When a result is stated for the field of complex numbers it can usually be extended to a result for an algebraically closed field of characteristic zero. I would like to see a list of results that hold for $\mathbb{C}$, but whose extension to the setting algebraically closed and characteristic zero fields is either (i) untrue or (ii) not known. In each case can we say what additional properties of the complex numbers allow for a proof of the result?
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1Relevant: https://mathoverflow.net/questions/90551/what-does-the-lefschetz-principle-in-algebraic-geometry-mean-exactly – Sergey Guminov Jul 07 '23 at 14:48
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2Welcome new contributor. There are quite a number of results that hold for algebraically closed fields of sufficiently large transcendence degree, but do not hold over the algebraic closure of $\mathbb{Q}$ (or, rather, are not known to hold over that field). – Jason Starr Jul 07 '23 at 15:14
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2An example of such a "result" alluded to by Jason is the following: Let $B$ be a smooth curve over an algebraically closed field $k$ of characteristic zero, and let $X\to B$ be a projective morphism with geometrically connected fibres, where $X$ is a smooth variety over $k$. Assume that, for all $b$ in $B(k)$, the projective variety $X_b$ contains a rational curve. Then, the variety $X_{K(B)}$ over the function field of $B$ contains a rational curve. This is not difficult to prove when $k$ is uncountable, but it is not known (although probably true) when $k=\overline{\mathbb{Q}}$. – Ariyan Javanpeykar Jul 07 '23 at 17:37
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5Any two algebraically closed fields have the same first order theory in the language of rings so you need to look at things not expressible in first order logic. – Benjamin Steinberg Jul 07 '23 at 18:37