Bott-Chern complexity of Kähler pairs

We introduce the Bott-Chern complexity of a compact Kähler pair $(X,B)$. This invariant compares $\dim (X)$, $\dim H_{\mathrm{B}\mathrm{C}}^{1,1}(X)$ and the sum of the coefficients of $B$. When $(X,B)$ is Calabi-Yau, we show that its Bott-Chern complexity is non-negative. We prove that the Bott-Chern complexity of a Calabi-Yau compact Kähler pair $(X,B)$ is at least three whenever $X$ is not projective. Furthermore, we show this value is optimal and is achieved by certain singular non-projective K3 surfaces.