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.










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  • 1




    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
    7 hours ago

















up vote
4
down vote

favorite
2












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.










share|cite|improve this question


















  • 1




    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
    7 hours ago















up vote
4
down vote

favorite
2









up vote
4
down vote

favorite
2






2





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.










share|cite|improve this question













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






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asked 10 hours ago









Piotr Hajlasz

5,67632153




5,67632153








  • 1




    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
    7 hours ago
















  • 1




    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
    7 hours ago










1




1




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
7 hours ago






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
7 hours ago












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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.






share|cite|improve this answer





















  • Thank you so much! That is exactly what I was looking for.
    – Piotr Hajlasz
    3 hours ago






  • 2




    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
    2 hours ago













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up vote
6
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.






share|cite|improve this answer





















  • Thank you so much! That is exactly what I was looking for.
    – Piotr Hajlasz
    3 hours ago






  • 2




    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
    2 hours ago

















up vote
6
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.






share|cite|improve this answer





















  • Thank you so much! That is exactly what I was looking for.
    – Piotr Hajlasz
    3 hours ago






  • 2




    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
    2 hours ago















up vote
6
down vote



accepted







up vote
6
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.






share|cite|improve this answer












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.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 4 hours ago









Robert Bryant

72.1k5211310




72.1k5211310












  • Thank you so much! That is exactly what I was looking for.
    – Piotr Hajlasz
    3 hours ago






  • 2




    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
    2 hours ago




















  • Thank you so much! That is exactly what I was looking for.
    – Piotr Hajlasz
    3 hours ago






  • 2




    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
    2 hours ago


















Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
3 hours ago




Thank you so much! That is exactly what I was looking for.
– Piotr Hajlasz
3 hours ago




2




2




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
2 hours ago






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
2 hours ago




















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