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Viro theorem and topology of real and complex combinatorial hypersurfaces. (English) Zbl 1050.14056

In this quite long and very well structured article the authors introduce the notion of combinatorial hypersurfaces, which are codimension 2 submanifolds of $ℂ{ℙ}^{n}$ invariant under complex conjugation whose real parts are codimension 1 submanifolds of $ℝ{ℙ}^{n}$. This concept appears after removing the convexity condition imposed by Viro to the lattice subdivisions of the Newton polytope, to construct real algebraic varieties with prescribed topology [see e.g. O. Ya. Viro, Russ. Math. Surv. 41, No. 3, 55–82 (1986; Zbl 0619.14015)].

The authors show that combinatorial hypersurfaces obey almost all known topological restrictions satisfied by real algebraic surfaces; among them let us quote that they satisfy the generalized Harnack inequality, the Gudkov-Rokhlin and the Gudkov-Krahnov-Kharlamov congruences, some kind of Comessati inequalities for combinatorial hypersurfaces in $ℂ{ℙ}^{3}$, and that those of degree $d$ in $ℂ{ℙ}^{3}$ are homeomorphic to nonsingular algebraic surfaces in $ℂ{ℙ}^{3}$ of the same degree.

The paper can be viewed as the first step trying to answer the following questions:

(i) How far are are combinatorial hypersurfaces from the algebraic ones?

(ii) What are the main differences between the combinatorial hypersurfaces and the notion of flexible curve introduced by O. Ya. Viro [in: Topology, general and algebraic topology, and applications. Proc. Int. Conf.,Leningrad 1982, Lect. Notes Math. 1060, 187–200 (1984; Zbl 0576.14031)]?

It must be pointed out that, as the authors recognize, the notion of combinatorial hypersurface was firstly introduced, with an slightly different language, in the pioneer work of F. Santos [“Improved counterexamples to the Ragsdale conjecture”, Univ. de Cantabria, Spain, Preprint 1994].

##### MSC:
 14P25 Topology of real algebraic varieties 14J70 Algebraic hypersurfaces 32C18 Topology of analytic spaces
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