Viro theorem and topology of real and complex combinatorial hypersurfaces

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. A part of the present work was done during the stay of the second author at the Fields Institute, Toronto, and at the NSF Science and Technology Research Center for the Computation and Visualization of Geometric Structures, funded by NSF/DMS89-20161. The work was completed during the stay of both authors at Max-Planck-Institu für Mathematik. The authors thank these funds and institutions for hospitality and financial support.