Definition: Let $k$ be a field and $\P^n$ projective $n$-space over $k$. An
algebraic variety $X\subset \P^n$ is the zero locus of a collection of homogeneous polynomials $f_i\in k[x_0,\dots,x_n]$.
Here we let $k=\C$, the complex numbers. Complex projective space $\C\P^n$ may be described as a complex manifold, with open sets $U_i = \{(x_0:\cdots:x_n)\ :\ x_i\neq 0\}$ and maps
\[
\begin{array}{r c l}
\varphi_i\ :\ U_i & \to & \C^n, \\
(x_0:\cdots:x_n) & \mapsto & \left(\frac{x_0}{x_i},\dots,\widehat{\frac{x_i}{x_i}},\dots,\frac{x_n}{x_i}\right),
\end{array}
\]
which can be quickly checked to agree on overlaps. In this context we assume all varieties are smooth, so they are submanifolds of $\C\P^n$.
Definition: An
almost complex manifold is a real manifold $M$ together with a vector bundle endomorphism $J:TM\to TM$ (called a
complex structure) with $J^2=-\id$.
Note that every complex manifold admits an almost complex structure on its underlying real manifold. Indeed, given standard coordinates $z_i=x_i+y_i$ for $i=1,\dots,n$ on $\C^n$, we get a basis $\partial/\partial x_1, \dots, \partial /\partial x_n$, $\partial/\partial y_1, \dots, \partial/\partial y_n$ on the underlying real tangent space $T_pU$, for $p\in M$ and $U\owns p$ a neighborhood. Then $J$ is defined by
\[
J\left(\frac\partial{\partial x_i}\right) = \frac\partial{\partial y_i}
\hspace{1cm},\hspace{1cm}
J\left(\frac\partial{\partial y_i}\right) = -\frac\partial{\partial x_i}.
\]
Write $T_\C M=TM\otimes_\R\C$ for the complexification of the tangent bundle, which admits a canonical decomposition $T_\C M = T^{1,0}M\oplus T^{0,1}M$, where $J|_{T^{1,0}}=i\cdot \id$ and $J|_{T^{0,1}}=(-i)\cdot \id$. We call $T^{1,0}M$ the
holomorphic tangent bundle of $M$ and $T^{0,1}M$ the
antiholomorphic tangent bundle of $M$, even though it is extraneous to consider any related map here as holomorphic. Define vector bundles (or sheaves, to consider sections on open sets)
\[
A^k_M = \textstyle \bigwedge^k(T_\C M)^*,
\hspace{1cm}
A^{p,q}_M = \textstyle \bigwedge^p(T^{1,0}M)^* \otimes_\C \bigwedge^q(T^{0,1}M)^*,
\]
where we drop the subscript $M$ when the context makes it clear. There is a canonical decomposition $A^k = \bigoplus_{p+q=k} A^{p,q}$, which yields projection maps $\pi^{p,q}:A^k \to A^{p,q}$. The exterior differential $d$ on $T^*M$ may be extended $\C$-linearly to $(T_\C M)^*$, and hence also to $A^k$. Define two new maps
\begin{align*}
\partial = \pi^{p+1,q}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p+1,q}, \\
\bar\partial = \pi^{p,q+1}\circ d|_{A^{p,q}}\ :\ &\ A^{p,q} \to A^{p,q+1}.
\end{align*}
These satisfy the Leibniz rule and (under mild assumptions) $\partial^2 = \bar\partial^2 = 0$ and $\partial \bar \partial = -\bar \partial \partial$.
From now on, the manifold $M$ will be complex with the natural complex structure described above.
Definition: A
Riemannian metric on $M$ is a function $g:TM\times TM \to C^\infty(M)$ such that for all $V,W\in TM$,
- $g(V,W)=g(W,V)$, and
- $g_p(V_p,V_p)\geqslant 0$ for all $p\in M$, with equality iff $V=0$.
A
Riemannian manifold is a pair $(M,g)$ where $g$ is Riemannian.
Locally we write $g_p:T_pM\times T_pM \to \R$, defined as $g_p(V_p,W_p)=g(V,W)(p)$. If $x_1,\dots,x_n$ are local coordinates on some open set $U\subset M$, then $g=\sum_{i,j}g_{ij}dx_i\wedge dx_j\in A^2(M)$, for $g_{ij} = g(\frac\partial{\partial x_i},\frac \partial{\partial x_j})\in C^\infty(U)$. Writing $V = \sum_if_i\frac\partial{\partial x_i}$ and $W=\sum_jg_j\frac\partial{\partial x_j}$, we get the local expression
\[
g_p(V_p,W_p) = \sum_{i,j}g_{ij}(p)f_i(p)g_j(p).
\]
Definition: A
Hermitian metric on a complex manifold $M$ is a Riemannian metric $g$ such that $g(JV,JW)=g(V,W)$ for all $V,W\in TM$. A
Hermitian manifold is a pair $(M,g)$ where $g$ is Hermitian.
There is an induced form $\omega:TM \times TM\to C^\infty(M)$ given by $\omega (V,W)=g(JV,W)$, called the
fundamental form. From $g$ being Hermitian it follows that $\omega\in A^{1,1}(M)\subset A^2(M)$. Note also that any two of the structures $J,g,\omega$ determine the remaining one.
Definition: A
Kähler metric on a complex manifold $M$ is a Hermitian metric whose fundamental form is closed (that is, $d\omega = 0$). A Kähler manifold is a pair $(M,g)$ where $g$ is Kähler.
Example: Recall the atlas given to $\C\P^n$ above. There is a metric (canonical in some sense) on each $U_j$ given by
\[
\omega_j = \frac i{2\pi} (\partial \circ \bar\partial) \left(\log\left(\sum_{\ell=0}^n \left|\frac{x_\ell}{x_j}\right|^2 \right)\right),
\]
called the
Fubini--Study metric. Each $\omega_j$ is a section of $A^{1,1}(U_j)$, and as a quick calculation shows that $\omega_j|_{U_j\cap U_k} = \omega_k|_{U_j\cap U_k}$, there is a global metric $\omega_{FS}\in A^{1,1}(\C\P^n)$ such that $\omega_{FS}|_{U_j} = \omega_j$ for all $j$.
Hence $\C\P^n$ is a Kähler manifold. If we have a smooth projective variety $X\subset \C\P^n$, then it is a submanifold of $\C\P^n$, so by restricting $\omega_{FS}$ to $X$, we get that $X$ is also a Kähler manifold. Therefore all smooth projective varieties are Kähler.
References: Huybrechts (Complex Geometry, Chapters 1.3, 2.6, 3.1), Lee (Riemannian manifolds, Chapter 3)
