On relative constructible sheaves and integral transforms

The aim of this note is threefold. The first is to obtain a simple characterization of relative constructible sheaves when the parameter space is projective. The second is to study the relative Fourier–Mukai transform for relative constructible sheaves and for relative regular holonomic D-modules and prove they induce relative equivalences of categories. The third is to introduce and study the notions of relative constructible functions and relative Euler–Poincaré index. We prove that the relative Euler–Poincaré index provides an isomorphism between the Grothendieck group of the derived category of complexes with bounded relative R-constructible cohomology and the ring of relative constructible functions.