Studies of K3 surfaces with symmetry

2018 Fiscal Year Final Research Report

• Project/Area Number: 15K17520
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Tokai University
• Principal Investigator: Taki Shingo 東海大学, 理学部, 准教授 (30609714)
• Project Period (FY): 2015-04-01 – 2019-03-31
• Keywords: K3曲面 / 自己同型 / 対数的有理曲面

Outline of Final Research Achievements

It is a fundamental problem to study automorphisms of algebraic varieties. In particular, studies on automorphisms of K3 surfaces are one of the important problems. By the definition of K3 surfaces, these have a nowhere vanishing holomorphic 2-form. A finite group which acts on K3 surfaces as automorphisms is called "symplectic" or "non-symplectic" if it acts trivially or non-trivially on a nowhere vanishing holomorphic 2-form, respectively.
In the research, we classify complex K3 surfaces with non-symplectic automorphism of order 16 in full generality, study log Enriques surfaces of index 7.

Free Research Field

代数幾何

Academic Significance and Societal Importance of the Research Achievements

幾何的対象を考察する際，それが持つ対称性に注目することで新たな世界が見えてくることがある．例えば，一般の三角形に対称性は無いが，二等辺三角形や正三角形のような特殊な三角形は「左右対称」や「120度の回転」など特別な対称性を持つ．二等辺三角形や正三角形の特殊性はこのような対称性の存在によって特徴付けられているとも言える．本研究のように対称性を表す自己同型を通して代数多様体を調べることで，新たな知見が得られた．