论文标题
几乎复杂的歧管，betti编号$ b_i = 0 $除外，$ i = 0，n/2，n $
Almost complex manifold with Betti number $b_i=0$ except $i=0, n/2, n$
论文作者
论文摘要
本文研究$ n = 4k（k> 1）的存在$尺寸简单连接的封闭封闭的几乎复杂的歧管，betti number $ b_i = 0 $ = 0 $，除$ i = 0，n/2，n $。我们表征了这种歧管的所有理性共同体戒指，并表明它们必须具有甚至具有欧拉的特征甚至签名，也就是说，中间的betti number $ b_ {n/2} $都必须是。与作者早期通过平滑的封闭歧管实现理性的共同体学戒指的工作并行，我们指出并证明了沙利文的理性手术实现了几乎复杂的歧管定理，并在我们的背景下证明了它的应用。当且仅当戒指结构支持封闭歧管的交叉形式时，并且只有一个封闭的歧管的相交形式，并且它具有满足签名方程和Riemann-Roch完整性关系，并且顶部Chern号等于Euler特征时，就可以通过简单连接的几乎复杂的歧管来实现规定的理性共同体学环的规定理性共同体，并且最高的Chern数字等于Euler的特征。根据Stong对$ U $和$ SU $ COBORDISM的特征，我们明确地计算了Chern数字之间的Riemann-Roch完整性关系，而只有中间和顶级Chern类才能非零。实现的必要条件表示为签名和欧拉（Euler）特征之间的一组一致关系，我们表明，签名的2-辅助顺序的下限以及相对于实现歧管的维度而言，欧拉（Euler）特征的增加。
This paper studies existence of $n=4k (k>1)$ dimensional simply-connected closed almost complex manifold with Betti number $ b_i=0$ except $i=0, n/2, n$. We characterize all the rational cohomology rings of such manifolds and show they must have even Euler characteristic and even signature, which is to say the middle Betti number $b_{n/2}$ must be even. Parallel to the author's earlier work on realizing rational cohomology ring by smooth closed manifolds, we state and prove Sullivan's rational surgery realization theorem for almost complex manifold and demonstrate its application in our context. A prescribed rational cohomology ring can be realized by a simply connected almost complex manifold if and only if the ring structure supports the intersection form of a closed manifold, and it holds Chern numbers that satisfy the signature equation and the Riemann-Roch integrality relations, and the top Chern number equals the Euler characteristic. According to Stong's characterization of $U$ and $SU$ cobordism, we explicitly compute the Riemann-Roch integrality relations among Chern numbers in the case when only the middle and top Chern classes can be nonzero. The necessary and sufficient conditions for realization are expressed as a set of congruence relations among the signature and Euler characteristic, we show that the lower bounds of the 2-adic order of the signature and the Euler characteristic increase with respect to the dimension of the realizing manifold.
