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!