Viro theorem and topology of real and complex combinatorial hypersurfaces

Details Viro theorem and topology of real and complex combinatorial hypersurfaces Viro theorem and topology of real and complex combinatorial hypersurfaces

2001-05-24

arxiv.org/abs/math/0105198v2

Mathematics

Algebraic Geometry

45 pages, Latex, 2 figures

Scientific paper

We introduce a class of combinatorial hypersurfaces in the complex projective space. They are submanifolds of codimension~2 in $\C P^n$ 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.

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