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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 \(75\).

14P05 Real algebraic sets
14P20 Nash functions and manifolds
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