Algebraic theory of Higgs sheaves and its applications

2019 Fiscal Year Final Research Report

• Project/Area Number: 16K05106
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Chuo University
• Principal Investigator: Miyaoka Yoichi 中央大学, 理工学部, 教授 (50101077)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Keywords: 可積分作用つき層 / ヒッグス層 / Mehta-Ramanathan 型制限定理 / テンソル積保存定理 / Bogomolov 不等式

Outline of Final Research Achievements

We tried to construct an elementary and purely algebraic theory of Higgs sheaves. We proved the following three results: 1) Mehta-Ramanathan theorem which asserts that the semi stability of a sheaf with an action of a vector bundle is preserved by restriction to general hypersurfaces of high degree, 2) Tensor product theorem to the effect that the tensor product of two semistable sheaves with integral actions of a bundle is again semisgtable, and 3) the Bogomolov inequality for the 1st and 2nd Chern classes of a semistable Higgs sheaf (a sheaf with an integrable action of the tangent bundle). These results shed a new light on, and give a reconstruction of, the theory of Higgs sheaves in terms of pure algebraic geometry.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

理論物理から生まれたヒッグス層は、微分方程式論、微分幾何学、代数幾何学など、純粋数学のさまざまな分野においても重要な役割を果たしつつある。しかしながら従来のヒッグス層理論は、ゲージ理論、すなわち非線形偏微分方程式論　 (C. Simpson, T. Mochizuki) や、 p 進ホッジ理論 (A. Langer) といった大道具を用いており、きわめて難解なものであった。われわれの研究はヒッグス層の基礎理論に明快かつ初等的な枠組みを与え、見通しをよくするものである。