I am finding some examples of a Riemann surface with non-constant holomorphic sectional curvature. Since any Riemann surface is of real dimension 2, such an example is reduced to an example with non-constant Gaussian curvature.
In fact, I want to find an example in a concrete way. I already know that any orientable (real) surface is Kahler, but such a fact is just about the existence of a Kahler metric. I think it does not provide any control over the curvature. So, I want to make a Hermitian metric on some surface so that the metric is determined by an explicit formula, and the Gaussian curvature is computable from that.
Do you know any such example?
Thanks!
Addition: I found an answer after some consideration, and I misunderstood some results that I mentioned. Actually, any oriented real surface with metric $g$ is itself Kahler by the existence of isothermal coordinates. Thus, any real surface with non-constant Gaussian curvature can be an example to my question.
There is a simple answer to my question. First, note that if $(M,g)$ is a Hermitian manifold, then for any conformal change $\tilde{g}$ of $g$ also provides a Hermitian metric for $M$. Thus, for example, consider $M = \mathbb{C} = \mathbb{R}^2$ with the Euclidean metric $g$. Write $$ \tilde{g} = e^{2f} g. $$ Then, the Gaussian curvature is given by $$ K_{\tilde{g}} = e^{-2f} (K_g - \Delta_g f) = - e^{-2f} \Delta_g f. $$ Thus, by choosing some suitable function $f$, we can make $K_{\tilde{g}}$ is not constant. For example, put $f (z) = \vert z\vert^2$. Note that $(M, \tilde{g})$ is a Hermitian manifold with complex dimension $1$, so this is Kahler, and the Gaussian curvature at $p \in M$ coincides with the holomorphic sectional curvature at $p$.