Let $M$ be a compact complex manifold equipped with a line bundle $L$ which has curvature which is non-negative and strictly positive outside of a measure zero set $Z$. In his paper <A HREF="https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/morse.pdf"> "Holomorphic Morse inequalities"</a> Demailly proved that $M$ is Moishezon (this was a new proof of the Grauert-Riemenschneider conjecture, originally due to Siu).

I want to show that if $M$ is projective, $L$ is semi-ample, that is, base point free. I have a very simple argument, which I guess should be known, but I was unable to find it. 

The proof is based on Kawamata BPF theorem, and Demailly's holomorphic Morse inequalities.

Kawamata BPF:<br>
Let $L$ be a nef line bundle on a projective manifold $M$ such that $L^{\otimes a}\otimes K_M^{-1}$ is big and nef for some $a$. Then $L$ is semiample.

<i> Here I was wrong, and I need to apply a correction, thanks to 
the user YangMills. The bundle $L^{\otimes a}\otimes K_M^{-1}$ 
should be big <b>and</b> nef. Then the argument I gave becomes invalid, unless $Z$ is very special.</i>

Demailly:
$$
h^1(E^k) - h^0(E^k) \leq -\frac{k^n}{n!} \int_U
\left(\frac{\sqrt 1}{2\pi} \Theta_E\right)^n + o(k^n),
$$
where $E$ is a line bundle, $\Theta_E$ its curvature, and $U\subset M$ the set where $\Theta_E$ has at least $n-1$ positive eigenvalues.

Semipositive line bundles are clearly nef.
To apply Kawamata to the semipositive line bundle $L$, we need only to check that $L^{\otimes a}\otimes K_M^{-1}$ is big. However, for very big $a$ the curvature of $L^{\otimes a}\otimes K_M^{-1}$ is strictly positive outside of a small neighbourhood of the zero set $Z$, which contributes very little to the integral $ \int_U (\frac{\sqrt 1}{2\pi} \Theta_E)^n$, hence this integral is positive, and $E=L^{\otimes a}\otimes K_M^{-1}$ is big, because $h^0(E^k)$ grows as $k^n$.

I would be very grateful to anyone who can give me the reference to this, or point out where I did an error (or give a counterexample).

Reference:<br>
Kawamata, Y., Pluricanonical systems on minimal algebraic varieties,
Invent. Math., 79, (1985), no. 3, 567-588.

J.-P. Demailly,
Holomorphic Morse inequalities,
Several complex variables and complex geometry,
Part 2 (Santa Cruz, CA, 1989), 93--114,
Proc. Sympos. Pure Math., 52, Part 2,
Amer. Math. Soc., Providence, RI, 1991.