Derived categories and their geometrical applications (part 4)

complexes of OX-modules with coherent cohomologies. 1 To what extent does D(X) determine X. Birational equiv-alences. On one hand we have a following resut: Theorem 1.1 ([BO01], Theorem 2.5). Let X be a smooth irreducible projective variety with an ample canonical or anti-canonical sheaf. If D(X) is equivalent to D(X ′) for some other smooth algebraic variety X ′, then X is isomorphic to X ′. On the other hand, we have a following conjecture: Conjecture 1 (Bondal, Orlov). If X1 and X2 are birational smooth projective Calabi-Yau varieties of dimension n, then there is an equivalence D(X1) ∼− → D(X2). Theorem 1.2. The conjecture above holds for n = 3. Proof. A minimal model programme result by Kollar [Kol89] states that any birational transform between smooth projective threefolds that preserves canonical class can be split in a finite sequence of flops. This reduces this theorem to proving a derived equivalence over a flop. It turns out that we can actually construct the flop Y ′ of a contraction Y → X as a fine moduli space of certain objects in D(Y) called perverse point sheaves. Then the Fourier-Mukai transform defined by the universal family gives the derived equivalence D(Y) → D(Y ′). We examine these techniques in details in our next section. 2 Fourier-Mukai transforms Let X and Y be smooth projective varieties and let E be an object of D(X×Y). Following Mukai, we define an integral transform ΦE from D(X) to D(Y) to be ΦE(−) = RpiY ∗(E L ⊗ pi∗X(−)) (2.1) where piX and piY denote projection