No abstract uploaded!

An odd Seiberg-Witten invariant imposes bounds on the signature of a closed, almost complex 4-manifold with vanishing first Chern class. This applies in particular to symplectic 4-manifolds of Kodaira dimension zero.

We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori’s Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifolds. As a by-product, we confirm the conjecture of Guedj [20] for automorphisms on 3-dimensional projective manifolds, and also determine π1(X).

We study minimal hypersurfaces in manifolds of non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay at infinity. By comparison with capped spherical cones, we identify a precise borderline for the Ricci curvature decay. Above this value, no complete area-minimizing hypersurfaces exist. Below this value, in contrast, we construct examples.

We construct Coleff–Herrera products and Bochner–Martinelli type residue currents associated with a tuple$f$of weakly holomorphic functions, and show that these currents satisfy basic properties from the (strongly) holomorphic case. This include the transformation law, the Poincaré–Lelong formula and the equivalence of the Coleff–Herrera product and the Bochner–Martinelli type residue current associated with$f$when$f$defines a complete intersection.