I hope this is not too trivial, but I think this may be well known to someone (not me).

1$\begingroup$ Just the classical form? $\endgroup$ – François G. Dorais♦ Oct 11 '13 at 15:53

4$\begingroup$ What do you mean by "reverse math"? $\endgroup$ – Liviu Nicolaescu Oct 11 '13 at 16:34

3$\begingroup$ en.wikipedia.org/wiki/Reverse_mathematics $\endgroup$ – Benjamin Steinberg Oct 11 '13 at 17:09

1$\begingroup$ At least over an algebraically closed field, any given instance will be provable in that complete theory. On the other hand, the usual assertion which quantifies over all line bundles is indirectly making some statement in the language of arithmetic, much as here: mathoverflow.net/questions/105478/… A priori, the same questions apply. $\endgroup$ – Adam Epstein Oct 11 '13 at 18:34