The harmonic volume for compact Riemann surfaces as a function on the moduli space of Riemann surfaces.

• Project/Area Number: 21740057
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Single-year Grants
• Research Field: Geometry
• Research Institution: Kisarazu National College of Technology
• Principal Investigator: TADOKORO Yuuki 木更津工業高等専門学校, 基礎学系, 准教授 (10435414)
• Project Period (FY): 2009 – 2011
• Project Status: Completed (Fiscal Year 2011)
• Budget Amount: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2011: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2009: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: 調和体積 / 反復積分 / リーマン面 / モジュライ空間 / 写像類群 / トポロジー

Research Abstract

Harris defined the harmonic volume for compact Riemann surfaces, using Chen's iterated integrals. It depends only on the complex structure of compact Riemann surfaces. We obtain the trace map images of the values of certain harmonic volumes for some cyclic quotients of Fermat curves. These provide the algorithm showing that the algebraic cycles called by the Ceresa cycles are not algebraically equivalent to zero in the Jacobian varieties. We apply the method to the case for the prime N<1000 with N=1 modulo 3.

Research Products

• [Journal Article] A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic (2009)
  • Author(s): Yuuki Tadokoro
  • Journal Title: Tsukuba J. Math. 33 Volume: no.1 Pages: 29-38
  • NAID: 120006309451
  • URL: http://projecteuclid.org/euclid.tkbjm/1251833205
  • Peer Reviewed
• [Journal Article] A nontrivial algebraic cycle in the Jacobian variety of the Fermat sextic (2009)
  • Author(s): 田所勇樹
  • Journal Title: Tsukuba Journal of Mathematics 1巻 Pages: 29-38
  • NAID: 120006309451
  • Peer Reviewed
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2012)
  • Author(s): 田所勇樹
  • Place of Presentation: 東京工業大学
  • Year and Date: 2012-02-07
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2012)
  • Author(s): 田所勇樹
  • Organizer: 代数幾何セミナー
  • Place of Presentation: 京都大学理学部
  • Year and Date: 2012-02-06
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2011)
  • Author(s): 田所勇樹
  • Organizer: 日本数学会トポロジー分科会一般講演
  • Place of Presentation: 早稲田大学
  • Year and Date: 2011-03-21
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2010)
  • Author(s): 田所勇樹
  • Organizer: 研究集会「トポロジーの現在と未来」
  • Place of Presentation: 文部省共済組合箱根合宿所
  • Year and Date: 2010-12-21
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2010)
  • Author(s): 田所勇樹
  • Organizer: 研究集会「トポロジーの現在と未来」
  • Place of Presentation: 箱根
  • Year and Date: 2010-12-21
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2010)
  • Author(s): 田所勇樹
  • Organizer: 研究集会「リーマン面に関連する位相幾何学」
  • Place of Presentation: 東京大学
  • Year and Date: 2010-09-05
• [Presentation] Nontrivial algebraic cycles in the Jacobian varieties of some quotients of Fermat curves (2010)
  • Author(s): 田所勇樹
  • Organizer: 研究集会「リーマン面に関する位相幾何学」
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2010-09-05
• [Presentation] A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic (2010)
  • Author(s): 田所勇樹
  • Organizer: Teichmuller Theory and its Interactions in Mathematics and Physics
  • Place of Presentation: Centre de Recerca Matematica, Bellaterra(スペイン)
• [Presentation] A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic (2010)
  • Author(s): 田所勇樹
  • Organizer: Teichmuller Theory and its Interactions in Mathematics and Physics
  • Place of Presentation: Centre de Recerca Matematica, Bellaterra, Spain
• [Presentation] リーマン面の調和体積とその応用 (2009)
  • Author(s): 田所勇樹
  • Organizer: 第52回函数論シンポジウム
  • Place of Presentation: 大阪府立大学
  • Year and Date: 2009-11-23
• [Presentation] Riemann 面の調和体積とその応用 (2009)
  • Author(s): 田所勇樹
  • Organizer: 第52回函数論シンポジウム
  • Place of Presentation: 大阪府立大学(大阪府)
  • Year and Date: 2009-11-23
• [Presentation] A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic (2009)
  • Author(s): 田所勇樹
  • Place of Presentation: 北海道大学
  • Year and Date: 2009-07-06