Let $M$ be a compact oriented even‐dimensional manifold. This note constructs a compact symplectic manifold $S$ of the same dimension and a map $f:S\to M$ of strictly positive degree. The construction relies on two deep results: the first is a theorem of Ontaneda that gives a Riemannian manifold $N$ of tightly pinched negative curvature which admits a map to $M$ of degree equal to 1; the second is a result of Donaldson on the existence of symplectic divisors. Given Ontaneda's negatively curved manifold $N$, the twistor space $Z$ is symplectic. The manifold $S$ is then a suitable multisection of the twistor space, found via Donaldson's theorem.

中文翻译：

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辛控制

让 $\mathrm{中号}$是紧凑型的偶数维流形。本笔记构造了一个紧凑的辛流形$\mathrm{小号}$ 尺寸和地图相同 $F：\mathrm{小号}\to \mathrm{中号}$严格肯定的程度。该构造依赖于两个深层结果：第一个是Ontaneda定理，它给出了黎曼流形$ñ$ 紧捏的负曲率，它允许映射为 $\mathrm{中号}$程度等于1；第二个是唐纳森关于辛除数存在的结果。鉴于Ontaneda的负弯曲流形$ñ$，扭曲空间 $ž$是辛的。歧管$\mathrm{小号}$ 然后是通过Donaldson定理找到的扭曲空间的合适多部分。