= Solution
For the <Calabi-Yau threefold> in the stated convention, the <Hodge decomposition theorem for compact Kähler manifolds> gives
$$
0=H^1(M,\mathbb C)=H^{1,0}_{\bar\partial}(M)\oplus H^{0,1}_{\bar\partial}(M).
$$
In particular, the <Dolbeault theorem> yields $H^1(M,\mathcal O_M)=0$. By <Serre duality for compact complex manifolds> in dimension three,
$$
H^2(M,\mathcal O_M)^*\cong H^1(M,K_M).
$$
The <canonical bundle> is trivial, so the group on the right equals $H^1(M,\mathcal O_M)=0$. Thus $H^2(M,\mathcal O_M)=0$, and part (c) gives
$$
\boxed{\text{Every Calabi-Yau threefold satisfying the given definition is projective}.}
$$
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