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Derived categories of coherent sheaves and motives. (English) Zbl 1146.18302

Russ. Math. Surv. 60, No. 6, 1242-1244 (2005); translation from Usp. Mat. Nauk 60, No. 6, 231-232 (2005).
From the text: The bounded category \({\mathcal D}^b(X)\) of coherent sheaves is, by definition, the triangulated category that can be naturally associated with an algebraic variety \(X\). It happens sometimes that the equivalence \({\mathcal D}^b(X)\simeq{\mathcal D}^b(Y)\) holds for two distinct varieties \(X\) and \(Y\). A natural question arises: can anything be said about the motives of the given varieties in such a situation? The first such example [S. Mukai, Nagoya Math. J. 81, 153–175 (1981; Zbl 0417.14036)] – an abelian variety \(A\) and its dual \(\widehat{A}\) – shows that the motives of the given varieties are not necessarily isomorphic. However, apparently in all known examples their motives with rational coefficients are isomorphic. We can prove this for smooth projective varieties.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F42 Motivic cohomology; motivic homotopy theory
18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)

Citations:

Zbl 0417.14036
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