"리만-로흐 정리"의 두 판 사이의 차이
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==메모== | ==메모== | ||
* 코쉬-리만 연산자의 index = 1-g | * 코쉬-리만 연산자의 index = 1-g | ||
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* Hitchin, Nigel. 2010. “The Atiyah–Singer Index Theorem.” In The Abel Prize, edited by Helge Holden and Ragni Piene, 117–152. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-642-01373-7_7. | * Hitchin, Nigel. 2010. “The Atiyah–Singer Index Theorem.” In The Abel Prize, edited by Helge Holden and Ragni Piene, 117–152. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-642-01373-7_7. | ||
** http://books.google.de/books?id=bzn2RIYwOXUC&pg=PA118&dq=abel+prize+riemann-roch+index&hl=en&sa=X&ei=P8WoUZbAL6GkigK25YHgAQ&ved=0CC4Q6AEwAA#v=onepage&q=abel%20prize%20riemann-roch%20index&f=false | ** http://books.google.de/books?id=bzn2RIYwOXUC&pg=PA118&dq=abel+prize+riemann-roch+index&hl=en&sa=X&ei=P8WoUZbAL6GkigK25YHgAQ&ved=0CC4Q6AEwAA#v=onepage&q=abel%20prize%20riemann-roch%20index&f=false | ||
* http://mathoverflow.net/questions/7689/why-is-riemann-roch-an-index-problem | * http://mathoverflow.net/questions/7689/why-is-riemann-roch-an-index-problem | ||
* https://www.math.ucdavis.edu/~kapovich/RS/RiemannRoch2.pdf | * https://www.math.ucdavis.edu/~kapovich/RS/RiemannRoch2.pdf | ||
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| + | ==관련논문== | ||
| + | * Mundy, Sam. ‘A New Proof of an Arithmetic Riemann-Roch Theorem’. arXiv:1410.8025 [math], 29 October 2014. http://arxiv.org/abs/1410.8025. | ||
| + | * Simha, R. R. 1981. “The Riemann-Roch Theorem for Compact Riemann Surfaces.” L’Enseignement Mathématique. Revue Internationale. IIe Série 27 (3-4): 185–196 (1982). | ||
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[[분류:리만곡면론]] | [[분류:리만곡면론]] | ||
2014년 10월 29일 (수) 19:45 판
개요
- X : genus 가 g인 컴팩트 리만곡면
- L : line bundle of degree d
- \(H^{0}(L),H^{1}(L)\) : $L$의 holomorphic section으로 주어지는 sheaf에 대한 코호몰로지 군. 유한차원 복소벡터공간
- $p>1$이면, $H^{p}(L)=0$
- $h^{p}(L)=\operatorname{dim}H^{p}(L)$
- 리만-로흐 정리
\[h^{0}(L)-h^{1}(L)=d-g+1\]
- 세르의 쌍대성을 이용하면, 다음과 같이 표현된다
\[h^{0}(L)-h^{0}(L^{-1}\otimes K)=d-g+1\] 여기서 $K$는 $X$에 정의된 canonical bundle
line bundle
- divisor $D=p_1+\cdots+p_d$, $p_1,\cdot, p_d$ distinct
- $L_D$ : line bundle
- $H^0(L)$ : space of meromorphic functions with at worst simple poles at the $p_i$
- $H^0(L^{-1}\otimes K)$ : space of holomorphic 1-forms vanishing at the $p_i$
메모
- 코쉬-리만 연산자의 index = 1-g
- Hitchin, Nigel. 2010. “The Atiyah–Singer Index Theorem.” In The Abel Prize, edited by Helge Holden and Ragni Piene, 117–152. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/978-3-642-01373-7_7.
- http://mathoverflow.net/questions/7689/why-is-riemann-roch-an-index-problem
- https://www.math.ucdavis.edu/~kapovich/RS/RiemannRoch2.pdf
관련논문
- Mundy, Sam. ‘A New Proof of an Arithmetic Riemann-Roch Theorem’. arXiv:1410.8025 [math], 29 October 2014. http://arxiv.org/abs/1410.8025.
- Simha, R. R. 1981. “The Riemann-Roch Theorem for Compact Riemann Surfaces.” L’Enseignement Mathématique. Revue Internationale. IIe Série 27 (3-4): 185–196 (1982).