Bondal-Polishchuk Conjecture for a Smooth Projective Variety
Prior state unknown→disproved
first counterexample of the form D^b(X) for X smooth projective
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algebra / Algebraic geometry
Bondal-Polishchuk Conjecture for a Smooth Projective Variety
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.
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Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form $D^b(X)$ for a smooth projective variety was known. A particular weak Fano threefold provides one.
first counterexample of the form D^b(X) for X smooth projective
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