Study of geometric and analytic monodromies and local zeta functions

• Project/Area Number: 22540172
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Basic analysis
• Research Institution: University of Tsukuba
• Principal Investigator: TAKEUCHI Kiyoshi 筑波大学, 数理物質系, 教授 (70281160)
• Co-Investigator(Renkei-kenkyūsha): MATSUI Yutaka 近畿大学, 理工学部, 准教授 (10510026)
• Project Period (FY): 2010 – 2012
• Project Status: Completed (Fiscal Year 2012)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2011: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2010: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: D-加群 / 超幾何関数 / 特異点理論 / モノドロミー / 偏屈層 / ミルナー束 / 代数解析学 / D-カロ群 / 代数解析 / ゼータ関数

Research Abstract

By using the theory of motivic Milnor fibers, we obtained some formulas which express the Jordan normal forms of monodromies at infinity of polynomial maps. Moreover we generalized this result to the case of polynomial maps from complete intersection varieties. By using the theory of rapid decay homologies we also obtained the integral representations of confluent A-hypergeometric functions. By this result we calculated their asymptotic expansions and Stokes multipliers at infinity.

Research Products

• [Journal Article] Meromorphic continuations of local zeta functions and their applications to oscillating integrals (2013)
  • Author(s): 竹内潔、岡田季久
  • Journal Title: Tohoku Math. Journal Volume: (印刷中)
  • NAID: 130005562146
  • Peer Reviewed
• [Journal Article] Monodromy at infinity of polynomial maps and Newton polyhedra (with Appendix by C. Sabbah) (2013)
  • Author(s): 竹内潔、松井優
  • Journal Title: Int. Math. Res. Not. Volume: Vol. 2013, No. 8 Pages: 1691-1746
  • DOI: 10.1093/imrn/rns092
  • Peer Reviewed
• [Journal Article] Meromorphic continuations of local zeta functions and their applications to oscillating integrals (2013)
  • Author(s): 竹内 潔
  • Journal Title: Tohoku Mathematical Journal Volume: 65, no.2
  • NAID: 130005562146
  • Peer Reviewed
• [Journal Article] Monodromy at infinity of polynomial maps and Newton polyhedra (2013)
  • Author(s): 竹内 潔
  • Journal Title: International Mathematics Research Notices Volume: 2013, no.8 Pages: 1691-1746
  • Peer Reviewed
• [Journal Article] Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers (2012)
  • Author(s): 竹内潔、A.Esterov
  • Journal Title: Int. Math. Res. Not. Volume: Vol. 2012, No. 15 Issue: 15 Pages: 3567-3613
  • DOI: 10.1093/imrn/rnr154
  • Peer Reviewed
• [Journal Article] Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers (2012)
  • Author(s): 竹内 潔
  • Journal Title: International Mathematics Research Notices Volume: 2012, no.15 Pages: 3567-3613
  • Peer Reviewed
• [Journal Article] Motivic Milnor fibers over complete intersection varieties and their virtual Betti numbers (2012)
  • Author(s): 竹内潔, Alexander ESTEROV
  • Journal Title: IMRN Volume: (印刷中)
  • Peer Reviewed
• [Journal Article] A geometric degree formula for A-discriminants and Euler obstructions of toric varieties (2011)
  • Author(s): 竹内潔、松井優
  • Journal Title: Advances in Mathematics Volume: Vol. 226 Pages: 2040-2064
  • URL: http://www.sciencedirect.com/science/article/pii/S0001870810003476
  • Peer Reviewed
• [Journal Article] MILNOR FIBERS OVER SINGULAR TORIC VARIETIES AND NEARBY CYCLE SHEAVES (2011)
  • Author(s): 竹内潔、松井優
  • Journal Title: Tohoku Mathematical Journal, Second Series Volume: 63 Issue: 1 Pages: 113-136
  • DOI: 10.2748/tmj/1303219938
  • NAID: 110008425971
  • ISSN: 0040-8735, 2186-585X
  • Peer Reviewed
• [Journal Article] Monodromy zeta functions at infinity, Newton polyhedra and constructible sheaves (2011)
  • Author(s): 竹内潔、松井優
  • Journal Title: Mathematische Zeitschrift Volume: Vol. 268 Issue: 1-2 Pages: 409-439
  • DOI: 10.1007/s00209-010-0678-5
  • Peer Reviewed
• [Journal Article] Milnor fibers over singnlar toric varieties and nearby cycle sheaves (2011)
  • Author(s): 竹内潔・松井優
