Is the anticanonical bundle of a compact Kähler manifold positive? I am working on the proof of the Kodaira embedding theorem and on one part of this proof you have "$−K_{X}+L^{\otimes k}$ is positive". Since it is part of the requirements, it is clear that $L$ is positive. But generally, is the following statement true?

Let $X$ be a compact Kähler manifold. Then the anticanonical bundle $−K_{X}$ (or $K_{X}^{*}$) is positive.

 A: A holomorphic line bundle $L$ on a compact complex manifold $X$ is positive if $c_1(L) \in H^2(X; \mathbb{Z})$ can be represented by a closed positive $(1, 1)$-form. It follows that $X$ admits a Kähler metric with Kähler form $\omega$ such that $c_1(L) = [\omega]$. In particular, if your claim were true, a compact complex manifold would be Kähler if and only if had positive anticanonical bundle - as we will see, this is far from true.
On a compact Riemann surface $X$, a holomorphic line bundle $L$ has a degree: if $c_1(L) = [\alpha]$, then 
$$\deg L = \langle c_1(L), [X]\rangle = \int_X\alpha.$$ 
In particular, if $L$ is positive, $c_1(L) = [\omega]$ so
$$\deg L = \int_X\omega > 0.$$
On a Riemann surface, $K_X^* = TX$ so 
$$\deg K_X^* = \deg TX = \langle c_1(TX), [X]\rangle = \langle e(TX), [X]\rangle = \chi(X)$$
where $e(TX)$ denotes the Euler class of $TX$. We see that if $X$ is a compact Riemann surface with positive anticanonical bundle,
$$\chi(X) > 0.$$
As $\chi(X) = 2 - 2g$ where $g$ is the genus of $X$, the only compact Riemann surface with positive anticanonical bundle is $\mathbb{CP}^1$. 
A compact complex manifold with positive anticanonical bundle is called Fano. By the Kodaira embedding theorem, such a manifold is projective, so any compact Kähler, non-projective manifold will also provide a counterexample to your claim. Furthermore, if $K_X^*$ is positive, $K_X$ is negative so $X$ has Kodaira dimension $-\infty$. This provides another class of counterexamples: Kähler manifolds with non-negative Kodaira dimension.
A: Just to complement Michael Albanese's nice answer:  one should think of positive anticanonical bundle as being unusual.  General type varieties have maximal (in particular, positive) Kodaira dimension, and so it is their canonical bundles that tend to be positive, rather than their anticanonical bundles.  (Literally, a variety is general type if its canonical bundle is big, which is weaker than being positive/ample, but is an assumption in a similar direction.  Thanks to Michael Albanese for correcting an earlier incorrect statement here.)
As the name suggests, you should imagine that a typical, or general, variety, is of general type.
