Deformation of K-theoretic cycles

TL;DR

This paper explores the deformation theory of K-theoretic cycles using higher K-theory, addressing how to define tangent spaces and identify obstructions, notably through negative K-groups.

Contribution

It introduces a framework for defining tangent spaces to cycle class groups and identifies negative K-groups as obstructions in deformation theory.

Findings

Negative K-groups detect obstructions to cycle deformation

Provides a new approach to tangent space definition in K-theoretic cycles

Answers open questions by Green and Griffiths

Abstract

By using higher K-theory, we study deformation theory of K-theoretic cycles. As an application, we answer two questions posed by Mark Green and Philip Griffiths: (1). How to define tangent spaces to cycle class groups in general? (2). Obstruction issues. The highlight is the appearance of negative K-groups which detects the obstructions to deforming cycles.