= Solution
An <almost complex structure> is a smooth endomorphism $J:TM\to TM$ with $J^2=-I$; in particular the real dimension is even. Complexifying the <tangent bundle> splits it into the $i$ and $-i$ eigensubbundles
$$
T_{\mathbb C}M=T^{1,0}M\oplus T^{0,1}M,
$$
interchanged by conjugation. On a <complex manifold>, these are locally spanned by $\partial_{z^j}$ and $\partial_{\bar z^j}$. An <integrable almost complex structure> is one arising from a holomorphic coordinate atlas. Its $T^{0,1}$ sections are closed under the <Lie bracket>. The obstruction is the <Nijenhuis tensor>
$$
N_J(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y].
$$
Expanding this expression in the two eigentypes shows that $N_J=0$ is equivalent to involutivity of $T^{0,1}$. Holomorphic coordinates make this condition necessary; the <Newlander-Nirenberg theorem> gives its sufficiency for smooth $J$. Integrability also gives $d=\partial+\bar\partial$, with $\partial^2=\bar\partial^2=0$ and $\partial\bar\partial+\bar\partial\partial=0$. The <Dolbeault operator> on $T^{1,0}M=T'_M$ defines its <holomorphic vector bundle> structure.
A real <connection on a vector bundle> $\nabla$ on $TM$ preserves $J$ when $\nabla J=0$. Its complexification then preserves both eigensubbundles. Restricting to $T'_M$ gives a complex connection $D$. Conversely any complex connection on $T'_M$ gives such a real connection by adjoining its conjugate on $T^{0,1}$ and restricting to the conjugation-fixed real tangent bundle. Equivalently, transfer $D$ through the real isomorphism $v\mapsto(v-iJv)/2$. These operations are inverse. \b[Preserving $J$ alone does not impose the holomorphic compatibility condition $D^{0,1}=\bar\partial_{T'}$.]
For integrable $J$, the latter condition has a precise real-connection interpretation: the mixed-type <torsion tensor> vanishes. In holomorphic coordinates it says $D_{\partial_{\bar z^i}}\partial_{z^j}=0$. Reality also gives $\nabla_{\partial_{z^j}}\partial_{\bar z^i}=0$. Since these coordinate fields commute, their torsion is zero. Conversely a $J$-preserving real connection with zero mixed torsion has these two derivatives equal; they belong to opposite eigentypes, so both vanish, proving $D^{0,1}=\bar\partial_{T'}$. This is the <holomorphic tangent connection and mixed torsion criterion>.
Full torsion-freeness imposes the additional symmetry $\Gamma^k_{ij}=\Gamma^k_{ji}$ in $D_{\partial_{z^i}}\partial_{z^j}=\Gamma^k_{ij}\partial_{z^k}$, and its conjugate. Such a $J$-preserving torsion-free connection exists on any <complex manifold>: local holomorphic-coordinate flat connections have these properties, and a real <partition of unity> combines them globally. Preservation of $J$ and zero torsion persist because both conditions are affine in the connection. Conversely if $J$ is merely almost complex, a torsion-free connection preserving it forces $N_J=0$: replace brackets by $\nabla_XY-\nabla_YX$ in the displayed tensor and use $\nabla J=0$; all terms cancel. Thus this connection condition detects integrability, without imposing any metric condition.
A <Hermitian metric> is a real <Riemannian metric> $g$ with $g(JX,JY)=g(X,Y)$. It induces a Hermitian metric $h$ on $T'_M$ and the fundamental two-form $\omega(X,Y)=g(JX,Y)$. The unique <Chern connection> satisfies both $D^{0,1}=\bar\partial_{T'}$ and metric compatibility. In a holomorphic frame with metric matrix $H$ and coefficient-column convention, its matrix is $H^{-1}\partial H$. It need not be torsion-free when transferred to $TM$.
In holomorphic coordinates its coefficients are $\Gamma^k_{ij}=h^{k\bar l}\partial_i h_{j\bar l}$. Their symmetry in $i,j$ is equivalent to $\partial_i h_{j\bar l}=\partial_j h_{i\bar l}$, which says $\partial\omega=0$. The conjugate gives $\bar\partial\omega=0$. The result that a <torsion-free Chern tangent connection characterizes a Kähler metric> follows: \b[the transferred Chern connection is torsion-free exactly when $\boxed{d\omega=0}$], the <Kähler metric> condition. In that case it equals the <Levi-Civita connection>, by uniqueness of the metric-compatible torsion-free connection. Equivalently a Hermitian metric is Kähler exactly when its <Levi-Civita connection> preserves $J$. In the reverse direction $\nabla J=0$ implies $\nabla\omega=0$, and zero torsion gives $d\omega=0$.
This explains the role of <Kähler metrics>: they make the real Riemannian and holomorphic connection theories coincide. A general <complex manifold> has torsion-free $J$-preserving connections and has Hermitian metrics, but need not have a connection satisfying both requirements simultaneously. On a <Kähler manifold>, the <Kähler identities> then link the <Dolbeault Laplacian> and <Hodge Laplacian>, giving $\Delta_d=2\Delta_{\bar\partial}=2\Delta_\partial$ and the resulting harmonic <Hodge decomposition theorem for compact Kähler manifolds>.
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