Rokhlin's inequalities for real algebraic curves and an extremal property
Let \(A\) be the real projective algebraic curve associated to a real homogeneous nonsingular polynomial of even degree \(2k\) whose complexification, which is denoted \(\mathbb{C} A\), is a submanifold of the complex projective plane \(\mathbb{C} \mathbb{P} (2)\) homeomorphic to a sphere with \(g= 2k^2- 3k+1\) handles. It is assumed that \(A\) is a dividing curve, i.e. \(\mathbb{C} A- A\) is disconnected. Let \(l\) be the number of ovals of \(A\) and let \(r= g+1-l\). Denote by \(p\) the number of even ovals of \(A\), and by \(n\) the number of odd ovals. A component of \(\mathbb{R} \mathbb{P} (2)- A\) is called even if each of the ovals bounding it from inside encloses an odd number of other ovals, and is called odd otherwise. The number of nonempty even ovals bounding an even component \(\mathbb{R} \mathbb{P} (2)- A\) from outside is denoted by \(\pi\), and the number of nonempty odd ovals bounding an even component of \(\mathbb{R} \mathbb{P}(2)- A\) from outside is denoted by \(\nu\). A famous result of \textit{V. A. Rokhlin} [Funct. Anal. Appl. 14, 29-33 (1980; Zbl 0478.14010)] states that if \(k\) is even, then \(n- 2\nu\geq \frac{3k-6-r} {2},\) while if \(k\) is odd, then \(p- 2\pi\geq \frac{3k-5-r} {2}.\) The extremal property referred in the title concerns those curves achieving equality in the first of the inequalities above. Namely, suppose \(A\) is oriented by complex conjugation, and let \(L\) be an oriented one-sided simple closed curve smoothly embedded in \(\mathbb{R} \mathbb{P} (2)- A\). Let \(\lambda^+(L)\) be the number of ovals of \(A\) which are oriented in the same direction as \(L\), \(\lambda^-(L)= 1- \lambda^+(L)\), and define \[ \omega= \omega(L)= [\lambda^+(L)- \lambda^-(L)]/ 2\pmod 2. \] An island of \(A\) is an outermost oval of \(A\) together with all ovals of \(A\) contained in it, and it is said to be even if it consists of an even number of ovals. The paper under review is devoted to give a new proof of the following result, already proved by the author using Rokhlin's original argument [\textit{S. P. Paris}, Math. Ann. 304, 613-620 (1996; Zbl 0841.14026)]: If every island of \(A\) is even and \(n- 2\nu= \frac{3k- 6-r} {2}\), then \(\frac k2+ \omega\equiv 0\pmod 2\). This new proof uses link cobordism.
scientific article; zbMATH DE number 1310837