Journal of the European Mathematical Society· 2026Q1
The Lichtenbaum–Quillen dimension of complex varieties
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- Q1SCImago
- 2026year
Short summary
A new invariant, the Lichtenbaum–Quillen dimension (LQD), is introduced for smooth complex varieties, quantifying the degree at which algebraic and topological K-theory with finite coefficients become isomorphic.
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Key points
- Introduced the Lichtenbaum–Quillen dimension (LQD) for smooth complex varieties.
- LQD measures the degree of isomorphism between algebraic and topological K-theory with finite coefficients.
- LQD is computable and provides an obstruction to rationality, though weaker than unramified cohomology.
- LQD's compatibility with semi-orthogonal decompositions proves new cases of the integral Hodge conjecture and computes K-theory for Fano varieties.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Journal of the European Mathematical Society, 2026 · DOI ↗
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Field: Geometry and Topology
Geometry and TopologyMathematics