Integral functors

Abstract

The first instance of an integral functor is to be found in Mukai’s 1981 paper on the duality between the derived categories of an Abelian variety and of its dual variety [224]. Integral functors have also been called “Fourier-Mukai functors” or “Fourier-Mukai transforms.” However, we shall give these terms specific meanings that we shall introduce in Chapter 2. The core idea in the de_nition of an integral functor is very simple: if we have two varieties X and Y, we may take some “object” on X, pull it back to the product X×Y, twist it by some object (“kernel”) in X×Y and then push it down to Y (i.e., we integrate on X). This is what happens with the Fourier transform of functions: one takes a function f(x) on Rn, pulls it back to Rn × Rn, multiplies it by the kernel e i x⋅y and then integrates over the first copy of Rn, thus obtaining a function f(y) on the second copy.