Mukai flops and derived categories

Abstract

In this note, we shall prove that two smooth projective varieties of dim 2n connected by a Mukai flop have equivalent bounded derived categories. More precisely, let $\varphi :X--\to X^{+}$ be a Mukai flop with centers $Y\subset X$ and $Y^{+}\subset X^{+}$. In our case, the natural fuctor $\Phi :D(X)\to D(X^{+})$ defined by the graph of $\varphi$ is not fully faithful. Instead, let $X\to \bar{X}$ and $X^{+}\to \bar{X}$ be the birational contraction maps of the centers, and put ${\hat X} := X \times_{{\bar X} X^+$. Then $\hat{X}$ is a normal crossing variety with two irreducible components. This $\hat{X}$ defines a functor $\Psi :D(X)\to D(X^{+})$. We shall prove that this $\Psi$ is an equivalence. Recently, Wierzba and Wisniewski have announced that two birationally equivalent, complex projective symplectic 4-folds are connected by a finite sequence of Mukai flops. Our result with this shows that $D(X)$ is a birational invariant for complex projective symplectic 4-folds.

revised version