A study of 4-dimensional manifolds from the symplectic-topological viewpoint and the algebraic-geometrical viewpoint

2014 Fiscal Year Final Research Report

• Project/Area Number: 23540096
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Kyushu Institute of Technology
• Principal Investigator: SATO Yoshihisa 九州工業大学, 大学院情報工学研究院, 教授 (90231349)
• Co-Investigator(Kenkyū-buntansha): ASHIKAGA Tadashi 東北学院大学, 工学部, 教授 (90125203) ; HIROSE Susumu 東京理科大学, 理工学部, 准教授 (10264144) ; ENDO Hisaaki 東京工業大学, 大学院理工学研究科, 教授 (20323777)
• Project Period (FY): 2011-04-28 – 2015-03-31
• Keywords: Ｌｅｆｓｃｈｅｔｚ ｆｉｂｒａｔｉｏｎ / Ｌｅｆｓｃｈｅｔｚ ｐｅｎｃｉｌ / ｓｙｍｐｌｅｃｔｉｃ ｔｏｐｏｌｏｇｙ / ｇｅｏｇｒａｐｈｙ / ｐｓｅｕｄｏｈｏｌｏｍｏｒｐｈｉｃ ｃｕｒｖｅ / Ｇｒｏｍｏｖ ｉｎｖａｒｉａｎｔ / Ｋｏｄａｉｒａ ｄｉｍｅｎｓｉｏｎ

Outline of Final Research Achievements

Symplectic 4-manifolds are important objects in smooth 4-manifolds. A class of symplectic 4-manifolds includes Kähler surfaces or complex projective　surfaces, which are important objects in algebraic geometry. In algebraic geometry, in order to classify complex surfaces, ones study the　characterization of pairs of Chern numbers or the slopes of pencils of algebraic curve, whose problem is called the geography problem.　Furthermore, symplectic 4-manifolds admit fibration structures as Lefschetz fibrations/pencils.
We study symplectic 4-manifolds from the symplectic-toplogical viewpoint and the algebraic-geometrical viewpoint and can obtain the answer of the geography problem for symplectic 4-manifolds. Furthermore, we can determine the canonical classes of nonminimal Lefschetz fibrations, and then we calculus the Kodaira dimension of those fibrations.

Free Research Field

4次元微分位相幾何学，シンプレクティックトポロジー，複素代数曲面論