Asian Journal of Mathematics

Volume 24 (2020)

Number 2

Deformation of $K$-theoretic cycles

Pages: 303 – 330

DOI: https://dx.doi.org/10.4310/AJM.2020.v24.n2.a6

Author

Sen Yang (Shing-Tung Yau Center, Southeast University, Nanjing, China; School of Mathematics, Southeast University, Nanjing, China; and Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Abstract

For $X$ a $d$-dimensional smooth projective variety over a field $k$ of characteristic $0$, using higher algebraic $K$-theory, we study the following two questions asked by Mark Green and Phillip Griffiths in chapter 10 of [9] (page 186-190):

(1) For each positive integer $p$ satisfying $1 \leq p \leq d$, can one define the tangent space $TZ^p (X)$ to the cycle group $Z^p (X)$?

(2) Obstruction issues. The highlight is the appearance of negative $K$-groups which detect the obstructions to deforming cycles.

Keywords

$K$-theory, algebraic cycles, deformation, tangent spaces, obstructions

2010 Mathematics Subject Classification

14C25

Full Text (PDF format)

Received 5 February 2018

Accepted 29 May 2019

Published 8 September 2020