复形概型的Lichtenbaum-Quillen维度
The Lichtenbaum-Quillen dimension of complex varieties
AI总结：
本文研究复形概型的Lichtenbaum-Quillen维度，定义该维度为代数K理论与拓扑K理论在高维时变得同构的点，并分析了其计算方法及应用，包括对有理性障碍的贡献和对Hodge猜想的新证明。
摘要：
对于光滑复形概型的Lichtenbaum-Quillen猜想指出，代数和拓扑K理论在高维时具有有限系数变得同构。本文定义了概型的Lichtenbaum-Quillen维度，即此同构发生的点，证明其计算令人惊讶，并分析了多个例子。该维度提供了有理性的障碍，但比Colliot-Thélène和Voisin利用Bloch-Ogus理论定义的相关分式不变量弱。然而，由于其与半正交分解兼容，它允许利用同调投影对偶性证明Hodge猜想的一些新情况，并计算某些Fano概型导出范畴的Kuznetsov组件的高代数K理论。
英文摘要：
The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic 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.
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