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Deformation classes of real four-dimensional cubic hypersurfaces. (English) Zbl 1225.14047
Hypersurfaces of degree $$d$$ in the projective space $$\mathbb{P}^n(k)$$ constitute a projective space $$\mathbb{P}^N(k)$$ of dimension $$N=\binom{n+d}{d}-1$$. The discriminant hypersurface $$\Delta(k)$$ of $$\mathbb{P}^N(k)$$ consists of those polynomials $$f\in\mathbb{P}^N(k)$$ such that the hypersurface $$\{f=0\}\subset\mathbb{P}^n(k)$$ is singular. In fact $$\Delta(k)$$ is the locus of critical values of the map $\pi:\Gamma=\{(f,x)\in\mathbb{P}^N(k)\times\mathbb{P}^n(k):\, f(x)=0\}\mapsto\mathbb{P}^N(k),\, (f,x)\mapsto f.$ The universal hypersurface $$\Gamma$$ is smooth and the restriction $\pi|_{\Gamma_0}:\Gamma_0=\pi^{-1}(\mathbb{P}^N(k)\setminus\Delta(k))\to\mathbb{P}^N(k)\setminus\Delta(k)$ is called, since Kodaira and Spencer, a deformation family of nonsingular varieties. This works better if $$k=\mathbb C$$ that if $$k=\mathbb R$$ because, as usually, the real case presents “pathologies” concerning connectedness, unless $$d=1$$. This is why a natural and classical problem is to understand the nature of the connected components of $$\mathbb{P}^N(\mathbb R)\setminus\Delta(\mathbb R)$$. Two real nonsingular hypersurfaces $$\{f=0\}$$ and $$\{g=0\}$$ in $$\mathbb{P}^n(\mathbb R)$$ are said to be deformation equivalent if they represent points in the same connected component of $$\mathbb{P}^N(\mathbb R)\setminus\Delta(\mathbb R)$$, while $$\{f=0\}$$ and $$\{g=0\}$$ are said to be coarse deformation equivalent if $$\{f=0\}$$ is deformation equivalent to a projective transformation of $$\{g=0\}$$. Indeed, since the group $$\text{PGL}(n+1,\mathbb R)$$ of real projective transformations of $$\mathbb{P}^n(\mathbb R)$$ is connected if $$n$$ is even and has two connected components if $$n$$ is odd, the coarse deformation equivalence coincides with the deformation equivalence if $$n$$ is even, while each coarse deformation class contains at most two deformation classes if $$n$$ is odd.
In the paper under review, the authors treat the case of cubic hypersurfaces in $$\mathbb{P}^5(\mathbb R)$$, and they answer the following questions: How many coarse deformation classes exist? How can one distinguish hypersurfaces of distinct coarse deformation classes? Newton answered both questions for cubic plane curves, and Schläfli and Klein solved this problem for cubic surfaces in $$\mathbb{P}^3(\mathbb R)$$. Much more recently, Krasnov treated the case of cubic threefolds in [V. A. Krasnov, Izv. Math. 70, No. 4, 731–768 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 4, 91–134 (2006; Zbl 1222.14125)].
The main result of this excellent article states that if two real nonsingular cubic hypersurfaces in $$\mathbb{P}^5(\mathbb R)$$ have the same homological type, then they are coarse deformation equivalent. In fact, the authors show that in the statement above the homological type can be replaced by the isomorphism class of the kernel of $$\text{id}+c$$, where $$c:H_4(X(\mathbb C))\to H_4(X(\mathbb C))$$ is the involution induced by complex conjugation for a real cubic fourfold $$X$$. In particular this allows them to count the number of coarse deformation classes of real nonsingular cubic hypersurfaces in $$\mathbb{P}^5(\mathbb R)$$, which is $$75$$.

##### MSC:
 14P05 Real algebraic sets 14P20 Nash functions and manifolds
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##### References:
  Alfred Aeppli, Modifikation von reellen und komplexen Mannigfaltigkeiten, Comment. Math. Helv. 31 (1957), 219 – 301 (German). · Zbl 0098.36403  A. Degtyarev, I. Itenberg, and V. Kharlamov, Real Enriques surfaces, Lecture Notes in Mathematics, vol. 1746, Springer-Verlag, Berlin, 2000. · Zbl 0963.14033  Alex Degtyarev, Ilia Itenberg, and Viatcheslav Kharlamov, Finiteness and quasi-simplicity for symmetric \?3-surfaces, Duke Math. J. 122 (2004), no. 1, 1 – 49. · Zbl 1073.14053  T. Fujita, Classification Theorems of Polarized Varieties, Lecture Notes Series, London Math. Society, vol. 155, 1990. · Zbl 0743.14004  Lucien Guillou and Alexis Marin, Une extension d’un théorème de Rohlin sur la signature, À la recherche de la topologie perdue, Progr. Math., vol. 62, Birkhäuser Boston, Boston, MA, 1986, pp. 97 – 118 (French). · Zbl 0562.57010  I. Itenberg, Plane projective real curves of degree $$6$$ with one non-degenerate double point, PhD Thesis (Leningrad, 1991). · Zbl 0762.14031  Felix Klein, Ueber Flächen dritter Ordnung, Math. Ann. 6 (1873), no. 4, 551 – 581 (German). · JFM 06.0386.02  V. A. Krasnov, Rigid isotopy classification of real three-dimensional cubics, Izv. Ross. Akad. Nauk Ser. Mat. 70 (2006), no. 4, 91 – 134 (Russian, with Russian summary); English transl., Izv. Math. 70 (2006), no. 4, 731 – 768. · Zbl 1222.14125  V.A. Krasnov, Topological classification of Fano surfaces of real three-dimensional cubics, Izvestiya: Mathematics 71:5 (2007), 863-894. · Zbl 1135.14046  I. Newton, Enumeratio linearum tertii ordinis, (1704). · Zbl 0079.24203  V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111 – 177, 238 (Russian). · Zbl 0408.10011  V. V. Nikulin, Remarks on connected components of moduli of real polarized K3 surfaces, ArXiV:Math. AG/0507197 (2006).  B. Saint-Donat, Projective models of \?-3 surfaces, Amer. J. Math. 96 (1974), 602 – 639. · Zbl 0301.14011  Claire Voisin, Théorème de Torelli pour les cubiques de \?$$^{5}$$, Invent. Math. 86 (1986), no. 3, 577 – 601 (French). · Zbl 0622.14009  C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281 – 298. · Zbl 0215.39903
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