Topological properties of real algebraic varieties: Du côté de chez Rokhlin.

*(English. Russian original)*Zbl 1014.14030
Russ. Math. Surv. 55, No. 4, 735-814 (2000); translation from Usp. Mat. Nauk 55, No. 4, 129-212 (2000).

From the paper: This survey gives an overview of achievements in topology of real algebraic varieties in the direction initiated in the early 1970s by V. I. Arnol’d and V. A. Rokhlin. An attempt is made to systematize the principal results in the subject. After an exposition of general tools and results, special attention is given to surfaces and curves on surfaces.

The breakthrough. To a great extent, the interest shown by V. A. Rokhlin in the topology of real algebraic varieties was motivated by results obtained in the late 1960s by Gudkov [see D. A. Gudkov and G. A. Utkin, “Topology of curves of order 6 and surfaces of order 4”, Uchen. Zap. Gor’kov Univ. 87 (1969)] and by a subsequent paper by V. T. Arnol’d [Sib. Math. J. 29, No. 5, 717-726 (1988); translation from Sib. Mat. Zh. 29, No. 5 (171), 36-47 (1988; Zbl 0721.57003)], which made a considerable contribution to the solution of Hilbert’s sixteenth problem. Gudkov disproved one of Hilbert’s conjectures on the arrangement of ovals (that is, two-sided components) of plane \(M\)-sextics (that is, sextics with the maximal number of ovals). He corrected the conjecture, proved it for degree 6, and suggested as a new conjecture an extension of his result to \(M\)-curves of any even degree. (We recall that an \(M\)-curve of genus \(g\) is a curve with the maximal number of connected components of the real part; due to Harnack’s bound the maximal number is \(M=g+1\).)

To state Gudkov’s conjecture we recall that an oval of a plane curve is called even (odd) if it lies inside an even (respectively, odd) number of other ovals. The number of even (odd) ovals is denoted by \(p\) (respectively, \(n)\). In this notation Gudkov’s conjecture claims that \(p-n=k^2 \bmod 8\) for any \(M\)-curve of degree \(2k\). – In his remarkable paper cited above, V. T. Arnol’d related the study of real plane curves to the topology of 4-manifolds and the arithmetic of integral quadratic forms. Among other results, he proved a weaker version of Gudkov’s conjecture \((p-n=k^2\bmod 4)\). After that, events began developing swiftly. First, Rokhlin proposed a proof of Gudkov’s conjecture based on Rokhlin’s formula relating the signature of a 4-manifold to the Arf-invariant of a characteristic surface (see appendix C). Then, after Kharlamov’s generalization of Arnol’d’s results to surfaces, Rokhlin found another proof, not using any specific tools of 4-dimensional topology, and extended the result to varieties of any dimension. The revolutionary breakthrough initiated by Arnol’d was over, and a period of systematic study began.

Rokhlin’s heritage. Rokhlin published six papers on topology of real algebraic varieties. This is not a very large number, but each of these papers originated a whole new direction in the subject. In this survey we discuss, more or less in detail, all the papers except one, where non-algebraic coverings are used to study algebraic curves.

Rokhlin’s formula of complex orientation and Hilbert’s sixteenth problem. A non-singular real curve \(A\) irreducible over \(\mathbb{R}\) (that is, either an irreducible non-singular complex curve with an antiholomorphic involution \(\text{conj}:A\to A\) or a non-singular complex curve consisting of two irreducible components transposed by an antiholomorphic involution) is said to be “dividing” or of “type I” if its real part \(\mathbb{R} A= \text{Fix conj}\) divides \(A\) into two halves: two connected 2-manifolds \(A_+\) and \(A_-\) having \(\mathbb{R} A\) as their common boundary. The complex conjugation \(\text{conj}: A\to A\) interchanges the two manifolds \(A_\pm\), and the complex orientations of \(A_\pm\) induce two opposite orientations on \(\mathbb{R} A\), called its complex orientations.

Let \(A\) be a non-singular dividing plane curve. Two ovals are said to form an injective pair if they bound an annulus in \(\mathbb{R}\mathbb{P}^2\). An injective pair is said to be positive if the complex orientations of the ovals are induced by an orientation of the annulus, and negative otherwise. We denote by \(\Pi^+\) and \(\Pi^-\) the numbers of positive and negative injective pairs, respectively, and let \(\Pi=\Pi^++\Pi^-\).

Rokhlin’s formula. \(2(\Pi^+ -\Pi^-)= l-k^2\) for a non-singular dividing plane curve of even degree \(2k\) with \(l\) real components.

