Integral functors

  • Claudio BartocciEmail author
  • Ugo Bruzzo
  • Daniel Hernández Ruipérez
Part of the Progress in Mathematics book series (PM, volume 276)


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 R n , pulls it back to R n × R n , multiplies it by the kernel e i x⋅y and then integrates over the first copy of R n , thus obtaining a function \(\hat{f}\) (y) on the second copy.


Line Bundle Spectral Sequence Algebraic Variety Integral Functor Coherent Sheave 
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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  • Claudio Bartocci
    • 1
    Email author
  • Ugo Bruzzo
    • 2
  • Daniel Hernández Ruipérez
    • 3
  1. 1.Dipartimento di MatematicaUniversità di GenovaGenovaItaly
  2. 2.Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica NucleareTriesteItaly
  3. 3.Departamento de Matemáticas and Instituto Universitario de Fisica Fundamental y MatemáticasUniversidad de SalamancaSalamancaSpain

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