리만 곡면의 주기 행렬과 겹선형 관계 (bilinear relation)

개요

• $X$ : 종수가 $g$인 컴팩트 리만 곡면
• 다음을 만족하는 \(H_1(X, \mathbb{Z}) \cong \mathbb{Z}^{2g}\)의 기저, 2g 개의 닫힌 곡선 \(a_1, \dots, a_g,b_1,\cdots,b_g\)이 존재

$$ \langle a_i,b_j \rangle = \begin{cases} 1, & \text{if }i=j\\ 0, & \text{if }i\neq j \\ \end{cases} $$

• 다음을 만족하는 \(H^0(X, K) \cong \mathbb{C}^g\)의 기저, holomorphic 1-form $\omega_1,\cdots,\omega_{g}$가 존재

$$ \int_{a_i}\omega_j=\delta_{ij} $$

• 지겔 상반 공간 $\mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\}$
• $\tau_{i,j}=\int_{b_i}\omega_j$로 두면, $\tau=(\tau_{i,j})_{1\leq i,j\leq g}$는 지겔 상반 공간 $\mathcal{H}_g$의 원소이며, $X$의 period 행렬이라 부른다
• $g=3$ 인 경우

$$ \begin{array}{c|ccc|ccc} \text{} & a_1 & a_2 & a_3 & b_1 & b_2 & b_3 \\ \hline \omega _1 & \left\langle a_1|\omega _1\right\rangle & \left\langle a_2|\omega _1\right\rangle & \left\langle a_3|\omega _1\right\rangle & \left\langle b_1|\omega _1\right\rangle & \left\langle b_2|\omega _1\right\rangle & \left\langle b_3|\omega _1\right\rangle \\ \omega _2 & \left\langle a_1|\omega _2\right\rangle & \left\langle a_2|\omega _2\right\rangle & \left\langle a_3|\omega _2\right\rangle & \left\langle b_1|\omega _2\right\rangle & \left\langle b_2|\omega _2\right\rangle & \left\langle b_3|\omega _2\right\rangle \\ \omega _3 & \left\langle a_1|\omega _3\right\rangle & \left\langle a_2|\omega _3\right\rangle & \left\langle a_3|\omega _3\right\rangle & \left\langle b_1|\omega _3\right\rangle & \left\langle b_2|\omega _3\right\rangle & \left\langle b_3|\omega _3\right\rangle \end{array} = \begin{array}{c|ccc|ccc} \text{} & a_1 & a_2 & a_3 & b_1 & b_2 & b_3 \\ \hline \omega _1 & 1 & 0 & 0 & \tau _{1,1} & \tau _{1,2} & \tau _{1,3} \\ \omega _2 & 0 & 1 & 0 & \tau _{2,1} & \tau _{2,2} & \tau _{2,3} \\ \omega _3 & 0 & 0 & 1 & \tau _{3,1} & \tau _{3,2} & \tau _{3,3} \end{array} $$ 여기서 $\left\langle \gamma|\omega\right\rangle=\int_{\gamma}\omega$

예

• 클라인의 4차곡선의 경우, $g=3$인 곡선
• 주기 행렬은 다음과 같이 주어진다

$$ \frac{1}{2} \left( \begin{array}{ccc} \rho & 1 & 1 \\ 1 & \rho & 1 \\ 1 & 1 & \rho \\ \end{array} \right) $$ 여기서 $\rho=\frac{-1+\sqrt{-7}}{2}$.

메모

• http://en.wikipedia.org/wiki/Riemann_bilinear_relations
• http://mathoverflow.net/questions/22286/intuition-behind-riemanns-bilinear-relations
• http://www-nonlinear.physik.uni-bremen.de/~prichter/pdfs/ThetaConst.pdf
• http://magma.maths.usyd.edu.au/magma/handbook/text/1402
• http://www.math.harvard.edu/~ctm/home/text/class/harvard/sem/html/home/notes/99/course.pdf
• \(\omega_i\in \Omega^{1,0}\)
• \((\omega_k,\omega_l)=i\int_{X} \omega_k \wedge \omega_l=0\)
• \(\omega\neq 0\)
• \((\omega,\bar{\omega})=i\int_{X} \omega \wedge \bar{\omega}>0\)

관련된 항목들

• 사교 행렬
• 아벨-야코비 정리

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxV1RCbzVJYWUwOEU/edit
• RiemannCycles
  • Some tools for working on homology cycles for Riemann surfaces presented in an algebraic manner
• http://www.maplesoft.com/support/help/Maple/view.aspx?path=algcurves/periodmatrix
• http://iml.univ-mrs.fr/ati/GeoCrypt2011/slides/molin.pdf‎

리뷰논문, 에세이, 강의노트

• James Carlson and Phillip Griffiths What is...a period domain?, December 2008
• Beginning compact Riemann surface theory

관련논문

• Braden, Harry W., and Timothy P. Northover. 2012. “Bring’s Curve: Its Period Matrix and the Vector of Riemann Constants”. ArXiv e-print 1206.6004. http://arxiv.org/abs/1206.6004.