Rokhlin’s formula has numerous applications. For example, the Arnol’d and Slepyan congruences are straightforward consequences of it. Given a dividing curve of degree \(2k\), the Arnol’d congruence states that \(p-n=k^2\bmod 4\), and the Slepyan congruence states that \(p-n=k^3 \bmod 8\) provided that each odd oval contains immediately inside itself an odd number of even ovals (so that it bounds from the outside a component of the complement of the curve with even Euler characteristic). Another consequence of Rokhlin’s formula is that \(\Pi\geq |l-k^2|\) and \(l\geq k\) for any dividing curve of degree \(2k\).

What is and what is not contained in this survey. We do not intend to give a complete overview of all known results in topology of real algebraic varieties. Instead, we have tried to select results more closely related to phenomena discovered and studied by Rokhlin and his followers and to illustrate how the approaches and the techniques developed work. Most proofs are either omitted or just sketched. Historically a great deal of attention was given to the case of plane curves; we concentrate on results in higher dimensions. However, at the end we return to surfaces and curves on surfaces and discuss a few later developments, especially those related to complex orientations. A separate topic are real 3-folds. For a long time they stayed outside the main scope of the subject, but recently the situation started changing due to J. Kollár, and now 3-folds merit a separate paper.

Topology of real algebraic varieties is developing in two directions: prohibitions and constructions. In this survey we completely ignore the latter and concentrate on prohibition-type results, that is, on the restrictions on the topology of the real point set of a real algebraic variety imposed by the topology of its complex point set. Some other important topics currently under development but ignored in this survey are: special polynomials, fewnomials, complexity, singularities and singular varieties, approximations, metric properties, the Ax principle, toric varieties, algebraic cycles, moduli spaces, minimal models, and relations to symplectic geometry. Although our main subject is real algebraic varieties, many results extend to much wider categories. In particular, instead of algebraic varieties defined over \(\mathbb{R}\) we often consider closed complex manifolds supplied with an antiholomorphic involution. In many cases the complex structure does not need to be integrable. Moreover, many prohibition results are topological in their nature and thus hold for arbitrary smooth manifolds with involution.

Contents: 1. Introduction; 2. General tools and results; 3. Surfaces; 4. Curves on surfaces; Appendix A. Topology of involutions; Appendix B. Integral lattices and quadratic forms; Appendix C. The Rokhlin-Guillou-Marin form; Bibliography.

The breakthrough. To a great extent, the interest shown by V. A. Rokhlin in the topology of real algebraic varieties was motivated by results obtained in the late 1960s by Gudkov [see D. A. Gudkov and G. A. Utkin, “Topology of curves of order 6 and surfaces of order 4”, Uchen. Zap. Gor’kov Univ. 87 (1969)] and by a subsequent paper by V. T. Arnol’d [Sib. Math. J. 29, No. 5, 717-726 (1988); translation from Sib. Mat. Zh. 29, No. 5 (171), 36-47 (1988; Zbl 0721.57003)], which made a considerable contribution to the solution of Hilbert’s sixteenth problem. Gudkov disproved one of Hilbert’s conjectures on the arrangement of ovals (that is, two-sided components) of plane \(M\)-sextics (that is, sextics with the maximal number of ovals). He corrected the conjecture, proved it for degree 6, and suggested as a new conjecture an extension of his result to \(M\)-curves of any even degree. (We recall that an \(M\)-curve of genus \(g\) is a curve with the maximal number of connected components of the real part; due to Harnack’s bound the maximal number is \(M=g+1\).)

To state Gudkov’s conjecture we recall that an oval of a plane curve is called even (odd) if it lies inside an even (respectively, odd) number of other ovals. The number of even (odd) ovals is denoted by \(p\) (respectively, \(n)\). In this notation Gudkov’s conjecture claims that \(p-n=k^2 \bmod 8\) for any \(M\)-curve of degree \(2k\). – In his remarkable paper cited above, V. T. Arnol’d related the study of real plane curves to the topology of 4-manifolds and the arithmetic of integral quadratic forms. Among other results, he proved a weaker version of Gudkov’s conjecture \((p-n=k^2\bmod 4)\). After that, events began developing swiftly. First, Rokhlin proposed a proof of Gudkov’s conjecture based on Rokhlin’s formula relating the signature of a 4-manifold to the Arf-invariant of a characteristic surface (see appendix C). Then, after Kharlamov’s generalization of Arnol’d’s results to surfaces, Rokhlin found another proof, not using any specific tools of 4-dimensional topology, and extended the result to varieties of any dimension. The revolutionary breakthrough initiated by Arnol’d was over, and a period of systematic study began.

