Characterization of harmonic $(1,1)$-forms Let $(X,\Omega)$ be a compact Kähler manifold. Then there is the "usual" definition of the vector space $\mathcal H^{p,q}_{\bar\partial}$ which is the space of $\bar\partial$-harmonic $(p,q)$-forms (with respect to $\Omega$).  It is defined in a rather involved way by using the Hodge $\ast$ operator.
When $p=q=1$ there should be  the following easy description of the elements of $\mathcal H^{1,1}_{\bar\partial}$, but I am not even sure if it is true:

The elements of $\mathcal H^{1,1}_{\bar\partial}$ are of the type $\alpha\Omega$ where $\alpha\in\mathbb R$

Is the claim true? And if yes, why?  It seems something like the  "standard" theorem which says that harmonic functions on compact subsets of the complex plane are constant.
Edit: As pointed out in the answer, the claim is false. But I wanted to explain the origin of my question. In literature I found two different notions of admissible hermitian line bubndle $(L,h)$ on $X$:

*

*$c_1(L,h)$ is $\bar\partial$-harmonic

*$c_1(L,h)=\alpha\Omega$ for $\alpha\in\mathbb R$
I would like to understand why such definitions are the same
 A: First note that $\mathcal{H}^{1,1}_{\bar{\partial}}(X)$ is a complex vector space, so you would have to replace $\mathbb{R}$ by $\mathbb{C}$, but the claim is false regardless. The Kähler form $\Omega$ is always harmonic, so it spans a one-dimensional subspace of $\mathcal{H}^{1,1}_{\bar{\partial}}(X)$, but in general $\dim\mathcal{H}^{1,1}_{\bar{\partial}}(X) > 1$, so it won't span $\mathcal{H}^{1,1}_{\bar{\partial}}(X)$. By the Hodge theorem, we have $H^{1,1}_{\bar{\partial}}(X) \cong \mathcal{H}^{1,1}_{\bar{\partial}}(X)$ and there are many compact Kähler manifolds with $h^{1,1}(X) > 1$. For example, $X = \mathbb{CP}^1\times\mathbb{CP}^1$ has $h^{1,1}(X) = 2$.
More explicitly, if the Kähler metric on $\mathbb{CP}^1\times\mathbb{CP}^1$ is a product metric $g = \pi_1^*g_1 + \pi_2^*g_2$, then $\mathcal{H}^{1,1}_{\bar{\partial}}(\mathbb{CP}^1\times\mathbb{CP}^1) = \{a\pi_1^*\omega_1 + b\pi_2^*\omega_2 \mid a, b \in \mathbb{C}\}$ where $\omega_i$ is the Kähler form associated to $g_i$. In this case, the Kähler form on $X$ is $\Omega = \pi_1^*\omega_1 + \pi_2^*\omega_2$, so it spans the subspace of $\mathcal{H}^{1,1}_{\bar{\partial}}(X)$ with $a = b$.
