Variation of cycles in projective spaces.

Lau, Siu Cheong.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 51-52).Abstracts in English and Chinese.Chapter 1 --- In search of minimal cycles --- p.9Chapter 1.1 --- What do we mean by cycles? --- p.9Chapter 1.2 --- Integral currents --- p.10Chapter 1.3 --- Calibration theory --- p.13Chapter 2 --- Motivation from the Hodge Conjecture --- p.17Chapter 2.1 --- Hodge theory on Riemannian manifolds --- p.17Chapter 2.2 --- Hodge decomposition in Kahler manifolds --- p.19Chapter 2.3 --- The Hodge conjecture --- p.22Chapter 3 --- Variation of cycles in symmetric orbit --- p.26Chapter 3.1 --- Variational formulae --- p.26Chapter 3.2 --- Stability of cycles in Sm and CPn --- p.29Chapter 3.3 --- Symmetric orbit in Euclidean space --- p.31Chapter 3.4 --- Projective spaces in simple Jordan algebra --- p.39Chapter 3.4.1 --- Introduction to simple Jordan algebra --- p.39Chapter 3.4.2 --- Projective spaces as symmetric orbits --- p.41Chapter 3.4.3 --- Computation of second fundamental form --- p.43Chapter 3.4.4 --- The main theorem --- p.45Chapter 3.5 --- Future directions --- p.49Bibliography --- p.5