Let $M$ be a compact complex manifold of dimension three.

Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that $$ h^0(M, \mathcal O_M(nD)) > C 3^n$$ for sufficiently large $n \in \mathbb N$.

Assume that $M$ has a big divisor. My question is

Is $M$ projective?

If the answer for the above question is no, can adding some topological conditions to $M$ change the answer?

What if $M$ is simply-connected and $h^{2,0}=h^{0,2}=0$ with trivial canonical class?

Let $M$ be a compact complex manifold of dimension three.

Let us say that a divisor $D$ on $M$ is big if there is a constant $C>0$ such that $$ h^0(M, \mathcal O_M(nD)) > C n^3 $$ for sufficiently large $n \in \mathbb N$.

Assume that $M$ has a big divisor. My question is

Is $M$ projective?

If the answer for the above question is no, can adding some topological conditions to $M$ change the answer?

What if $M$ is simply-connected and $h^{2,0}=h^{0,2}=0$ with trivial canonical class?