and perhaps in some reference I can find an elegant presentation of
such a proof.
This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except the one by Huybrechts.
linear-algebra sg.symplectic-geometry
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
2
Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37
add a comment |
up vote
5
down vote
favorite
up vote
5
down vote
favorite
Let $omega=sum_{i=1}^n dx_iwedge dy_iinbigwedge^2mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $kleq n$, the Lefschetz operator
$L^{n-k}:bigwedge^kmathbb{R}^{2n}to
bigwedge^{2n-k}mathbb{R}^{2n}$ defined by $$
L^{n-k}alpha=alphawedgeunderbrace{omegawedgeldotswedgeomega}_{n-k}=alphawedgeomega^{n-k}
$$ is an isomorphism.
This follows from Proposition 1.2.30 in Huybrechts' Complex Geometry. The proof is slick and it uses representation theory.
Question: What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
proof by brute force. A proof by brute force is not difficult but ugly
and perhaps in some reference I can find an elegant presentation of
such a proof.
This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except the one by Huybrechts.
linear-algebra sg.symplectic-geometry
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
Let $omega=sum_{i=1}^n dx_iwedge dy_iinbigwedge^2mathbb{R}^{2n}$ be a standard symplectic form. The following result is due to Lefschetz:
For $kleq n$, the Lefschetz operator
$L^{n-k}:bigwedge^kmathbb{R}^{2n}to
bigwedge^{2n-k}mathbb{R}^{2n}$ defined by $$
L^{n-k}alpha=alphawedgeunderbrace{omegawedgeldotswedgeomega}_{n-k}=alphawedgeomega^{n-k}
$$ is an isomorphism.
This follows from Proposition 1.2.30 in Huybrechts' Complex Geometry. The proof is slick and it uses representation theory.
Question: What are other standard references for this result? I want to quote it properly and I would like to see a reference to a
proof by brute force. A proof by brute force is not difficult but ugly
and perhaps in some reference I can find an elegant presentation of
such a proof.
This result seems so fundamental that it should be available in many textbooks, but I am not aware of any except the one by Huybrechts.
linear-algebra sg.symplectic-geometry
linear-algebra sg.symplectic-geometry
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
asked Nov 29 at 17:19
Piotr Hajlasz
5,89642253
5,89642253
2
Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37
add a comment |
2
Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37
2
2
Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37
Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37
add a comment |
1 Answer
1
active
oldest
votes
up vote
10
down vote
accepted
There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.
I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
10
down vote
accepted
There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.
I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
add a comment |
up vote
10
down vote
accepted
There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.
I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
add a comment |
up vote
10
down vote
accepted
up vote
10
down vote
accepted
There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.
I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
There is an elementary proof in our 2003 book Exterior Differential Systems and Euler-Lagrange Partial Differential Equations (Bryant, et al, University of Chicago Press). It does not use any representation theory and is not 'brute force'; it only takes a couple of paragraphs using elementary facts about exterior algebra. See Proposition 1.1a, with the proof on page 13.
I learned that proof from Eugenio Calabi more than 30 years ago, and he told me that he had found it sometime back in the 50s.
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
answered Nov 29 at 23:25
Robert Bryant
72.5k5213313
72.5k5213313
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
add a comment |
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
Nov 30 at 0:47
5
5
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
Wow. That's the easiest shortest brute force proof I've ever seen. It can be found on arxiv: arxiv.org/abs/math/0207039
– Deane Yang
Nov 30 at 1:07
add a comment |
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Another older reference for this result is André Weil Introductions à l'étude des variétés kählériennes, Herman 1958. (The corollary on page 28 is the reuslt you mentioned.The proof is the standard proof using representations of $sl_2$.
– Liviu Nicolaescu
Nov 29 at 20:37