Stability conditions on triangulated categories and counting invariants

• Project/Area Number: 22684002
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Algebra
• Research Institution: The University of Tokyo
• Principal Investigator: TODA Yukinobu 東京大学, カブリ数物連携宇宙研究機構, 特任准教授 (20503882)
• Project Period (FY): 2010-04-01 – 2014-03-31
• Project Status: Completed (Fiscal Year 2013)
• Budget Amount: ¥10,140,000 (Direct Cost: ¥7,800,000、Indirect Cost: ¥2,340,000); Fiscal Year 2013: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2012: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2011: ¥2,470,000 (Direct Cost: ¥1,900,000、Indirect Cost: ¥570,000); Fiscal Year 2010: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
• Keywords: 連接層の導来圏 / 安定性条件 / Donaldson-Thomas不変量 / 双有理幾何学 / 行列分解 / 導来圏 / OSV予想 / Gromov-Witten不変量

Research Abstract

A 3-dimensional Calabi-Yau manifold is expected to appear as an extra dimension in string theory, and an important space both in mathematics and physics. The Donaldson-Thomas (DT) invariants counting curves on that space are conjectured to satisfy several properties originating with string theory. I constructed invariants counting semistable objects in derived categories of coherent sheaves, and applied them for several conjectures on DT invariants.

