projectivity with assumption of big and semi-amplness

Question

Let $X$ be a compact Kaehler manifold with $D$ be an effective divisor on $X$ such that $K_X+D$ is semi-ample and big then $X$ is projective?

Answer 1

Edit: I forgot about the Kaehler condition, so my answer is not relevant to the OP's question. As the comment of Henri shows, the correct answer is "yes". I will leave my original answer here (in modified form) in case anyone finds it useful.

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The comment of Henri shows the answer is yes. One could ask: is the same true without the Kaehler condition? Let me give an example showing that is not true, based on Jason Starr's idea in the comment.

In $\mathbf P^3$ take 4 points $p_1, p_2, p_3, p_4$ in general position. Let $X$ be the blowup at these 4 points.

Denote by $L_{ij}$ the proper transform on $X$ of the line through $p_i$ and $p_j$. Note that these curves are all smooth and rational, have normal bundle $O(-1) \oplus O(-1)$, and are disjoint.

Next, there is a birational contraction morphism $X \rightarrow Z$ to a projective variety $Z$ that contracts all the $L_{ij}$. This is given by the linear system of quadrics through all the points $p_i$ (or more precisely, the proper transforms of those quadrics on $X$).

Now let $X'$ be the flop of the lines $L_{12}$ and $L_{34}$. (Note that the evident rational map $X' \dashrightarrow Z$ is in fact a morphism $X' \rightarrow Z$, because the indeterminacy locus of the flop is contracted to a pair of points on $Z$.)

Let $A$ be an ample divisor on $Z$, $\Delta$ its pullback to $X'$, and $D=-K_{X'}+\Delta$. Then $D$ is effective and $K_{X'}+D=\Delta$ is semi-ample and big, as required.

I claim that $X'$ cannot be projective. The key point is that the numerical classes $[L_{ij}]$ satisfy the identities

$$[L_{ij}]+[L_{km}]=[L_{ik}]+[L_{jm}]$$ where $(i,j,k,m)$ is any permutation of $(1,2,3,4)$. This is easy to see: just write everything in the standard basis of $N_1(X)$.

Now the two "glued-in" curves on $X'$, which we denote by $\Lambda_{12}$ and $\Lambda_{34}$, have numerical classes $[\Lambda_{ij}]=-[L_{ij}]$. On the other hand the flops do not change the numerical class of curves disjoint from the flopping curves, so the proper transforms of $L_{13}$ and $L_{24}$ on $X'$ still have classes $[L_{13}]$ and $[L_{24}]$. So the union of these four curves on $X'$ is an effective curve with numerical class

$$[\Lambda_{12}]+[\Lambda_{34}]+[L_{13}]+[L_{24}] = 0.$$

So $X'$ cannot have an ample divisor.