Scalar curvature and deformations of complex structures

Deformations of Complex Structures

In this section, we discuss briefly how to compare different almost complex structures on a manifold. Let M be a fixed compact manifold, TM its tangent bundle. Let J ∈ End(TM ) be an almost complex structure on M , with associated decomposition TM ⊗ C ' TM 0,1 ⊕ TM 1,0, and projections π0,1 and π1,0 to the two summands. Now suppose J ′ is a second almost complex structure. The reader should thi...

On the Scalar Curvature of Complex Surfaces

Let (M, J) be a minimal compact complex surface of Kähler type. It is shown that the smooth 4-manifold M admits a Riemannian metric of positive scalar curvature iff (M,J) admits a Kähler metric of positive scalar curvature. This extends previous results of Witten and Kronheimer. A complex surface is a pair (M,J) consisting of a smooth compact 4-manifold M and a complex structure J on M ; the la...

Curvature from quantum deformations

A Poisson coalgebra analogue of a (non-standard) quantum deformation of sl(2) is shown to generate an integrable geodesic dynamics on certain 2D spaces of nonconstant curvature. Such a curvature depends on the quantum deformation parameter z and the flat case is recovered in the limit z → 0. A superintegrable geodesic dynamics can also be defined in the same framework, and the corresponding spa...

Positive Scalar Curvature and Surgery

Positive Scalar Curvature

One of the striking initial applications of the Seiberg-Witten invariants was to give new obstructions to the existence of Riemannian metrics of positive scalar curvature on 4– manifolds. The vanishing of the Seiberg–Witten invariants of a manifold admitting such a metric may be viewed as a non-linear generalization of the classic conditions [12, 11] derived from the Dirac operator. If a manifo...