The Chow norm and the existence problem of extremal metrics
Student No.:50
Time:Wed/Fri 9:50-11:25, 2017-08-23~2017-09-01
Instructor:Toshiki Mabuchi  [Osaka University]
Place:Conference Room 1, Floor 1, Jinchunyuan West Building
Starting Date:2017-8-23
Ending Date:2017-9-1





The Yau-Tian-Donaldson conjecture for anti-canonical polarization was recently solved affirmatively by Chen-Donaldson-Sun and Tian. However, this conjecture is still open for general polarizations or more generally in extremal Kähler cases. In this course, the unsolved cases of the conjecture will be discussed. It will be shown that the problem is closely related to the geometry of moduli spaces of test configurations for polarized algebraic manifolds.


Another important tool in our approach is the Chow norm introduced in [3]. This is closely related to Ding’s functional, and plays a crucial role in our differential geometric study of stability. By discussing the Chow norm from various points of view, we shall make a systematic study of the existence problem of extremal Kähler metrics.





Some basic knowledge of Kähler (or algebraic) geometry and algebraic groups





[1] S.K. Donaldson: Scalar curvature and projective embeddings I, J. Diff. Geom. 59 (2001), 479-522

[2] G. Tian: On a set of polarized Kähler metrics on algebraic manifolds, J. Diff. Geom. 32 (1990), 99-130

[3] S. Zhang: Heights and reductions of semi-stable varieties, Compos. Math. 104 (1996), 77-105

[4] R. Berman, S. Boucksom and M. Jonsson: A variational approach to the Yau-Tian-Donaldson conjecture,
arxiv: 1509.0456, math.DG (2015)