On the definition/notation for pseudoholomorphic curves

Question

A pseudoholomorphic curve is a map $u:(\Sigma,j) \to (M,J)$ from a Riemann surface $\Sigma$ with an almost complex structure $j$ to a manifold $M$ with an almost complex structure $J.$ We require moreover that $u$ satisfies the "Cauchy-Riemann equations"

$$ J \circ du = du \circ j. \qquad (*)$$

I assume that this is to be understood as an equality of maps $T_p\Sigma \to T_{u(p)}M$ for every $p \in \Sigma$ (or even as a bundle homomorphism). However, choosing local $j$-holomorphic coordinates $(s,t)$ on $\Sigma$, so that $j(\partial_s) = \partial_t,$ many authors state that the equation above is equivalent to

$$ \partial_s(u) + J(u)\partial_t(u) = 0.$$

I don't understand this statement. In fact, I don't even understand what the symbols above are supposed to represent. What is $J(u)?$ I understand that, taking (for simplicity) $M = \mathbb{C}^n$ with its standard complex structure, the equation $(*)$ above reduces to a system of $2n$ equations, corresponding to the usual Cauchy-Riemann equations. But then $J$ simply corresponds to a matrix, and doesn't depend in any way on $u.$ Why would one write $J(u),$ then?

Any clarification would be greatly appreciated.

Answer 1

$J$ is regarded as a section of the bundle of endomorphisms of $TM$, i.e. as a map $M\to\text{End}(TM$).

Rewrite the equation as $du+J\circ du\circ j=0$, and now write $du$ locally in terms of $ds$ and $dt$.

Answer 2

What Chris said is correct and completely to the point, but would have been too abstract for me when I was first learning. Apologies if his answer sufficed.

Consider the case of $M = \mathbb{C}^n$. For each point $z \in \mathbb{C}^n$, $T_zM$ can be identified with $\mathbb{C}^n$. An almost complex structure gives you a complex structure on each tangent space. This means that we should see it (in this case) as a $\mathbb{C}^n$ parametric family of matrices $J(z)$, where each $J(z)$ should be thought of as an endomorphism of $T_zM$. (In this case, the global trivialization of $T_zM$ is what allows us to identify this with a matrix.)

Now, consider a map $u \colon D \to M$. $J \circ du$ is then, very explicitly, given by $J(u(z)) du(z)$, where we now think of $du(z)$ as the Jacobian matrix of partial derivatives of $u$.

Everything now follows as Chris and you observe.