리만-로흐 정리

개요

• $X$ : 종수 (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://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
• 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).