I start learning complex manifold by myself and hard to lift my previous intuition of differential geometry over the complex structure.
Let $M$ be a real $2m$-dimensional manifold. We define an almost complex structure $J$ on $M$ to be a smooth tensor field $J \in \Gamma(TM \otimes T^*M)$ on $M$ satisfying $J_a^bJ_b^c=-\delta_a^c$. Where given any vector bundle $E$, we will always denote its space of sections by $\Gamma(E)$.
Here I understand that $J$ is a generalization of the usual multiplication by $\pm i$ in complex analysis, But I couldn't grasp what they mean by an almost complex structure $J$ on $M$ to be a smooth tensor field $J \in \Gamma(TM \otimes T^*M)$ on $M$ satisfying $J_a^bJ_b^c=-\delta_a^c$. Like how the role of $J$ integrated in the manifold? In fact, I haven't any clear idea about what it mean a tensor field over a manifold except the bookish definition, a tensor field on a manifold is a section of a tensor bundle over the manifold where tensor bundle is a fiber bundle whose fibers are vector spaces of tensors of a fixed type and rank.
From this M.SE I got,
Notice that a complex structure determines a unique (linear) complex structure on each tangent space $T_p M$, this time by declaring $J X := i X$ (or, more precisely, $J X := T \phi_\alpha^{-1} \cdot (i T\phi_\alpha \cdot X)$). The hypothesis that the transition maps are holomorphic ensures that $J$ so defined does not depend of the choice of chart $(U_\alpha, \phi_\alpha)$ in which one works.
I understand what it mean a complex structure on a real vector space like it give a notion to multiplication of complex number $i$, $$(a+ib).v=a.v+b.J(v),\text{where }a,b\in\mathbb R,v\in V_{\mathbb R}$$But I couldn't understand what does it mean "a complex structure determines a unique (linear) complex structure on each tangent space", like what kind of structure in the tangent space? I understand the atlas definition version of complex manifold but couldn't visualize or interpret what benefit/role actually it brings here.
Let the basis for $T_pM\cong\mathbb R^{2m}$ is $\left\{\frac{d}{dx^1}|_p,\frac{d}{dx^2}|_p,\cdots,\frac{d}{dx^m}|_p,\frac{d}{dy^1}|_p\cdots,\frac{d}{dy^m}|_p\right\}$, then how $J_p:T_pM\rightarrow T_pM$ will act on this basis? In Geometry, topology and physics- M. Nakahara write it as, $$J_p\left(\frac{d}{dx^{\mu}} \right)=\frac{d}{dy^\mu},\quad J_p\left(\frac{d}{dy^{\mu}} \right)=-\frac{d}{dx^\mu}$$
Where $z^\mu=x^\mu+iy^\mu$ are the coordinates of $p$ in a chart $(U,\phi)$, but I didn't see why is should be define like this?
Best Answer
An almost complex structure on a manifold $M$ is a bundle endomorphism $J : TM \to TM$ such that $J\circ J = -\operatorname{id}_{TM}$. Note, this is just a different way of expressing the definition you gave. So for each $p \in M$, we have a linear map $J_p : T_pM \to T_pM$ which satisfies $J_p\circ J_p = -\operatorname{id}_{T_pM}$. The map $J_p$ is what is being referred to as a linear complex structure (on $T_pM$).
As the comment you quoted indicates, given a complex structure on $M$, there is a natural almost complex structure. The tangent bundle of a complex manifold has a complex vector bundle structure, so $J$ is nothing more than multiplication by $i$.