Stability conditions on derived categories and Donaldson-Thomas invariants

2018 Fiscal Year Final Research Report

• Project/Area Number: 26287002
• Research Category: Grant-in-Aid for Scientific Research (B)
• Allocation Type: Partial Multi-year Fund
• Section: 一般
• 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, 教授 (20503882)
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 連接層の導来圏 / 安定性条件 / Donaldson-Thomas不変量 / Gopakumar-Vafa不変量

Outline of Final Research Achievements

We studied stability conditions on derived categories of coherent sheaves on Calabi-Yau 3-folds, and constructed moduli spaces of semistable objects. We applied this result to the study of Donaldson-Thomas invariants which count stable sheaves or curves on Calabi-Yau 3-folds. Moreover we gave mathematical definitions of Gopakumar-Vafa invariants on Calabi-Yau 3-folds and 4-folds, described conjectural relationships with Donaldson-Thomas invariants explicitly, and proved them in several cases.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

３次元カラビヤウ多様体は物理学の超弦理論においても考察される数学的対象であり、特にこの上のGopakumar-Vafa不変量は物理学者のGopakumarとVafaによって提唱された重要な不変量である。よってその不変量の数学的に厳密な定義を与える事は数学・物理双方にとって意義深いものである。Maulik氏との共同研究でGopakumar-Vafa不変量の数学的定義を与えることに成功し、この不変量の更なる理解を深めることができた。