The Lichtenbaum–Quillen dimension of complex varieties

The Lichtenbaum–Quillen conjecture for smooth complex varieties states that algebraic K-theory and topological K-theory with finite coefficients become isomorphic in high degrees. We define the “Lichtenbaum–Quillen dimension” of a variety in terms of the point where this happens, show that it is surprisingly computable, and analyze many examples. It gives an obstruction to rationality, but one that turns out to be weaker than unramified cohomology and some related birational invariants defined by Colliot-Thélène and Voisin using Bloch–Ogus theory. Because it is compatible with semi-orthogonal decompositions, however, it allows us to prove some new cases of the integral Hodge conjecture using homological projective duality, and to compute the higher algebraic K-theory of the Kuznetsov components of the derived categories of some Fano varieties.