  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.
