I am reading about Shimura Varieties and I am at odds with some of the claims being made (I realize this problem really has nothing to do with number theory, but perhaps some number theorists have come across this). Here are the definitions that I am working with:
Let $M$ be a smooth manifold.
A riemannian metric on $M$ is a smooth 2-tensor field $g$ such that $$g_{p} : T_{p} M \times T_{p}M \rightarrow \mathbb{R}$$ is symmetric and positive definite for all $p$.
An almost-complex structure on $M$ is a smooth tensor field $J = (J_{p})_{p \in M}$ $$ J_{p} : T_{p}M \rightarrow T_{p}M$$ such that $J^2 = -1$ for all $p \in M$. If $M$ is a complex manifold and $z^1 , \dots , z^n$ are local coordinates about $p$ and $x^1 , \dots , x^n , y^1, \dots, y^n$ are the corresponding real coordinates, then $J_{p}$ acts by $$\frac{\partial}{\partial x^j} \mapsto \frac{\partial}{\partial y^j} \quad \text{ and } \quad \frac{\partial}{\partial y^j} \mapsto -\frac{\partial}{\partial x^j}$$
A hermitian metric on a complex (or almost complex) manifold $M$ is a riemannian metric $g$ such that $$ g(JX , JY) = g(X , Y) \quad \text{for all vector fields } X , Y.$$ A hermitian manifold $(M , g)$ is a complex manifold $M$ with a hermitian metric $g$.
My problem is that the notes claim that the complex upper half plane $\mathcal{H}$ becomes a hermitian manifold (among other things) when endowed with the metric $g: = \frac{dxdy}{y^2}$.
While $g$ is certainly symmetric, my calculations show that $g$ is not positive definite; I have worked it out a few different ways, but maybe the easiest way to see this is that the matrix representation of $g$ with respect to the basis $\left\{ \frac{\partial}{\partial x} , \frac{\partial}{\partial y} \right\}$ is $$ [g] = \begin{pmatrix} 0 & \frac{1}{2y^2} \\ \frac{1}{2y^2} & 0 \end{pmatrix},$$ which is certainly not positive definite. Also I am getting that $g(JX, JY) = - g(X, Y)$ for $X,Y$ vector fields on $\mathcal{H}$.
I can outline my calculations if necessary, but is anything I have stated incorrect? I am assuming that $dxdy$ is denoting the symmetric product of $dx$ and $dy$, i.e. $dxdy = \frac{1}{2} ( dx \otimes dy + dy \otimes dy)$, perhaps the notation is indicating something else?
There is a typo in those notes. The metric should be given by $$\frac{dx\otimes dx + dy\otimes dy}{y^2}.$$ In complex coordinates this can be written as $\dfrac{|dz|^2}{y^2} = \dfrac{dz\otimes d\bar z + d\bar z\otimes dz}{2y^2}$. I suspect that they're writing down the Kähler form (which should be a $(1,1)$-form, and not a symmetric tensor) associated to the hermitian metric.