Conformal dimensions of the energy-momentum tensor

Currently I am reading the Di Francesco-Mathieu-Sénéchal textbook on conformal field theory. Above the equation 5.52, the author argues that the EM tensor should have scaling dimension 2 and spin 2. But why did the author then conclude that both the holomorphic and anti-holomorphic dimension of the tensor are 2? Isn't it that the holomorphic dimension is 2 while the anti-holomorphic one is 0?

Another question: why does the EM tensor transform like a primary field?

For the holomorphic part $T(z)$, we should have weights $(h,\bar h) =(2,0)$, and for the anti-holomorphic part $\tilde T(\bar z)$, we should have weights $(h,\bar h) =(0,2)$, so for each part, we should have a scaling dimension $2$, and a spin $\pm 2$
However, the holomorphic and anti -holomorphic parts of the Energy-momentum tensor are not generally primary fields (see $5.124$ p $136$, $5.121$ p $135$) because of a possible central charge.
• @LiXinghe : In fact, the discussion in 5.52 implies the transformation $z \to w=\frac{1}{z}$. Now, looking at $5.125, 5.124$, you can calculate that the Schwartzian derivative {$w;z$} is zero, so, in fact, for this particular transformation, the energy-momentum behaves as a primary field, but this is an exception, it does not work with general transformations $w(z)$. – Trimok Sep 14 '13 at 12:03