un 2 00 2 Mukai flops and derived categories

Derived categories possibly give a new significant invariant for algebraic varieties. In particular, when the caninical line bundle is trivial, there are varieties which are not birationally equivalent, but have equivalent derived categories. The most typical example can be found in the original paper [Mu]. On the other hand, for such varieties, it is hoped that the birationally equivalence should imply the equivalence of derived categories. One of the important classes for testing this question is the class of complex symplectic manifolds (for varieties in other classes, see [B2, Ka 1, Ch]). A Mukai flop is a typical birational map between complex symplectic manifolds. In this note, we shall prove that two smooth projective varieties of dim 2n connected by a Mukai flop have equivalent bounded derived categories of coherent sheaves. More precisely, let X and X be smooth projective varieties of dimension 2n such that there is a birational map φ : X − − → X obtained as the Mukai flop along a subvariety Y ⊂ X which is isomorphic to P. By definition, φ is decomposed into the blowing-up p : X̃ → X along Y and the blowing down p : X̃ → X contracting the p-exceptional divisor Ỹ to the subvariety Y + ∼= P of X. On the other hand, there are birational morphisms X → X̄ and X → X̄ which contract Y and Y + to points respectively. We put X̂ := X ×X̄ X + and let q : X̂ → X and q : X̂ → X be the natural projections. Note that X̂ is a normal crossing variety with two irreducible components X̃ and P × P. Let D(X) (resp. D(X)) be the bounded derived category of coherent sheaves on X (resp. X). We shall consider natural two functors