  • Journal Title: Tohoku Math.Journal Volume: 63 Pages: 113-136
  • Peer Reviewed
• [Journal Article] Monodromy 30th functions at infinity, Newton polyhedra and constructible sheaves (2011)
  • Author(s): 竹内潔・松井優
  • Journal Title: Mathematische Zeitschrift Volume: 268 Pages: 409-439
  • Peer Reviewed
• [Journal Article] A geometric degree formula for A-discriminants and Eular obstructions of toric varieties (2011)
  • Author(s): 竹内潔
  • Journal Title: Adv.Math Volume: 226, No2 Pages: 2040-2064
  • Peer Reviewed
• [Journal Article] Microlocal study of Lafschetz fixed-point formulas for higher-dimensional fixed point sets (2010)
  • Author(s): 竹内潔
  • Journal Title: IMRN Volume: 2010, No5 Pages: 882-913
  • Peer Reviewed
• [Journal Article] Monodromy at infinity of A-hypergeometric functions and toric compactifications (2010)
  • Author(s): 竹内潔
  • Journal Title: Math.Ann Volume: 348, No4 Pages: 815-831
  • Peer Reviewed
• [Presentation] Confluent A-hypergeometric functions and rapid decay homology cycles (2013)
  • Author(s): 竹内潔
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学理学部
  • Year and Date: 2013-03-15
• [Presentation] 合流型 A-超幾何関数の積分表示と漸近展開 (2012)
  • Author(s): 竹内潔
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学大学院理学研究科
  • Year and Date: 2012-12-06
• [Presentation] Monodromies at infinity of tame and non-tame polynomials (2012)
  • Author(s): 竹内潔
  • Organizer: Workshop on Free Divisors and Differential Equations
  • Place of Presentation: 東京農工大
  • Year and Date: 2012-11-06
• [Presentation] Monodromies at infinity of tame and non-tame polynomials (2012)
  • Author(s): 竹内潔
  • Organizer: Topology of real singularities and motivic aspects
  • Place of Presentation: MFO, Oberwolfach (ドイツ)
  • Year and Date: 2012-10-01
• [Presentation] A-discriminants and A-hypergeometric functions (2011)
  • Author(s): 竹内潔
  • Organizer: 第4回日豪特異点研究集会
  • Place of Presentation: 兵庫教育大学神戸サテライト
  • Year and Date: 2011-11-22
• [Presentation] On confluent A-hypergeometric functions (2011)
  • Author(s): 竹内潔
  • Organizer: 関口次郎教授還暦記念集会
  • Place of Presentation: 東京大学数理科学研究科
  • Year and Date: 2011-09-22
• [Presentation] Motivic Milnor fibers and their virtual Betti numbers (2011)
  • Author(s): 竹内潔
  • Organizer: 第6回日仏特異点シンポジウム
  • Place of Presentation: 福岡西新プラザ
  • Year and Date: 2011-09-07
• [Presentation] Motivic Milnor fibers and their virtual Betti numbers (2011)
  • Author(s): 竹内潔
  • Organizer: The 6th France-Japanese Symposium on Singularities
  • Place of Presentation: 九州大学西新プラザ(福岡市)(招待演講)
  • Year and Date: 2011-09-07
• [Presentation] Motivic Milnor fibers and Newton polyhedra (2011)
  • Author(s): 竹内潔
  • Organizer: Singularity theory conference
  • Place of Presentation: 中国科学技術大学(中国、合肥)
  • Year and Date: 2011-07-26
• [Presentation] Motivic Milnor fibers and Jordan normal forms of monodromies (2011)
  • Author(s): 竹内潔
  • Organizer: Workshop on geometry and analysis of discriminants
  • Place of Presentation: IPMU(東京大学) 招待講演
  • Year and Date: 2011-02-07
• [Presentation] Motivic Milnor fibers, mixed Hodge modules and Newton polyhedrons (2010)
  • Author(s): 竹内潔
  • Organizer: 特異点と多様体の幾何学
  • Place of Presentation: 山形大学理学部
  • Year and Date: 2010-09-17
• [Presentation] Monodromies at infinity of tame and non-tame polynomials
  • Author(s): 竹内 潔
  • Organizer: Topology of real singularities and motivic aspects
  • Place of Presentation: MFO, Oberwolfach （ドイツ）
  • Invited
• [Presentation] Monodromies at infinity of tame and non-tame polynomials
  • Author(s): 竹内 潔
  • Organizer: Workshop on Free Divisors and Differential Equations
  • Place of Presentation: 東京農工大
  • Invited
• [Presentation] 合流型 A-超幾何関数の積分表示と漸近展開
  • Author(s): 竹内 潔
  • Organizer: 微分方程式の総合的研究
  • Place of Presentation: 京都大学大学院理学研究科
  • Invited
• [Presentation] Confluent A-hypergeometric functions and rapid decay homology cycles
  • Author(s): 竹内 潔
  • Organizer: アクセサリー・パラメーター研究会
  • Place of Presentation: 熊本大学理学部
  • Invited