Towards a deeper understanding of K-stability

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14321
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11020:Geometry-related
• Research Institution: Tokyo University of Science
• Principal Investigator: Saito Shunsuke 東京理科大学, 理学部第一部数学科, 助教 (10846752)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: K安定性 / 漸近的Chow安定性 / トーリック多様体 / 一様相対Ding安定性 / 満渕定数 / 相対K安定性 / 超平面切断

Outline of Final Research Achievements

I have obtained the following results related to the stability of polarized varieties: (1)For a polarized toric surface on which the obstructions of asymptotic Chow semistability vanish, K-polystability implies asymptotic Chow polystability; (2)A new definition of relative Chow stability by means of a quantization of extremal vector fields has been given; (3)Examples of relatively Ding unstable toric Fano manifolds which are uniformly relatively K-stable for any polarization have been given for all dimensions greater than or equal to three; (4)Errors in Yotsutani-Zhou's classification of relative K-stability of smooth toric Fano 3-folds have been found;　(5)K-stability of hyperplane sections of Segre varieties for anticanonical polarization have been determined completely.

Free Research Field

複素幾何学

Academic Significance and Societal Importance of the Research Achievements

偏極代数多様体は数学において基本的な研究対象であり、その分類を行う上で標準計量の存在・非存在あるいは安定性・不安定性といった情報は重要な役割を担う。本研究課題の研究成果は、当該研究分野の中心的な問題について決定的な解答を与えるといった類のものではないが、トーリック多様体や超平面切断などの具体的な代数多様体の分類に関して素朴だが興味深い問題をいくつか提示できたという点で発展性や意義のあるものだと言える。