On S.-T. Yau's pseudonorm project
Student No.:40
Time:Tue & Thu 13:30-15:05, Aug.21,23,28,30
Instructor:Toshiki Mabuchi  [Osaka University]
Place:Conference Room 1, Jin Chun Yuan West Bldg.
Starting Date:2018-8-21
Ending Date:2018-8-30

In this course, the pseudonorm project towards birational classification proposed by S.-T. Yau [5] (see also [2]) will be discussed in detail from slightly different viewpoints.
In contrast to the GIT-limit in algebraic geometry (or to the Gromov-Hausdorff limit in Riemannian geometry), we have some straightforward compactification [4] of the moduli space of psedonormed spaces. This construction allows us to obtain natural limits for sequences of pseudonormed graded algebras.
The notion of the orthogonal direct sum [4] for psedonormed spaces will also make sense. Actually, associated to a stable curve, an orthogonal direct sum of some suitable psedonormed spaces will appear naturally.

Prerequisite: Some basic knowledge of Kähler (or algebraic) geometry.


[1]C.-Y. Chi: Pseudonorms and theorems of Torelli type, J. Differential Geom. 104 (2016), 239-273.
[2] C.-Y. Chi and S.-T. Yau: A geometric approach to problems in birational geometry, Proc. Natl. Acad. Sci. USA, 105 (2008), no. 48, 18696-18701.
[3] Y. Imayoshi and T. Mabuchi: A Torelli-type theorem for stable curves, in “Geometry and Analysis on Complex Manifolds”, World Scientific Publ. Co. (1994), 75-95.
[4] T. Mabuchi: Orthogonality in the geometry of Lp-spaces, in "Geometric Complex Analysis", World Scientific Publ. Co. (1996), 409-417.
[5] S.-T. Yau: On the pseudonorm project project of birational classification of algebraic varieties, in "Geometry and Analysis on Manifolds", Progr. in Math. 308 (2015), Birkhäuser, 327-339.