Research on fundamental group actions on derived categories of coherent sheaves and spaces of stability conditions

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K14502
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Kyoto University
• Principal Investigator: Hirano Yuki 京都大学, 理学研究科, 助教 (50804225)
• Project Period (FY): 2019-04-01 – 2023-03-31
• Keywords: Bridgeland安定性条件 / 連接層の導来圏 / 基本群の作用

Outline of Final Research Achievements

There are two main results of this research. One is on Bridgeland stability conditions. Spaces of Bridgeland Stability conditions on derived categories of coherent sheaves on algebraic varieties are important in the string theory. We describe the space of Bridgeland Stability conditions on certain triangulated categories associated to arbitrary 3-fold flopping contractions. As the other result, we show that the fundamental group actions, which are constructed by Halpern-Leistner and Sam, on the derived category of GIT quotient of certain representations of linear reductive groups correspond to compositions of equivalences induced by iterated Iyama--Wemyss mutations.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

3次元フロップ収縮に付随する三角圏上の安定性条件の空間は, これまで技術的仮定を置いた場合でしか記述されていなかったが, 本研究では一般の場合に記述することができた. また, そのアプローチは, 伊山--Wemyssによって近年発展した非可換代数の表現論を用いたものであり, その理論の有用性を示す意味でも意義のある研究であったといえる. また, 擬対称表現に付随するGIT商の連接層の導来圏上の基本群作用の研究においても, 伊山--Wemyssの理論が有用であることを確かめることができた.