# Terminology confusion in complex geometry (hermitian geometry)

I was reading the book by Huybrechts and he writes the definition of an Hermitian structure as follows:

Let $$X$$ be a complex manifold and $$I$$ be the induced almost complex structure. A Riemannian metric $$g$$ on $$X$$ is said to hermitian structure on $$X$$ if for any point $$x \in X$$, the scalar product $$g_x$$ on $$T_x X$$ is compatible with the almost complex structure $$I$$ (i.e $$g_x(v,w) = g_x(I(v),I(w)$$ for all $$v,w \in T_xM$$).

And I found the definition of Hermitian metric in Griffiths and Harris's Principles of Algebraic geometry as follows:

Let $$X$$ be a complex manifold of dimension $$n$$. A hermitian metric on $$X$$ is given by a positive definite hermitian inner product $$(\cdot, \cdot)_z : T^{'}_{z}(X) \otimes \overline{T^{'}_{z}(X)} \longrightarrow \mathbb{C}$$ on the holomorphic tangent space at $$z$$ for each $$z \in X$$, depending smoothly on $$z$$.

My question: I am facing difficulty in understanding the following terminologies:

1. Hermitian metric
2. Hermitian structure
3. Hermitian form
4. Hermitian inner product

Giving really stupid input, I think 1) and 2) must be the same things (if it is, still I don't get how).

Can anyone please clarify these terminologies? I am really confused looking at different other sources (in fact, I am getting more confused after referring to other references).

Thanks!

So let's start with a real vector space $$V$$, and suppose we have a complex structure $$I \in \operatorname{End}(V)$$, i.e. $$I^2 = - \operatorname{id}_V$$. In that way we may also consider $$V$$ as a complex vector space. Let $$g: V \times V \to \mathbb R$$ be a real inner product, i.e. a positive definite, symmetric real-bilinear form, and suppose that $$g$$ is hermitian in the sense of Huybrechts, i.e. $$g(v,w) = g(I(v), I(w)) \tag{I}$$ for all $$v,w \in V$$. Then we may define $$h(v,w) = g(v,w) + i g(v, I(w)).$$ Claim. The form $$h$$ is a Hermitian inner product on $$V$$, i.e. it is a positive definite, sesquilinear form $$V \times V \to \mathbb C$$.
Proof. First note that by (I), we have $$g(v, I(v)) = g(I(v), I^2(v)) = - g(I(v), v) = - g(v, I(v))$$, so that $$g(v, I(v)) = 0$$. In particular $$h(v,v) = g(v,v) \geq 0$$ with equality if and only if $$v = 0$$. Hence $$h$$ is positive definite.
By a similar application of (I) and the symmetry of $$g$$ we have $$h(v, w) = g(v, w) + i g(v, I(w)) = g(w, v) - ig(w, I(v)) = \overline{h(w, v)}.$$ So it remains to show that $$h$$ is $$\mathbb C$$-linear in the first component. It is clearly additive, since $$g$$ is additive in the first component. Now if $$\lambda = \mu + i \nu \in \mathbb C$$ is a scalar, we get \begin{align*} h(\lambda v, w) & = g(\lambda v, w) + i g(\lambda v, I(w)) \\ & = \mu g(v, w) + \nu g(I(v), w) + i \mu g(v, I(w)) + i \nu g(I(v), I(w)) \\ & \stackrel{\text{(I)}}{=} \left(\mu g(v, w) + i \nu g(v, w) \right) + i\left( \mu g(v, I(w)) + i \nu g(v, I(w)) \right)\\ & = \lambda g(v, w) + i \lambda g(v, I(w)) \\ & = \lambda h(v, w). \end{align*}
Now the same applies to complex manifolds, where you replace $$V$$ with the real tangent space $$T_x X$$, and the real inner product with the Riemannian metric. So we get that every Riemannian metric, that is hermitian in the sense of (I), defines a Hermitian inner product on all tangent spaces. This is called a hermitian form, I think. Conversely, given a hermitian inner product $$h$$, you can always obtain $$g$$ as the real part of $$h$$, and by sesquilinearity, (I) will hold for such $$g$$.