The Douady space of compact subvarieties a Kähler manifold is equipped with the Weil–Petersson current, which everywhere positive local continuous potentials, and class C ∞ when restricted to locus smooth fibers. There Quillen metric known exist, whose Chern form equal form. In algebraic case, we show that can be extended determinant line bundle as singular hermitian metric. On other hand in su...

We extend previous results on boundedness of sets coherent sheaves a compact K\"ahler manifold to the relative and not necessarily smooth case. This enlarged context allows us prove properness properties Douady space as well related semistability such existence Harder-Narasimhan filtrations.

We review several applications of Douady-Earle section to holomorphic motions over infinite dimensional parameter spaces. Using DouadyEarle section we study group-equivariant extensions of holomorphic motions. We also discuss the relationship between extending holomorphic motions and lifting holomorphic maps. Finally, we discuss several applications of holomorphic motions in complex analysis.

A systematic consideration of the problem of the reduction and extension of the structure group of a principal bundle is made and a variety of techniques in each case are explored and related to one another. We apply these to the study of the DixmierDouady class in various contexts including string structures, Ures bundles and other examples motivated by considerations from quantum field theory.