Characteristic classes of algebraic stacks and application to enumerative problems

• Project/Area Number: 24340007
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: Ohmoto Toru 北海道大学, 理学(系)研究科(研究院), 教授 (20264400)
• Co-Investigator(Kenkyū-buntansha): NARUSE Hiroshi 山梨大学, 総合研究部, 教授 (20172596) ; SUWA Tatsuo 北海道大学, 名誉教授 (40109418) ; YOKURA Shoji 鹿児島大学, 学術研究院理工学域理学系, 教授 (60182680)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥16,250,000 (Direct Cost: ¥12,500,000、Indirect Cost: ¥3,750,000); Fiscal Year 2015: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2014: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2013: ¥3,380,000 (Direct Cost: ¥2,600,000、Indirect Cost: ¥780,000); Fiscal Year 2012: ¥6,110,000 (Direct Cost: ¥4,700,000、Indirect Cost: ¥1,410,000)
• Keywords: 特異点論 / 特性類理論 / チャーン類 / トム多項式 / 数え上げ幾何 / ホイットニー類 / 半代数的集合 / 特性類 / 国際研究者交流（フランス，ベトナム） / 写像の特異点論 / チャーン・シュワルツ・マクファーソン類 / 国際研究者交流（フランス，ドイツ，米国） / 国際情報交換（フランス，ブラジル） / 特異点 / Chern-Schwartz-MacPherson class / Hirzebruch class / Todd class / Thom polynomials / Hilbert scheme

Outline of Final Research Achievements

Seeking for the complete foundation of the enumerative theory of singularities of maps, I have developed singular Chern class theory on various moduli spaces; in particular, together with collaborators I completed a sequence of works on zeta functions of homology characteristic classes of Hilbert schemes of points on a non-singular variety; I applied the theory of equivariant singular Chern classes to the classification theory of map-germs, that made a new breakthrough for computing singular Milnor numbers.

