A simple question about a statement of Kähler Manifold and Moishezon Manifold

I saw a statement in a question Non-compact Kähler manifolds which admit a positive line bundle, which says that any compact complex manifold that admits a positive line bundle must be a Kähler manifold automatically. Can someone explain to me why?

The second confusion appears in J.P. Demailly's book "Analytic Methods in Algebraic Geometry", Chapter 8, Corollary 8.3 here. How can he directly prove a compact complex manifold carrying a holomorphic big line bundle is a Moishezon manifold? There's not any hints and proof below Corollary 8.3 doesn't cover this statement.

(This is the first time I ask question on MathOverflow, I hope I can express my question accurately enough, and show my great gratitude to anyone who is willing to answer my question.)

• Since this seems to be two questions, it might be better to ask them in two separate posts. (For example, you will eventually need to mark only one answer as accepted, and it is easier to do so when there is only one question to answer.) Commented Dec 10, 2023 at 19:08
• Any complex manifold who admits a positive line bundle is automatically Kahler, compactness is unnecessary. Commented Apr 12 at 2:23

The answer probably depends a lot on how exactly you define a Kähler manifold as there are multiple equivalent definitions. In any case, the definition of a positive line bundle is that it has a metric whose curvature 2-form is positive. The curvature 2-form is skew-Hermitian so this means that multiplying by $$i$$ to make it Hermitian gives a positive definite Hermitian form. This positive definite Hermitian form then gives the Kähler structure.
• @KennyS For a line bundle $L$ on $X$ a section of $H^0( X, L^n)$ divided by another section of $H^0(X,L^n)$ gives a meromorphic function. A big line bundle has a "lot" of sections and a Moishezon manifold has a "lot" of meromorphic functions, so roughly the point is to check that if you have enough sections then dividing them gives you enough meromorphic functions. But I can't think of exactly how to do this. Commented Nov 8, 2023 at 14:17