Algebraic vs. analytic: a survey

2026/07/28 by Aravind Asok, Tom Bachmann, Michael J. Hopkins
Mathematics · #math.AG #math.AT #math.CV #math.HO #math.KT #msc:01A60 #msc:01A85 #msc:14C30 #msc:14F42 #msc:32L10 #msc:55R25

paper · pdf

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

arxiv created 2026/07/28 · arxiv updated 2026/07/30

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?

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Related

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• Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces
• On algebraic representatives of homeomorphism types of analytic hypersurface germs
• Covers of Abelian varieties as analytic Zariski structures
• Moishezon morphisms