2021 Fiscal Year Annual Research Report
| Project/Area Number |
21J10242
|
|---|
| Research Institution | The University of Tokyo |
|---|
Principal Investigator |
WANG Long 東京大学, 数理科学研究科, 特別研究員(DC2)
|
|---|
| Project Period (FY) |
2021-04-28 – 2023-03-31
|
| Keywords | Cone conjecture / Calabi－Yau varieties / Birational automorphisms / Arithmetic degrees / Dynamical degrees |
|---|
| Outline of Annual Research Achievements |
The birational cone conjecture predicts that there is a rational polyhedral fundamental domain for the action of birational automorphisms on the movable effective cone of a Calabi－Yau variety. I verified this conjecture for certain complete intersections. The study of birational automorphisms is of independent interest. One attractive area is the dynamics of birational automorphisms from both geometric and arithmetic viewpoints. In this direction, I solved Kawaguchi－Silverman conjecture for automorphisms of smooth quasi-projective surfaces.
|
| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
I solved the birational cone conjecture of Morrison and Kawamata for certain complete intersections in a product of projective spaces. This is a part of my study plan. Moreover, I solved this conjecture for anti-canonical hypersurfaces in certain Fano varieties. I also studied the birational automorphisms from the viewpoint of dynamics with emphasis on a conjecture of Kawaguchi and Silverman that connects the dynamical and arithmetic degrees of dominant rational self-maps. In particular, I verified this conjecture for automorphisms of smooth quasi-projective surfaces.
|
| Strategy for Future Research Activity |
I will investigate the general case of Morrison－Kawamata cone conjecture and choose some important and interesting classes that are directly related to mirror symmetry to begin with, including the Borcea－Voisin construction and the Batyrev construction. As for the birational cone conjecture, one should also start from specific examples, for instance, anti-canonical hypersurfaces in Fano varieties. On the other hand, I would like to continue studying arithmetic degrees of an arbitrary dominant rational self-maps towards the conjecture of Kawaguchi and Silverman. I hope the tools from birational geometry may help to the study of dynamics.
|
