Rationally Connected Varieties and Loop Spaces

We consider rationally connected complex projective manifolds M and show that their loop spaces—infinite dimensional complex manifolds—have properties similar to those of M . Furthermore, we give a finite dimensional application concerning holomorphic vector bundles over rationally connected complex projective manifolds. 0 Introduction LetM be a complex manifold and r = 0, 1, . . . ,∞. The space Cr(S1,M) of r times continuously differentiable maps x : S1 → M , the (free) C Research partially supported by NSF grant DMS0203072 and OTKA grants T046378, T42769 This research was done while the first author, on leave from Purdue University, visited the Department of Analysis, Eötvös University, Budapest. He is grateful to both institutions.