Rigidity of superminimal surfaces in complex projective spaces
The rigidity phenomenon of superminimal surfaces in \(\mathbb{C} P^ n\) was studied by the first author in [(*) ibid. 42, No. 3, 439-455 (1990; Zbl 0716.53055)]. One sees that certain classes of nonsingular plane algebraic curves as well as their quadratic transformations to generate rigid superminimal immersions in \(\mathbb{C} P^ 2\). It was also proved that a superminimal immersion generated by a generic rational curve in \(\mathbb{C} P^ 2\) of any degree is rigid. For superminimal immersions generated by plane cubics, rigidity up to finiteness was shown. These supporting evidences lead naturally to the question: Is an arbitrary superminimal surfaces in \(\mathbb{C} P^ n\) rigid? Following the line of thought in (*) together with further use of algebraic geometry, we are able to solve positively the above rigidity problem under general conditions. More precisely, we have the following: Theorem 1. Any compact branched superminimal immersion \(M\to \mathbb{C} P^ n\) is rigid up to finiteness. In other words, there are at most finitely many such immersions in \(\mathbb{C} P^ n\) which are isometric to the first one but mutually inequivalent under the isometries of \(\mathbb{C} P^ n\). Furthermore, the number of possible inequivalent immersions is bounded by a constant depending only on the area of \(M\). Theorem 2. Let \(f: M\to \mathbb{C} P^ n\) be any holomorphic map, where \(M\) is a compact Riemann surface. To a generic \(A\) in \(\text{PGL}(n+1, \mathbb{C})\) regarded as a real algebraic variety, all the branched superminimal immersions generated by \(Af\) are rigid. Slightly more general versions of the above theorems can be found at the end of Sections 4 and 5. When \(n=2\), letting the Riemann surface to vary in the moduli space, we obtain: Theorem 3. Branched superminimal immersions of genus \(g\), degree \(d\) and area \(m\pi\) in \(\mathbb{C} P^ 2\) form an analytic space with a natural action of \(\text{PGL} (3,\mathbb{C})\), which contains a real Zariski open dense set consisting of rigid superminimal surfaces. Moreover, this Zariski open dense set intersects each orbit of the \(\text{PGL} (3,\mathbb{C})\) action in a real Zariski open dense set of the orbit.
scientific article; zbMATH DE number 570305