Algebraic vs. analytic: a survey

Aravind Asok; Tom Bachmann; Michael J. Hopkins

Algebraic vs. analytic: a survey

Algebraic Geometry 2026-07-28 v1 Algebraic Topology Complex Variables History and Overview K-Theory and Homology

Authors: Aravind Asok , Tom Bachmann , Michael J. Hopkins

Abstract

Given a pair of complex algebraic varieties, we can ask: does every analytic map admit an algebraic representative? Such questions are most interesting when the source and target are non-compact varieties, in which case the problems are closely linked with Hodge-type conjectures. Concrete avatars of this kind of problem, which we discuss from several points of view, include: which holomorphic vector bundles on an affine variety admit an algebraic structure, and which homotopy classes of maps admit algebraic representatives?

Cite

@article{arxiv.2607.26190,
  title  = {Algebraic vs. analytic: a survey},
  author = {Aravind Asok and Tom Bachmann and Michael J. Hopkins},
  journal= {arXiv preprint arXiv:2607.26190},
  year   = {2026}
}

Comments

42 pages; to appear Proceedings of the Summer Algebraic Geometry Research Institute