Rokhlin’s heritage. Rokhlin published six papers on topology of real algebraic varieties. This is not a very large number, but each of these papers originated a whole new direction in the subject. In this survey we discuss, more or less in detail, all the papers except one, where non-algebraic coverings are used to study algebraic curves.

Rokhlin’s formula of complex orientation and Hilbert’s sixteenth problem. A non-singular real curve \(A\) irreducible over \(\mathbb{R}\) (that is, either an irreducible non-singular complex curve with an antiholomorphic involution \(\text{conj}:A\to A\) or a non-singular complex curve consisting of two irreducible components transposed by an antiholomorphic involution) is said to be “dividing” or of “type I” if its real part \(\mathbb{R} A= \text{Fix conj}\) divides \(A\) into two halves: two connected 2-manifolds \(A_+\) and \(A_-\) having \(\mathbb{R} A\) as their common boundary. The complex conjugation \(\text{conj}: A\to A\) interchanges the two manifolds \(A_\pm\), and the complex orientations of \(A_\pm\) induce two opposite orientations on \(\mathbb{R} A\), called its complex orientations.

Let \(A\) be a non-singular dividing plane curve. Two ovals are said to form an injective pair if they bound an annulus in \(\mathbb{R}\mathbb{P}^2\). An injective pair is said to be positive if the complex orientations of the ovals are induced by an orientation of the annulus, and negative otherwise. We denote by \(\Pi^+\) and \(\Pi^-\) the numbers of positive and negative injective pairs, respectively, and let \(\Pi=\Pi^++\Pi^-\).

Rokhlin’s formula. \(2(\Pi^+ -\Pi^-)= l-k^2\) for a non-singular dividing plane curve of even degree \(2k\) with \(l\) real components.

Rokhlin’s formula has numerous applications. For example, the Arnol’d and Slepyan congruences are straightforward consequences of it. Given a dividing curve of degree \(2k\), the Arnol’d congruence states that \(p-n=k^2\bmod 4\), and the Slepyan congruence states that \(p-n=k^3 \bmod 8\) provided that each odd oval contains immediately inside itself an odd number of even ovals (so that it bounds from the outside a component of the complement of the curve with even Euler characteristic). Another consequence of Rokhlin’s formula is that \(\Pi\geq |l-k^2|\) and \(l\geq k\) for any dividing curve of degree \(2k\).

What is and what is not contained in this survey. We do not intend to give a complete overview of all known results in topology of real algebraic varieties. Instead, we have tried to select results more closely related to phenomena discovered and studied by Rokhlin and his followers and to illustrate how the approaches and the techniques developed work. Most proofs are either omitted or just sketched. Historically a great deal of attention was given to the case of plane curves; we concentrate on results in higher dimensions. However, at the end we return to surfaces and curves on surfaces and discuss a few later developments, especially those related to complex orientations. A separate topic are real 3-folds. For a long time they stayed outside the main scope of the subject, but recently the situation started changing due to J. Kollár, and now 3-folds merit a separate paper.

Topology of real algebraic varieties is developing in two directions: prohibitions and constructions. In this survey we completely ignore the latter and concentrate on prohibition-type results, that is, on the restrictions on the topology of the real point set of a real algebraic variety imposed by the topology of its complex point set. Some other important topics currently under development but ignored in this survey are: special polynomials, fewnomials, complexity, singularities and singular varieties, approximations, metric properties, the Ax principle, toric varieties, algebraic cycles, moduli spaces, minimal models, and relations to symplectic geometry. Although our main subject is real algebraic varieties, many results extend to much wider categories. In particular, instead of algebraic varieties defined over \(\mathbb{R}\) we often consider closed complex manifolds supplied with an antiholomorphic involution. In many cases the complex structure does not need to be integrable. Moreover, many prohibition results are topological in their nature and thus hold for arbitrary smooth manifolds with involution.

Contents: 1. Introduction; 2. General tools and results; 3. Surfaces; 4. Curves on surfaces; Appendix A. Topology of involutions; Appendix B. Integral lattices and quadratic forms; Appendix C. The Rokhlin-Guillou-Marin form; Bibliography.

##### MSC:

14P25 | Topology of real algebraic varieties |

57S25 | Groups acting on specific manifolds |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14J25 | Special surfaces |