Confirm Mistake in Nakahara's Geometry, Topology, Physics

[Bballer152]

Can someone please confirm that there is a typo in Nakahara's Geometry, Topology, and Physics, on page 319 (the last line of the page) in the line following equation 8.27. There is a string of equalities, all of which make sense except the last one. I believe there should be no (1/2) in front of dim_C M, or the C should be replaced by R (reals). You don't necessarily need even need to follow the whole argument to see this (I think). This equality implies that the complex dimension of M is equal to the complex dimension of the complexified tangent space of a point in M (with the tangent space being viewed already as a 2m dimensional real vector space). Plus, there is no reason for the complex dimension of M to be even, so this equation really means trouble. I didn't see this in the errata online, however, so I'm still a little worried that my understanding is completely flawed (hence why I'm hoping this is indeed a typo!). Thanks in advance for any clarification!

 

[mathwonk]

it depends on the meaning of "complexified". I don';t have the book, but if you have a complex manifold, then as a real manifold it has even dimension, so its real tangent spaces are even real dimensional.

now if "complexify: means to tensor with the complex numbers, then you do get complex spaces of even complex dimension. In this approach one then decomposes that even complex dimensional space into (1,0) and (0,1) summands, i.e. holomorphic and antiholomorphic summands.

then the complex holomorphic tangent bundle is the (1,0) part, which now no longer need have even complex dimension. so read the definitions to see if this is what is going on.

 

[Bballer152]

Yes, that splitting is exactly what's going on, but that should still mean that dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=dim_ℂM as opposed to dim_ℂT_pM^+=dim_ℂT_pM^-=\frac{1}{2}dim_ℂT_pM^ℂ=\frac{1}{2}dim_ℂM, right? (The only difference is in the last equality). The +'s and -'s correspond to the (1,0) and (0,1) summands, respectively. The point is that if dim_ℂM=m, then dim_ℂT_pM^ℂ=2m, correct? We're taking the real 2m dimensional tangent space and then complexifying it, making it 4m real dimensional. Since m doesn't have to be even, I don't even see how the text's equation is meaningful, (let alone the fact that it simply doesn't equal what it says it should).

 

[mathwonk]

i think you are right and somebody put in an extra (1/2). i.e. obviously the complex dimension of the holomorphic tangent bundle should the complex dimension of the manifold. just take M = C for an example. or as you say, the number they have isn't even an integer in this case.