K-equivalence in Birational Geometry

Abstract

We give a survey of the background and recent development on the $K$-equivalence relation among birational manifolds. After a brief historical sketch of birational geometry, we define the $K$-partial ordering and $K$-equivalence in a birational class and discuss geometric situations that lead to these notions. One application to the filling-in problem for threefolds is given. We discuss the 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. We also formulate the {\it Main Conjectures} and prove a weak version of it. Namely, up to complex cobordism, $K$-equivalence can be decomposed into composite of classical flops. Finally we review some other current researches that are related to the study of $K$-equivalence relation.

18 pages. Minor changes, references updated

References in corpus (4)

• A New Cohomology Theory for Orbifold
• Francia's flip and derived categories
• Small Contractions of Symplectic 4-folds
• D-equivalence and K-equivalence

Cited by in corpus (6)

• Relative orbifold Gromov-Witten theory and degeneration formula
• Transformation of algebraic Gromov-Witten invariants of three-folds under flops and small extremal transitions, with an appendix from the stringy and the symplectic viewpoint
• K-correspondences and intrinsic pseudovolume forms
• Ruan's Conjecture on Singular symplectic flops
• Mukai Flop and Ruan Cohomology
• Integration of Voevodsky motives