Mukai flops and derived categories

By Y. Namikawa. 2002

70 citations recorded by the source corpus.

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 $\phi : X - - \to X^+$ 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 $\phi$ 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f8b91ba8199e0134aec8c87625d63baf1985b349

Citations in the corpus

• On the Chow ring of birational irreducible symplectic varieties (2013, unknown): “In [27], Namikawa studies the functor OZ := R(q ′ ◦ ιZ )∗ ◦ L(q ◦ ιZ )∗ : Db(X) → Db(X ′), which coincides with the Fourier–Mukai transform with kernel OZ .”
• Derived categories of coherent sheaves (2006, unknown): “Kawamata and Namikawa have also studied certain stratified Mukai flops [28] [39].”