Bondal, Alexey; Orlov, Dmitri Reconstruction of a variety from the derived category and groups of autoequivalences. (English) Zbl 0994.18007 Compos. Math. 125, No. 3, 327-344 (2001). There exist examples of different varieties \(X\) with equivalent derived categories \(D^b_{coh}(X)\) of coherent sheaves. The authors show that \(D^b_{coh}(X)\) neet not to be a weak invariant of \(X\). They prove that \(X\) is uniquely determined by its category \(D^b_{coh}(X)\) if its anticanonical (Fano case) or canonical (general type case) sheaf is ample. Among other results, they prove that, for a smooth algebraic variety with either ample canonical or anticanonical sheaf, the group of exact autoequivalences is the semidirect product of the group of automorphisms of the variety and the Picard group of translations. Reviewer: B.M.Schein (Fayetteville) Cited in 20 ReviewsCited in 120 Documents MSC: 18E30 Derived categories, triangulated categories (MSC2010) 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Keywords:derived categories of coherent sheaves; autoequivalences × Cite Format Result Cite Review PDF Full Text: DOI arXiv