FOURIER-MUKAI TRANSFORMS AND FOLIATIONS

FRANZ KAMBER (UNIVERSITY OF ILINOIS)

Abstract: We construct a Fourier–Mukai transform for smooth complex vector bundles E over a torus bundle π : M → B , the vector bundles being endowed with various structures of increasing complexity. At a minimum, we consider vector bundles E with a flat partial unitary connection, that is families or deformations of flat vector bundles on the torus T . This leads to a correspondence between such objects on M and relative skyscraper sheaves S supported on a spectral variety Σ ↪ M̂ , where π̂ : M̂ → B is the flat dual fiber bundle. Additional structures on E will be reflected by corresponding data on the transform (S, Σ) . Several variations of this construction will be presented, emphasizing the aspects of foliation theory which enter into this picture.