Israel Journal of Mathematics

, Volume 133, Issue 1, pp 189–238 | Cite as

Viro theorem and topology of real and complex combinatorial hypersurfaces

  • Ilia Itenberg
  • Eugenii Shustin


We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension 2 inℂP n and are topologically “glued” out of algebraic hypersurfaces in (ℂ*) n . Our construction can be viewed as a version of the Viro gluing theorem relating topology of algebraic hypersurfaces to the combinatorics of subdivisions of convex lattice polytopes. If a subdivision is convex, then according to the Viro theorem a combinatorial hypersurface is isotopic to an algebraic one. We study combinatorial hypersurfaces resulting from non-convex subdivisions of convex polytopes, show that they are almost complex varieties, and in the real case, they satisfy the same topological restrictions (congruences, inequalities etc.) as real algebraic hypersurfaces.


Euler Characteristic Toric Variety Double Covering Real Algebraic Variety Coordinate Hyperplane 
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© Hebrew University 2003

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique de RennesCNRSRennes CedexFrance
  2. 2.School of Mathematical SciencesTel Aviv University Ramat AvivTel AvivIsrael

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