Research Products

• [Journal Article] Bogomolov-Gieseker-type inequality and counting invariants (2013)
  • Author(s): Y. Toda
  • Journal Title: J. Topol Volume: 6, no. 1 Pages: 217-250
• [Journal Article] Curve counting theories via stable objects II : DT/ncDT flop formula (2013)
  • Author(s): Y. Toda
  • Journal Title: J. Reine Angew. Math Volume: 675 Pages: 1-51
• [Journal Article] Stability conditions and extremal contractions (2013)
  • Author(s): Yukinobu Toda
  • Journal Title: Math. Ann. Volume: 357 Pages: 631-685
  • Peer Reviewed
• [Journal Article] Bogomolov-Gieseker type inequality and counting invariants (2013)
  • Author(s): Yukinobu Toda
  • Journal Title: J. Topol. Volume: 6 Pages: 217-250
  • Peer Reviewed
• [Journal Article] Curve counting theories via stable objects II: DT/ncDT flop formula (2013)
  • Author(s): Yukinobu Toda
  • Journal Title: J. Reine. Angew. Math. Volume: 675 Pages: 1-51
  • Peer Reviewed
• [Journal Article] Introduction and open problems of Donaldson-Thomas theory. Derived categories in algebraic geometry (2012)
  • Author(s): Y. Toda
  • Journal Title: EMS Ser. Congr. Rep., Eur. Math. Soc
• [Journal Article] Stable pairs on local K3 surfaces (2012)
  • Author(s): Y. Toda
  • Journal Title: J. Differential Geom Volume: 92, no.2 Pages: 285-371
• [Journal Article] Stability conditions and curve counting invariants on Calabi-Yau 3-folds (2012)
  • Author(s): Y. Toda
  • Journal Title: Kyoto J. Math Volume: 52, no.1 Pages: 1-50
• [Journal Article] Stable pairs on local K3 surfaces (2012)
  • Author(s): Yukinobu Toda
  • Journal Title: Journal of Differential Geometry Volume: 92 Pages: 285-371
  • Peer Reviewed
• [Journal Article] Stability conditions and curve counting invariants on Calabi-Yau 3-folds (2012)
  • Author(s): Yukinobu Toda
  • Journal Title: Kyoto Journal of Mathematics Volume: 52 Pages: 1-50
  • Peer Reviewed
• [Journal Article] Curve counting invariants around the conifold point (2011)
  • Author(s): Y. Toda
  • Journal Title: J. Differential Geom Volume: 89, no.1 Pages: 133-184
• [Journal Article] Moduli spaces of stable quotients and wall-crossing phenomena (2011)
  • Author(s): Y. Toda
  • Journal Title: Compos. Math Volume: 147, no.5 Pages: 1479-1518
• [Journal Article] Curve counting invariants around the conifold point (2011)
  • Author(s): Yukinobu Toda
  • Journal Title: Journal of Differential Geometry Volume: 89 Pages: 133-184
  • Peer Reviewed
• [Journal Article] Moduli spaces of stable quotients and wall-crossing phenomena (2011)
  • Author(s): Yukinobu Toda
  • Journal Title: Compositio Math Volume: 147 Pages: 1479-1518
  • Peer Reviewed
• [Journal Article] Generating functions of stable pair invariants via wall-crossings in derived categories. New developments in algebraic geometry, integrable systems and mirror symmetry (RIMS, Kyoto, 2008) (2010)
  • Author(s): Y. Toda
  • Journal Title: Adv. Stud. Pure Math, Math. Soc. Japan, Tokyo Volume: 59 Pages: 389-434
• [Journal Article] Curve counting theories via stable objects I. DT/PT correspondence (2010)
  • Author(s): Y. Toda
  • Journal Title: J. Amer. Math. Soc Volume: 23, no.4 Pages: 1119-1157
• [Journal Article] Derived categories of coherent sheaves on algebraic varieties (2010)
  • Author(s): Y. Toda
  • Journal Title: Triangulated categories, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge Volume: 375 Pages: 408-451
• [Journal Article] On a computation of rank two Donaldson-Thomas invariants (2010)
  • Author(s): Y. Toda
  • Journal Title: Commun. Number Theory Phys Volume: 4, no.1 Pages: 49-102
• [Journal Article] Tilting generators via ample line bundles (2010)
  • Author(s): Y. Toda and H. Uehara
  • Journal Title: Adv. Math Volume: 223, no.1 Pages: 1-29
• [Journal Article] Generating functions of stable pair invariants via wall-crossings in derived categories (2010)
  • Author(s): Yukinobu Toda
  • Journal Title: Adv.Stud.Pure Math. Volume: 59 Pages: 389-434
  • Peer Reviewed
• [Journal Article] On a computation of rank two Donaldson-Thomas invariants (2010)
  • Author(s): Yukinobu Toda
  • Journal Title: Commun.Number Theory Phys. Volume: 4 Pages: 49-102
  • Peer Reviewed
• [Journal Article] Curve counting theories via stable objects I, DT/PT correspondence (2010)
  • Author(s): Yukinobu Toda
  • Journal Title: J.Amer.Math.Soc. Volume: 23 Pages: 1119-1157
  • Peer Reviewed
• [Presentation] Flops and S-duality conjecture (2014)
  • Author(s): Yukinobu Toda
  • Organizer: Calabi-Yau geometry and mirror symmetry conference
  • Place of Presentation: National Taiwan University, Taiwan
  • Invited
• [Presentation] Derived category of coherent sheaves and counting invariants (2014)
  • Author(s): Yukinobu Toda
  • Organizer: ICM 2014
  • Place of Presentation: ソウル, 韓国
  • Invited
• [Presentation] Gepner type stability conditions on graded matrix factorizations (2013)
  • Author(s): 戸田 幸伸
  • Organizer: Pacific Rim conference
  • Place of Presentation: 札幌コンベンションセンター
  • Year and Date: 2013-07-05
  • Invited
• [Presentation] Gepner type stability conditions on graded matrix factorizations (2013)
  • Author(s): 戸田 幸伸
  • Organizer: Birational Geometry and Geometric Invariant Theory
  • Place of Presentation: ウィーン大学ESI, オーストリア
  • Year and Date: 2013-05-23
  • Invited
• [Presentation] Gepner type stability conditions on triangulated categories (2013)
  • Author(s): 戸田 幸伸
  • Organizer: Higher Dimensional Algebraic Geometry (in honour of Professor Yujiro Kawamata's sixtieth birthday)
