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Title:
A failed attempt at irrationality via algebraic K-theory
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Abstract:
The Quillen-Lichtenbaum conjecture, proved by Voevodsky, states that for smooth complex n-folds, the map from algebraic to topological K-theory with finite coefficients is an isomorphism in degree n-1 and higher, and injective in degree n-2. From this and some blow-up formulas one can construct a birational invariant. Surprisingly, it is somewhat computable. But sadly it vanishes for all cubic fourfolds; the proof repackages substantial cycle-theoretic results of Voisin, M. Shen, and others. Or to say it in a positive way, Kuznetsov's K3 category behaves from this perspective like the derived category of an honest surface. This is work in progress with Elden Elmanto.
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