Research Products

• [Int'l Joint Research] ローマ大学数学教室(イタリア)
• [Int'l Joint Research] フランス国立科学研究センター(フランス)
• [Int'l Joint Research] ミュンスター大学数学教室(ドイツ)
• [Journal Article] Singularities of Maps and Characteristic Classes (2016)
  • Author(s): T. Ohmoto
  • Journal Title: Advanced Studies in Pure Mathematics, MSJ (Proc. Real and Complex Singularities (Sao Carlos, Brazil, 2012)) Volume: 68 Pages: 191-265
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Bifurcation of plane-to-plane map-germs with corank 2 of parabolic type (2016)
  • Author(s): T. Yoshida, Y. Kabata and T. Ohmoto
  • Journal Title: RIMS Koukyuroku Bessatsu Volume: 55
  • Peer Reviewed
• [Journal Article] MOtivic and derived motivic Hirzebruch classes (2016)
  • Author(s): J-P. Brasselet, J. Schuemann and S. Yokura
  • Journal Title: Homology, Homotopy and Applications Volume: -
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Localized intersection of currents and Lefschetz coincidence point theorem (2016)
  • Author(s): C. Bisi, F. Bracci, T. Izawa and T. Suwa
  • Journal Title: Ann. Mat. Pura Appl. Volume: 195 Pages: 601-621
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bifurcation of plane-to-plane map-germs of corank two (2015)
  • Author(s): T. Yoshida, Y. Kabata and T. Ohmoto
  • Journal Title: Quarterly Journal of Math. Volume: 66 Issue: 1 Pages: 369-391
  • DOI: 10.1093/qmath/hau013
  • Peer Reviewed
• [Journal Article] Perturbation of Baum-Bott residues (2015)
  • Author(s): F. Bracci and T. Suwa
  • Journal Title: Asian J. Math. Volume: 19 Issue: 5 Pages: 871-885
  • DOI: 10.4310/ajm.2015.v19.n5.a4
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Singularities of Maps and Characteristic Classes (2015)
  • Author(s): T. Ohmoto
  • Journal Title: Advanced Studies in Pure Mathematics, Math. Soc. Japan Volume: 未定
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] Characteristic classes of symmetric products of complex quasi-projective varieties (2015)
  • Author(s): Sylvain Cappell, Laurentiu Maxim, Joerg Schuermann, Julius Shaneson and Shoji Yokura
  • Journal Title: J. fur Reine und Angewandte Mathematik Volume: 未定 Issue: 728 Pages: 35-63
  • DOI: 10.1515/crelle-2014-0114
  • Peer Reviewed / Open Access
• [Journal Article] Localized intersection of currents and the Lefschetz coincidence point theorem (2015)
  • Author(s): C. Bisi, F. Bracci, T. Izawa and T. Suwa
  • Journal Title: Annali di Matematica Pura ed Applicata Volume: 未定 Issue: 2 Pages: 601-621
  • DOI: 10.1007/s10231-015-0480-4
  • Peer Reviewed
• [Journal Article] Finite sum Cauchy identity for dual Grothendieck polynomials (2014)
  • Author(s): A. Lascoux and H. Naruse
  • Journal Title: Proceedings of the Japan Academy, Series A, Mathematical Sciences Volume: 90 Issue: 7 Pages: 87-91
  • DOI: 10.3792/pjaa.90.87
  • NAID: 40020185467
  • Peer Reviewed / Open Access
• [Journal Article] Naive motivic Donaldson-Thomas type Hirzebruch classes and some problems (2014)
  • Author(s): Vittoria Bussi and Shoji Yokura
  • Journal Title: Journal of Singularities Volume: 10 Pages: 26-53
  • DOI: 10.5427/jsing.2014.10b
  • Peer Reviewed / Open Access
• [Journal Article] GSV-indices as residues (2014)
  • Author(s): T. Suwa
  • Journal Title: J. of Singularities Volume: 9 Pages: 206-218
  • DOI: 10.5427/jsing.2014.9p
  • Peer Reviewed / Open Access
• [Journal Article] Singularities of Maps and Characteristic Classes (2014)
  • Author(s): T. Ohmoto
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 印刷中
  • Peer Reviewed
• [Journal Article] Characteristic classes of Hilbert schemes od points via symmetrics products (2013)
  • Author(s): S Cappel, L Maxim, T Ohmoto, J Schurmann, S. Yokura
  • Journal Title: Geometry & Topology Volume: 17 Issue: 2 Pages: 1165-1198
  • DOI: 10.2140/gt.2013.17.1165
  • Peer Reviewed
• [Journal Article] A note on Chern-Schwartz-MacPherson class (2012)
  • Author(s): T. Ohmoto
  • Journal Title: IRMA Lectures in Mathematics and Theoretical Physics, European Math. Soc., Volume: 20 Pages: 117-132
  • Peer Reviewed
• [Journal Article] Vassiliev type invariants for genric maps, revisited (2012)
  • Author(s): T. Ohmoto
