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  "body": "\n## Summary\n\nA Fourier–Mukai transform is an exact functor between the [[derived categories of coherent sheaves|2VP2D51KB8]] of two varieties $X$ and $Y$.\nIt is given by an object on $X \\times Y$ called the kernel.\nMukai showed that the Poincaré bundle gives an equivalence between the derived categories of an abelian variety and of its dual.\nBy a theorem of Orlov, all fully faithful exact functors between such derived categories of smooth projective varieties are of this form.\n\n## Explanations\n\n### L1 human\n\nThe classical Fourier transform turns a function $f(x)$ into a function of a new variable by integrating $f(x)$ against a kernel $e^{-2\\pi i x y}$.\nA Fourier–Mukai transform does the same with sheaves instead of functions: pull back, multiply by a kernel on the product, and push forward, which is the analogue of integration.\nDifferent varieties can have equivalent derived categories, and Fourier–Mukai transforms are the tool that detects and produces such equivalences.\nThe simplest kernel is the structure sheaf of the diagonal, which gives the identity functor.\n",
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