The Definition of an Almost Complex Manifold (Nakahara)

I am having problems making sense of Michio Nakahara's definition of the Almost Complex Structure/Almost Complex Manifold, such as it appears in Geometry, Topology and Physics (2nd Edition).

On p. 318-319, he writes

The tensor field $J$ is called the almost complex structure of a complex manifold $M$. Note that any $2m$-dimensional manifold locally admits a tensor field $J$ which squares to $-I_{2m}$. However, $J$ may be patched across charts and defined globally only on a complex manifold.

What confuses me is when he later on defines the almost complex manifold on p. 342:

Let $M$ be a differentiable manifold. The pair $(M,J)$, or simply $M$, is called an almost complex manifold if there exists a tensor field $J$ of type (1,1) such that at each point $p$ of $M$, $J_p^2 = - \text{id}_{T_p M}$. The tensor field $J$ is also called the almost complex structure.

What confuses me is that this definition seems to be in contradiction to the first one. As far as I can understand it, $J$ in the second definition is defined globally in the sense that it is defined at every single point on the manifold $M$. Yet if this was the case, then by the first definition, it would not only be an almost complex manifold we were looking at, but would in fact be a full complex manifold.

My question therefore is, when Nakahara on p. 318-319 writes that $J$ can only be defined globally on a complex manifold, in what sense does he mean globally? And what does he mean by "be patched across charts"? In what sense cannot $J$ for an almost complex but still non-complex manifold not be "patched across charts and defined globally"?

Many thanks.

• I don't know what text is around those quotes. The difference is in the transition function between charts. Whether they are required to be holomorphic or just smooth. In the almost case J is just a smooth field. – OR. Jun 18 '17 at 11:16

1 Answer

$\newcommand{\Cpx}{\mathbf{C}}$tl; dr: The second extract is the standard definition. As for resolving the first extract, if Nakahara was talking about "a complex manifold $(M, J)$" just before, then the first is essentially correct. The extract does read a little strangely in isolation, and "only" is not quite what was meant, or else Nakahara meant, "...only on an almost complex manifold."

As a matter of terminology, it's more logically consistent (and, I think, suggestive) to speak of a complex structure as a tensor field $J$ satisfying $J^{2} = -I$, and to call an integrable complex structure a holomorphic structure.

The major snag is, the term "complex" has incompatible meaning depending whether it contrasts with "almost-complex" or "holomorphic". The bottom line is, when you hear someone say "almost-complex...", gently persuade them to say "complex" instead, and to say "holomorphic" if that's what they mean.

In the interest of setting a good example, here are Nakahara's "translated" extracts:

On p. 318-319:

The tensor field $J$ is called the complex structure of a holomorphic manifold $M$. Note that any $2m$-dimensional manifold locally admits a tensor field $J$ which squares to $-I_{2m}$. However, $J$ may be patched across charts and defined globally only on a holomorphic manifold.

On p. 342:

Let $M$ be a differentiable manifold. The pair $(M,J)$, or simply $M$, is called a complex manifold if there exists a tensor field $J$ of type (1,1) such that at each point $p$ of $M$, $J_p^2 = - \text{id}_{T_p M}$. The tensor field $J$ is also called the complex structure.

There are two logical issues threaded through these extracts:

1. Is the tensor field $J$ (with $J^{2} = -I$ pointwise) locally defined or globally defined?

2. Is the tensor field $J$ integrable (in the sense of having vanishing Nijenhuis tensor, i.e., being locally induced by multiplication by $i$ in local holomorphic coordinates)?

If $M$ is even-dimensional, then locally $M$ is modeled on $\Cpx^{m}$ for some positive integer $m$, so locally $M$ admits a holomorphic structure, i.e., an integrable complex structure $J$. (This fact is trivial, and uninteresting because generally one cannot choose the overlap maps to be holomorphic, or even to respect $J$.)

If $M$ is even-dimensional and oriented (i.e., equipped with a distinguished orientation), $M$ may or may not admit a complex structure compatible with the orientation. The two-sphere $S^{2}$ and six-sphere $S^{6}$ are known at admit (global) complex structures (for either orientation), and all other even-dimensional spheres are known not to admit a complex structure (for either orientation).

If $M$ is complex, $M$ may or may not admit a holomorphic structure, i.e., an integrable complex structure, i.e., a holomorphic atlas. The six-sphere is the most famous complex manifold not known to admit a holomorphic structure (recent claims notwithstanding).

• I find this answer suitable for anyone who is familiar and has experience with (almost) complex manifolds. However, the OP is clearly new to the field, and so, I would imagine this only increases their confusion. – Amitai Yuval Jun 18 '17 at 14:11
• Well, I can hardly complain, but I do need some help clearing up the last little things so that I can properly appreciate the answer given. First of all, am I right in then understanding that what you refer to as a complex manifold/structure is what Nakahara and other sources refer to as an almost complex manifold/structure, and what you refer to as a holomorphic manifold/structure is what other sources would refer to as a complex manifold. – StormyTeacup Jun 19 '17 at 9:58
• Second, am I right then in understanding it that J is in fact defined globally on an almost complex/complex manifold, but the transition functions between the various charts are then not guaranteed to be holomorphic, only to be smooth? That for that to be the case, you need the manifold to actually be complex/holomorphic? – StormyTeacup Jun 19 '17 at 10:03
• @StormyTeacup: Yes, exactly right. Note that people speak of complex vector bundles when the fibres have the structure of complex vector spaces and the transition functions are complex-linear (smooth, $GL(m, \mathbf{C})$-valued), and of holomorphic vector bundles when the transition functions are holomorphic. With "20th Century" terminology, therefore, a manifold is "almost-complex" when its tangent bundle is complex, and a manifold is "complex" when its tangent bundle is (possibly up to a complex-linear isomorphism) holomorphic. – Andrew D. Hwang Jun 19 '17 at 10:32
• You're welcome, and it's good to be a stickler for details in this type of situation. :) To clarify the tl;dr in Nakahara's terminology, I would say either, "However, $J$ may be patched across charts and defined globally only on an almost-complex manifold", or else (particularly if he was discussing complex (i.e., holomorphic) manifolds just before the first extract, "However, $J$ may be patched across charts and defined globally [delete "only] on a complex manifold". – Andrew D. Hwang Jun 20 '17 at 10:38