Rational parallelism on complex manifolds

On Rational Homotopy of Four-manifolds

We give explicit formulas for the ranks of the third and fourth homotopy groups of all oriented closed simply connected four-manifolds in terms of their second Betti numbers. We also show that the rational homotopy type of these manifolds is classified by their rank and signature.

Remarks on Complex Contact Manifolds

1. Statement of results. Let M be a complex manifold of complex dimension 2w + l. Let { Ui} be an open covering of M. We call M a complex contact manifold if the following conditions are satisfied: (1) On each [/,there exists a holomorphic 1-form w,such that Wi/\(do3i)n is different from zero at every point of Ui. (2) If UiC\Uj is nonempty, then there exists a nonvanishing holomorphic function ...

Twistor Spaces and Balanced Metrics on Complex Manifolds Complex manifolds of complex dimension

Complex manifolds of complex dimension 1 (Riemann surfaces) are of course always Kähler, that is admit Kähler metrics, on account of the obvious dimension situation: dω=0 simply because it is a 3-form! This dimensional necessity naturally does not apply in complex dimension 2 or higher, but as it happens, most compact complex surfaces in fact are Kähler. Moreover, the non-Kähler examples occurr...

Complex Manifolds

is holomorphic. Thus P has the structure of a complex manifold, called complex projective space. The “coordinates” Z + [Z0, . . . , Zn] are called homogeneous coordinates on P. P is compact, since we have a continuous surjective map from the unit sphere in C to P. Note that P is just the Riemann sphere C ∪ {∞}. Any inclusion C → C induces an inclusion P → P; the image of such a map is called a ...

Symmetries on Manifolds, Deformations and Rational Homotopy: a Survey