  • Place of Presentation: 東京大学
  • Year and Date: 2013-01-07
  • Invited
• [Presentation] Introduction to Donaldson-Thomas theory, Bogomolov-Gieseker type inequality and DT invariants, Gepner type stability conditions on graded matrix factorizations (2013)
  • Author(s): 戸田 幸伸
  • Organizer: Graduate Student Workshop on Moduli Spaces and Bridgeland Stability
  • Place of Presentation: イリノイ大学シカゴ校, アメリカ
  • Invited
• [Presentation] Gepner type stability conditions on graded matrix factorizations (2013)
  • Author(s): Yukinobu Toda
  • Organizer: Birational geometry and geometric invariant theory
  • Place of Presentation: University of Illinois at Chicago, US
  • Invited
• [Presentation] Gepner type stability conditions on graded matrix factorizations (2013)
  • Author(s): Yukinobu Toda
  • Organizer: Graduate scholl workshop on moduli and Bridgeland stability
  • Place of Presentation: University of Illinois, Chicago, USA
  • Invited
• [Presentation] Stability condition at the Gepner point (2012)
  • Author(s): 戸田 幸伸
  • Organizer: Homological projective duality and quantum gauge theory
  • Place of Presentation: 東京大学Kavli IPMU
  • Year and Date: 2012-11-14
  • Invited
• [Presentation] Stability conditions and Donaldson-Thomas type invariants on Calabi-Yau 3-folds (2012)
  • Author(s): 戸田 幸伸
  • Organizer: 幾何学賞受賞記念講演
  • Place of Presentation: 九州大学
  • Year and Date: 2012-09-19
  • Invited
• [Presentation] Stability conditions and birational geometry (2012)
  • Author(s): 戸田 幸伸
  • Organizer: 日韓共同会議
  • Place of Presentation: 九州大学医学部百年講堂
  • Year and Date: 2012-09-16
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and OSV Conjecture I, II (2012)
  • Author(s): 戸田 幸伸
  • Organizer: Relation to string theory to gauge theories and moduli problems of branes
  • Place of Presentation: ステクロフ数学研究所, ロシア
  • Year and Date: 2012-09-13
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and counting invariants (2012)
  • Author(s): 戸田 幸伸
  • Organizer: The Conference of mirror symmetry and related topics
  • Place of Presentation: 昆明大学, 中国
  • Year and Date: 2012-08-24
• [Presentation] Stability conditions and birational geometry of projective surfaces (2012)
  • Author(s): 戸田 幸伸
  • Organizer: Workshop on Moduli and Birational geometry
  • Place of Presentation: 釜山, 韓国
  • Year and Date: 2012-07-10
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and counting invaraints (2012)
  • Author(s): 戸田 幸伸
  • Organizer: Arithmetic and Algebraic geomegry 2012
  • Place of Presentation: 東京大学
  • Year and Date: 2012-02-16
  • Invited
• [Presentation] Stability conditions and birational geometry of projective surfaces (2012)
  • Author(s): Yukinobu Toda
  • Organizer: Workshop on moduli and birational geometry
  • Place of Presentation: Busan, Korea
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and counting invariants (2012)
  • Author(s): Yukinobu Toda
  • Organizer: The conference of mirror symmetry and related topics
  • Place of Presentation: Kunming, China
  • Invited
• [Presentation] Bogomolov-Gieseker type inequality and counting invariants (2012)
  • Author(s): Yukinobu Toda
  • Organizer: Relation of string theory to gauge theory and moduli problems of branes
  • Place of Presentation: Steklov institute, Moscow, Russia
  • Invited
• [Presentation] Multiple cover formula of generalized DT invariants (2011)
  • Author(s): 戸田 幸伸
  • Organizer: Derived categories in Algebraic Geometry
  • Place of Presentation: ステクロフ数学研究所, ロシア
  • Year and Date: 2011-09-05
  • Invited
• [Presentation] Parabolic stable pairs on Calabi-Yau 3-folds (2011)
  • Author(s): 戸田 幸伸
  • Organizer: Double Affine Hecke Algebras, the Langlands Program, Affine flag varieties, Conformal Field Theory, Super Yang Mills Theory
  • Place of Presentation: Luminy, フランス
  • Year and Date: 2011-06-29
  • Invited
• [Presentation] Curve counting invariants on Calabi-Yau 3-folds (2011)
  • Author(s): 戸田 幸伸
  • Organizer: Moduli spaces, Spitalfields Day
  • Place of Presentation: ニュートン数理科学研究所, イギリス
  • Year and Date: 2011-05-05
  • Invited
• [Presentation] Stable pairs on local K3 surfaces (2011)
  • Author(s): 戸田 幸伸
  • Organizer: Derived categories
  • Place of Presentation: ニュートン数理科学研究所, イギリス
  • Year and Date: 2011-04-12
  • Invited
• [Presentation] Stable pairs on local K3 surfaces (2011)
  • Author(s): Yukinobu Tbda
  • Organizer: Derived categories
  • Place of Presentation: Newton Institute, Cambridge, UK(招待講演)
  • Year and Date: 2011-04-12
• [Presentation] Stability conditions and Bogomolov-Gieseker type inequalities I (2011)
  • Author(s): 戸田 幸伸
  • Organizer: Derived categories 2011 Tokyo
  • Place of Presentation: 東京大学
  • Year and Date: 2011-01-24
  • Invited
• [Presentation] Introduction of Bridgeland stability conditions and construction of Bridgeland stability on 3-folds (2011)
  • Author(s): 戸田 幸伸
  • Organizer: KIAS summer school on derived categories
  • Place of Presentation: Alpensia Resort, 韓国
  • Invited
• [Presentation] Curve counting invariatns via stable objects, I, II, III (2010)
  • Author(s): 戸田 幸伸
  • Organizer: Derived categories of Algebro-Geometric Origin and Integrable Systems
  • Place of Presentation: IAS, イスラエル
  • Invited
• [Presentation] Curve counting theories via stable objects (2010)
  • Author(s): Yukinobu Toda
  • Organizer: Derived categories of algebro geometric origin and integrable systems
  • Place of Presentation: エルサレム,イスラエル