闭辛流形几何中的几个问题

We consider the following questions:.A. (1) Give a topological characterization of non-elliptically closed symplectic manifolds, if a non-elliptically closed symplectic manifolds has hard Lefschetz property, we have obtained the corresponding result..(2) It is well known that any non-elliptically closed symplectic manifold is a symplectically closed aspherical manifold, give a topological characterization of closed aspherical manifolds...B. (1) Solve Donaldson’s type Calabi-Yau equations on closed symplectic four-manifolds..(2) Consider higher dimension cases...C. (1) Donaldson posed the following question: If an almost complex structure on a closed four-manifold is tamed by a symplectic form, must it be compatible with a new symplectic form? When h_J^-=b^+-1, in particular b^+=1, we have obtained a positive answer. Hence we will consider general cases..(2) Consider applications of the Donaldson’s question for "tamed to compatible".

我们考虑以下几个问题：.A. (1) 对非椭圆闭辛流形给出拓扑刻画，当非椭圆闭辛流形具强Lefschetz性质，我们已得到相应结果。.(2) 众所周知，任一非椭圆闭辛流形是无球辛流形，期望给出闭无球流形的拓扑刻画。..B. (1) 解四维辛流形上Donaldson型Calabi-Yau方程。.(2) 考虑高维情形。..C. (1) Donaldson提出以下问题：闭四维流形上的近复结构被一辛形式驯化，给定近复结构是否一定与一新的辛形式相容？如果h_J^-=b^+-1（特别，b^+=1），我们已给出肯定回答。我们将考虑一般情形。.(2) 考虑Donaldson关于“由驯化到相容”问题的应用。

A.对非椭圆闭辛流形在一定条件下给出一个拓扑刻画，即在此条件下证明了Hopf猜想。最后证明了经典Hopf猜想。.B.给出了关于闭辛流形上的Donaldson型Calabi-Yau方程解的存在性。.C.给出解Donaldson“驯服”问题的充分条件。