I suspect that the question under consideration is whether or not $VP=VNP$; this is the problem directly studied by geometric complexity theorists, as I understand their work.  This project is described in some detail [here][1] (this paper is by Bürgisser, Landsberg, Manivel, and Weyman, describing work of Mulmuley and related people)--it is aimed at algebraic geometers; so you will likely be comfortable with it.  The description of the complexity problem under consideration is in section 9.

The algebraic $VP$ vs. $VNP$ conjeture is due to Valiant, in [this paper][2] and his paper "Reducibility by algebraic projections" which I can't find online at the moment, unfortunately; these are references [63] and [64] in the paper I link to above.  Valiant is, as I recall, a very clear writer, so hopefully you will find these papers readable as well.

Essentially, Valiant argues that some algebraic properties of the permanent and related varieties should have complexity-theoretic implications; a reasonable heuristic for this might be the many combinatorial interpretations of the permanent.  Unfortunately, as far as I know there are few implications between these algebraic versions of P vs. NP and the problem itself; there are some results assuming GRH.  See e.g. [this paper by Bürgisser][3].

Hopefully, this is the algebraic analogue of P vs. NP you were looking for.


  [1]: https://doi.org/10.1137/090765328 "An overview of mathematical issues arising in the geometric complexity theory approach to VP≠VNP. SIAM J. Comput. 40, No. 4, 1179–1209 (2011). zbMATH review at https://zbmath.org/1252.68134"
  [2]: https://doi.org/10.1145/800135.804419 "Completeness classes in algebra. In: Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing (STOC '79). Association for Computing Machinery, New York, NY, USA, 249–261 (1979)."
  [3]: https://doi.org/10.1016/S0304-3975(99)00183-8 "Cook's versus Valiant's hypothesis. Theor. Comput. Sci. 235, No. 1, 71–88 (2000). zbMATH review at https://zbmath.org/0943.68070"