Summary: 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 M. Green and P. Griffiths [On the tangent space to the space of algebraic cycles on a smooth algebraic variety. Princeton, NJ: Princeton University Press (2005; Zbl 1076.14016), Chapter 10]:

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(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 \(Z^p (X)\)?

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(2) Obstruction issues.

The highlight is the appearance of negative \(K\)-groups which detect the obstructions to deforming cycles.