I encountered the problem of trying to find the examples of complex varieties which are not the complexification of a real variety, when I considered the Exercise II.4.7.(a) in Hartshorne's book. I wonder whether there are indeed some interesting complex varieties which don't have the so called semi-linear involution in Hartshorne's book.
For rational affine curve, I have found the following example $$A=\mathbb{C}[x,\frac{1}{x-1},\frac{1}{x-i},\frac{1}{x-i-1}]$$ by a little hard calculation, considering the possible semi-linear involutions of the field $\mathbb{C}\left(x\right)$ induced by the complex conjugation of $\mathbb{C}$ .
But I really wonder
whether there are some interesting examples of complex projective varieties without real descent.
I have poor knowledge on the classical topics in algebraic geometry, therefore I find it difficult to work out the problem by myself. Could anyone shed some light on my confusion? Thanks a lot!