K-equivalence in Birational Geometry

In this article we survey the background and recent development on the Kequivalence relation among birational manifolds. The content is based on the author’s talk at ICCM-2001 at Taipei. I would like to dedicate this article to Professor Chern, Shiing-shen to celebrate his 90th birthday. For manifolds, K-equivalence is the same as c1-equivalence. In this sense, a major part of birational geometry is really to understand the geometry of the first chern class. After a brief historical sketch of birational geometry in §1, we define in §2 the K-partial ordering and K-equivalence in a birational class and discuss geometric situations that will lead to these notions. One application to the filling-in problem for threefolds is given. In §3 we discuss motivic aspect of K-equivalence relation. We believe that K-equivalent manifolds have the same Chow motive though we are unable to prove it at this moment. Instead we discuss various approaches toward the corresponding statements in different cohomological realizations. §4 is devoted to the Main Conjectures and the proof of a weak version of it. Namely, up to complex cobordism, K-equivalence can be decomposed into composite of classical flops. Finally in §5 we review some other current researches that are related to the study of K-equivalence relation.