New applications of the theory of mixed Hodge modules and mixed sheaves

• Project/Area Number: 21540037
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Kyoto University
• Principal Investigator: SAITO Morihiko 京都大学, 数理解析研究所, 准教授 (10186968)
• Project Period (FY): 2009 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: ホッジ加群 / ネロン模型 / チャウ・キュネット分解 / 超平面配置 / ヒルツェブルフ類 / 消滅輪体 / コンツェヴィチ予想 / コンツェビッチ予想 / ミルナー類 / 混合ホッジ加群 / 局所モノドロミー / 対称積 / ヒルツェブルフ特性類 / 仮想特性類 / 双対加群 / ホッジ予想 / 許容法函数 / 柏原予想 / 非確定型微分方程式系

Research Abstract

By applying the theory of mixed Hodge modules, many new results are obtained in various fields of algebraic geometry such as Hodge structures, algebraic cycles, singularities, characteristic classes, etc. In case of characteristic classes, for instance, a completely new formula for the Hirzebruch-Milnor classes of complete intersections of smooth projective varieties is obtained by using vanishing cycles of mixed Hodge modules inductively. Many of these results cannot be proved without using the theory of mixed Hodge modules.

Research Products

• [Journal Article] Hirzebruch-Milnor classes of complete intersections (2013)
  • Author(s): Maxim, Laurentiu; Saito, Morihiko; Schurmann, Jorg
  • Journal Title: Adv.Math Volume: 241 Pages: 220-245
  • DOI: 10.1016/j.aim.2013.04.001
  • Peer Reviewed
• [Journal Article] Relative Chow-Kunneth decompositions for morphisms of threefolds (2012)
  • Author(s): Muller-Stach, Stefan; Saito, Morihiko
  • Journal Title: J. Reine Angew. Math Volume: 666 Issue: 666 Pages: 141-161
  • DOI: 10.1515/crelle.2011.119
  • Peer Reviewed
• [Journal Article] Symmetric products of mixed Hodge modules (2011)
  • Author(s): Maxim, Laurentiu; Saito, Morihiko; Schurmann, Jorg
  • Journal Title: J. Math. Pures Appl Volume: (9) 96 Issue: 5 Pages: 462-483
  • DOI: 10.1016/j.matpur.2011.04.003
  • Peer Reviewed
• [Journal Article] On the local zeta functions and the b-functions of certain hyperplane arrangements, with an appendix by Willem Veys (2011)
  • Author(s): Budur, Nero; Saito, Morihiko; Yuzvinsky, Sergey
  • Journal Title: J. Lond. Math. Soc. Volume: (2) 84 Issue: 3 Pages: 631-648
  • DOI: 10.1112/jlms/jdr025
  • Peer Reviewed
• [Journal Article] First Milnor cohomology of hyperplane arrangements, Contemp (2011)
  • Author(s): Budur, Nero; Dimca, Alexandru; Saito, Morihiko
  • Journal Title: Amer. Math. Soc., Providence, RI Volume: 538 Pages: 279-292
  • DOI: 10.1090/conm/538/10606
  • Peer Reviewed
• [Journal Article] A variant of Neron models over curves (2011)
  • Author(s): Saito, Morihiko; Schnell, Christian
  • Journal Title: Manuscripta Math Volume: 134 Issue: 3-4 Pages: 359-375
  • DOI: 10.1007/s00229-010-0398-5
  • Peer Reviewed
• [Presentation] Hirzebruch-Milnor classes of complete intersections (招待講演)、幾何学とトポロジーにおける特異点 (2011)
  • Author(s): 斉藤盛彦
  • Organizer: 第6回 日仏特異点シンポジウム
  • Place of Presentation: 九州大学西新会館
  • Year and Date: 2011-09-06
• [Presentation] Jumping coefficients, spectra and b-functions of hyperplane arrangements (2009)
  • Author(s): 斉藤盛彦
  • Organizer: Oberwolfach workshop
  • Place of Presentation: Oberwolfach, Germany
  • Year and Date: 2009-09-23
• [Presentation] Generalization of Neron models of Green, Griffiths, Kerr (2009)
  • Author(s): 斉藤盛彦
  • Organizer: Hodge theory and algebraic geometry
  • Place of Presentation: 京都大学数理解析研究所
  • Year and Date: 2009-06-30
• [Presentation] Generalization of Neron models of Green, Griffiths, Kerr (2009)
  • Author(s): 斉藤盛彦
  • Organizer: Hodge Theory and Motives
  • Place of Presentation: Institut Mathematiques de Bourgogne, Dijon, France
  • Year and Date: 2009-05-01