D-Equivalence and K-Equivalence

Abstract

Let X and Y be smooth projective varieties over C. They are called D-equivalent if their derived categories of bounded complexes of coherent sheaves are equivalent as triangulated categories, while K-equivalent if they are birationally equivalent and the pull-backs of their canonical divisors to a common resolution coincide. We expect that the two equivalences coincide for birationally equivalent varieties. We shall provide a partial answer to the above problem in this paper. for useful discussions or comments and the anonymous referee for suggestions. From D-equivalence to K-equivalenceWe need the concept of Fourier-Mukai transformation:Definition 2.1. Let X and Y be smooth projective varieties, and let p 1 : X ×Y → X and p 2 : X ×Y → Y be projections. For an object e ∈ D(X ×Y ), we define an integral functor Φ e X→Y : D(X) → D(Y ) by Φ e X→Y (a) = p 2 * (p * 1 (a) ⊗ e)

Homological Mirror Symmetry for manifolds of general type

“…This can be done in two steps -first pass to an orbifold cover of Y aff with mild singularities, and then construct a crepant resolution of that cover. Concretely, consider the quadric cone Z aff : 1 [6] and [16] applies to this situation and allows us to identify the derived categories of and the crepant resolution Y . Combined with our argument from section 7 this implies that we have an equivalence of categories…”

“…We expect that an anlogue of Conjecture 3.3 statement should hold in these cases as well. In paticular we expect that Conjecture 3.3 should hold when F is the Fano threefold X 16 and G is a curve of genus 3, or when F is a Fano threefold X 12 and G is a curve of genus 7. Some calculations for the mirrors of reductive Fano varieties give strong evidence for that.…”

Relative integral functors for singular fibrations and singular partners

“…The relevance of derived categories and Fourier-Mukai functors in birational geometry is nowadays well known [8,18]. One of the most important problems in this context is the minimal model problem.…”

“…Thus, one can get some information about the derived categories in dimension 3 from the derived categories of the corresponding smooth fourfold. The results in [13] can be generalized to Q-Gorenstein teminal threefolds (see [1,18,19]). In this situation, the associated Gorenstein stack allows us to reduce the problem to the Gorenstein case.…”

Fourier-Mukai transforms

“…In higher dimension it is still open. Following (and slightly generalizing) [11] (see also [39]) let us say that a birational map X X + between Q-Gorenstein varieties is a generalized flip if there is a commutative diagram withX smooth…”