Geometry Seminar


Capacity, Green's functions, and Intersection Theory


Dr. Lau Chi Fong

Market Risk Management, HSBC Global Markets



Chinburg originally introduced sectional capacity as an arithmetic measure for sets on an algebraic variety over a global field. The sets in question are adelic. There is a component set for each place of the ground field. An ample divisor produces the size measure. The Gillet-Soulé Arithmetic Amplitude Theorem motivated sectional capacity.  This gives asymptotics for measuring the space of sections with sup norm at most 1.


We discuss work of Chinburg, Rumely, and the speaker relating sectional capacity and arithmetic intersection theory. The main theorem asserts that the sectional capacity is a limit of top self-intersection numbers of metrized line bundles. This limit is relative to a canonical sequence of models determined by the nonarchimedean part of the set. The metrics come from smoothings of plurisubharmonic extremal functions attached to archimedean components of the set. This is the strongest result of its type with a formulation within Gillet-Soulé theory. Conjecturally, however, the sectional capacity should be a top intersection number of an adelic line bundle with singular metrics. The talk will discuss evidence for such an intersection theory.



1.     T. Chinburg, Capacity Theory on Varieties, Compositio Math 80 (1991), 71-84.

2.     H. Gillet and C. Soulé, Amplitude Arithmétique, CRAS Paris 307 (1988), Série I (Math), 887-890.

3.     R. Rumely, C.F. Lau and R. Varley, Existence of the Sectional Capacity, Memoires of the AMS 690 (2000), AMS, Providence, R.I.

4.     T. Chinburg, C.F. Lau and R. Rumely, Capacity Theory and Arithmetic Intersection Theory, Duke Mathematical Journal  117, no 2 (2003), 229-285.



November 17, 2005 (Thursday)


4:00 – 5:00pm


Room 517, Meng Wah Complex




All are welcome