Dolbeault homotopy theory

Abstract:

For complex manifolds, we define “complex homotopy groups” in terms of the Dolbeault complex. Many theorems of classical homotopy theory are reflected in the properties of complex homotopy groups. Analytic fibre bundles yield long exact sequences of complex homotopy groups and various Hurewicz theorems relate complex homotopy groups to the Dolbeault cohomology. In a more analytic vein, the classical Fröhlicher spectral sequence has a complex homotopy analogue. We compute these complex homotopy invariants for such examples as Calabi-Eckmann manifolds, Stein manifolds, and complete intersections.