2026/08/03 by Song-Yan Xie
Mathematics · #math.AG #math.CV #msc:32E10 #msc:32Q56 #msc:14J26 #msc:14J45 #msc:14J70
20 pages
arxiv created 2026/08/03 · arxiv updated 2026/08/04
We classify complements of cubic hypersurfaces in complex projective spaces of dimension at least two, including reducible and nonreduced cubics. The complement is Oka unless the cubic is the union of three distinct hyperplanes containing a common projective subspace of codimension two; this exceptional complement is not Oka. For a smooth cubic, we pass to an unramified cyclic triple cover of the complement and construct complement-preserving sprays along residual affine conics. On cubic surfaces the twenty-seven lines give a finite dominating family. In higher dimensions we rescale the sprays so that they descend from blowups along lines, and use the tangent-transfer geometry of Kaliman--Zaidenberg to make their tangent directions span at every point. For an irreducible singular cubic, projection from a singular point, Kusakabe's localization theorem, and Hanysz's theorem on meromorphic-graph complements yield an induction on dimension. Reducible cubics reduce to complements of affine hypersurfaces defined by polynomials of degree at most two; explicit complete vector fields and elementary normal forms isolate the exceptional configuration.