International Congress of Mathematicians

50 A.Bondal D.OrlovDefinition 4 [BK] A sequence of triangulated subcategories (BQ,...,B U ) in a triangulatedcategory T> is said to be semiorthogonal if Bj C B(- whenever 0 < j and denote this as follows:V=(B 0 ,....,B rExamples of semiorthogonal decompositions are provided by exceptional sequencesof objects [Bo]. These arise when all B,'s are equivalent to the derived categoriesof finite dimensional vector spaces V b (k — mod). Objects which correspond tothe 1-dimensional vector space under a fully faithful functor F :V b (k — mod) —¥ T>can be intrinsically defined as exceptional, i.e. those satisfying the conditionsHom'(B , E) = 0, when i ^ 0, and Hom°(ii', E) = k. There is a natural actionof the braid group on exceptional sequences [Bo] and, under some conditions,on semiorthogonal sequences of subcategories in a triangulated category [BK].We propose to consider the derived category of coherent sheaves as an analogueof the motive of a variety, and semiorthogonal decompositions as a tool for simplificationof this 'motive' similar to splitting by projectors in Grothendieck motivictheory.Let X and M be smooth algebraic varieties of dimension n and m respectivelyand E an object in V b (X x M). Denote by p and IT the projections of M x X to Mand X respectively. With E one can associate the functor # : V b (M) —y V b (X)defined by the formula: E (-):=B,iT*(E(êp*(-)).It happens that any fully faithful functor is of this form.Theorem 5 [Or2] Let F : V b (M) -+ V b (X) be an exact fully faithful functor,where M and X are smooth projective varieties. Then there exists a unique up toisomorphism object E £ V b (M x X) such that F is isomorphic to the functor #.The assumption on existence of the right adjoint to F, which was originally in[Or2], can be removed in view of saturatedness of V b (M) due to [BK], [BVdB].This theorem is in conjunction with the following criterion.Theorem 6 [BOI] Let M and X be smooth algebraic varieties and E £ V b (M x X).Then # is fully faithful functor if and only if the following orthogonality conditionsare verified:i) E.om x ($ E (O tl ) , $ E (O t2 )) = 0 for every i andti^t 2 .it) Eom x (^E(O t ),^E(O t )) = k,Eom x (^E(O t ) , & E (O t )) = 0, for i