Given a smooth, compact complex surface with ample canonical bundle satisfying $c\_1^2=3c\_2$, is it true that every Kahler class is a multiple of $c\_1$? This seems to be the case for fake projective planes, because $b\_2=1$.



Kähler classes for surfaces of general type with $c_1^2=3c_2$
The answer is **no**.


A counterexample is provided by the so-called *[Cartwright-Steger surface](https://arxiv.org/pdf/1412.4137v2.pdf)* $X$, namely a surface of general type with $p\_g(X)=q(X)=1$ and $K\_X^2=9$.


Such a surface has Picard rank $3$, and moreover $\mathrm{NS}(X)=H^{1, 1}(X)$, in other words all Hodge classes in $X$ are algebraic. In particular, the vector space $$H^{1, 1}\_{\mathbb{R}}(X):=H^{1,1}(X) \cap H^1(X, \, \mathbb{R})$$ has real dimension $3$, because its complexification is $H^{1, 1}(X).$


On the other hand, by general facts the Kähler cone $\mathcal{K}\_X$ of $X$ is an open, convex cone in $H^{1,1}\_{\mathbb{R}}(X)$, hence it cannot be of dimension $1$. 


