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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!

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1 Answer 1

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This stuff is really confusing, so don't feel bad about it. I find it always helpful to think of real and complex vector spaces first, before passing to complex manifolds.

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 $g$.

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