6 M ay 2 00 3 Mukai flops and derived categories II

If Problem 2 is affirmative, then, which functor gives the equivalence ? The following examples suggest that the functor Ψ defined by the fiber product X ×X̄ X + would be a correct one. Examples. (1) ([B-O]): Let X be a smooth quasi-projective variety of dimension 2h−1 which contains a subvarietyM ∼= P with NM/X ∼= O(−1) . One can blow up X along M and blow down the exceptional divisor in another direction. In this way, we have a new varietyX with a subvarietyM ∼= P. Let s : X → X̄ and s : X → X̄ be the birational contraction maps of M and M to points respectively. Let μ : X ×X̄ X + → X and μ : X ×X̄ X + → X be the projections. Then Ψ(•) := Rμ∗Lμ (•) is an equivalence.