Symplectic structures and cohomologies on some solvmanifolds.
Given a symplectic fibration , with compact and symplectic and the fibres complex projective, we produce symplectic submanifolds of analytic in the vertical direction, and apply this to complex vector bundles on symplectic manifolds.
It is proved that if is a weakly 1-complete Kähler manifold with only one end, then or there exists a proper holomorphic mapping of onto a Riemann surface.
We study the Calabi functional on a ruled surface over a genus two curve. For polarizations which do not admit an extremal metric we describe the behavior of a minimizing sequence splitting the manifold into pieces. We also show that the Calabi flow starting from a metric with suitable symmetry gives such a minimizing sequence.