Integral transforms for coherent sheaves

The theory of integral, or Fourier-Mukai, transforms between derived categories of sheaves is a well established tool in noncommutative algebraic geometry. General kernel theorems represent all reasonable linear functors between categories of perfect complexes (or their large version, quasi-coherent complexes) on schemes and stacks over some fixed base as integral kernels in the form of complexes (of the same nature) on the fiber product. However, for many applications in mirror symmetry and geometric representation theory one is interested instead in the bounded derived category of coherent sheaves (or its large version, ind-coherent sheaves), which differs from perfect complexes (and quasi-coherent complexes) once the underlying variety is singular. In this paper, we prove general kernel theorems for linear functors between derived categories of coherent sheaves over a base in terms of integral kernels on the fiber product. Namely, we identify coherent kernels with functors taking perfect complexes to coherent complexes (an analogue of the classical Schwartz kernel theorem), and kernels which are coherent relative to the source with functors taking all coherent complexes to coherent complexes. The proofs rely on key aspects of the functional analysis of derived categories, namely the distinction between small and large categories and its measurement using t-structures. These are used in particular to correct the failure of integral transforms on ind-coherent complexes to correspond to ind-coherent complexes on a fiber product. The results are applied in a companion paper to the representation theory of the affine Hecke category, identifying affine character sheaves with the spectral geometric Langlands category in genus one.