  • Journal Title: Contemporary Math. Amer.Math. Soc. Volume: 569 Pages: 143-160
  • Peer Reviewed
• [Presentation] Hunting invariants of discriminants (2016)
  • Author(s): T. Ohmoto
  • Organizer: Enumerative Geometry of Moduli Spaces of Sheaves
  • Place of Presentation: ローザンヌ（スイス）
  • Year and Date: 2016-05-10
  • Int'l Joint Research / Invited
• [Presentation] C^1-triangulations of semialgebraic sets and de Rham homotopy theory (2015)
  • Author(s): T. Ohmoto
  • Organizer: The 6h Japanese-Australian workshop on Real and Complex Singularities
  • Place of Presentation: 鹿児島大学理学部（鹿児島県鹿児島市）
  • Year and Date: 2015-11-23
  • Int'l Joint Research / Invited
• [Presentation] Counting spaces and related topics (2015)
  • Author(s): S. Yokura
  • Organizer: International Conference on Set-Theoretic Topology and its Applications 2015
  • Place of Presentation: 神奈川大学（神奈川県横浜市）
  • Year and Date: 2015-08-24
  • Int'l Joint Research / Invited
• [Presentation] Geometry of plane-to-plane germs with corank two (2015)
  • Author(s): T. Ohmoto
  • Organizer: Singularities in Generic Geometry and its Applications - Kobe/Kyoto 2015 (Valencia IV)
  • Place of Presentation: 京都大学数理解析研究所（京都府京都市）
  • Year and Date: 2015-06-08
  • Int'l Joint Research / Invited
• [Presentation] Classical enumerative geometry and Thom polynomials (2015)
  • Author(s): T. Ohmoto
  • Organizer: Workshop IMPANGA 15
  • Place of Presentation: ベドレボ（ポーランド）
  • Year and Date: 2015-04-16
  • Int'l Joint Research / Invited
• [Presentation] 特異空間の特性類についてー数え上げの観点からー (2015)
  • Author(s): 與倉昭治
  • Organizer: 日本数学会2015年度年会
  • Place of Presentation: 明治大学（東京都文京区）
  • Year and Date: 2015-03-21
  • Invited
• [Presentation] Characteristic classes of singular varieties (2015)
  • Author(s): S. Yokura
  • Organizer: Local and Global Invariants of Singularities
  • Place of Presentation: エクスマルセイユ大学（フランス）
  • Year and Date: 2015-02-26
  • Invited
• [Presentation] C^1-triangulations of semialgebraic sets and de Rham homotopy theory (2015)
  • Author(s): 大本 亨
  • Organizer: 研究集会「接触構造、特異点、微分方程式及びその周辺」
  • Place of Presentation: 旭川市ときわ市民ホール（旭川市）
  • Year and Date: 2015-01-22
  • Invited
• [Presentation] Fiberwise bordism groups and their bivariant analogues (2014)
  • Author(s): S. Yokura and J. Schurmann
  • Organizer: Analysis and Topology in Interaction 2014
  • Place of Presentation: コルトナ（イタリア）
  • Year and Date: 2014-06-14
  • Invited
• [Presentation] Thom class in localized intersections and the Riemann-Roch (2014)
  • Author(s): T. Suwa
  • Organizer: Workshop on semi-algebraic sets
  • Place of Presentation: 東京大学（東京都目黒区）
  • Year and Date: 2014-05-01
  • Invited
• [Presentation] Chern Classes of Image and Discriminant of Stable Maps (2014)
  • Author(s): T. Ohmoto
  • Organizer: Branched Coverings, Degenerations, and Related Topics 2014
  • Place of Presentation: 広島大学(広島県東広島市)
  • Invited
• [Presentation] Image and Discriminant Chern Classes of Stable Maps (2013)
  • Author(s): T. Ohmoto
  • Organizer: 筑波大学大学院数理物質科学研究科数学教室談話会
  • Place of Presentation: 筑波大学(茨城県つくば市)
  • Invited
• [Presentation] Shoji Yokura's works and Kagoshima (2013)
  • Author(s): T. Ohmoto
  • Organizer: 1st Franco-Japanese-Vietnamese Symposium on Singularities of the CNRS-VAST-JSPS
  • Place of Presentation: ニース大学(フランス)
  • Invited
• [Presentation] Image and Discriminant Chern Classes of Stable Maps (2013)
  • Author(s): T. Ohmoto
  • Organizer: Mini-workshop on singularities 2013
  • Place of Presentation: 東京理科大学(東京都新宿区)
  • Invited
• [Presentation] Geometry of plane-to-plane map-germ (2012)
  • Author(s): T. Ohmoto
  • Organizer: 可微分写像の特異点論とその応用
  • Place of Presentation: 日本大学文理学部(東京都)(招待講演)
  • Year and Date: 2012-12-12
• [Presentation] Chern class for singular varieties and enumerative geometry (2012)
  • Author(s): T. Ohmoto
  • Organizer: The 8th HU and SNU Symposium on Mathematics
  • Place of Presentation: 北海道大学(札幌市)(招待講演)
  • Year and Date: 2012-12-07
• [Book] IRMA Lectures in Mathematics and Theoretical Physics,, Vol. 20 (2012)
  • Author(s): V. Blanloeil, T. Ohmoto, (eds.)
  • Total Pages: 362
  • Publisher: ユーロッパ数学会(European Math. Soc.,)