Line bundles, regular mappings and the underlying real algebraic structure of complex algebraic varieties (Q1568704)

The main theme of the paper is the comparison of the topological structure of a real algebraic variety, say \(X\), with its algebraic structure. One task in this direction is to study the factor group \(G(x)=VB^1_\mathbb{C} (X)/VB^1_{{\mathbb{C}} \text{-alg}} (X)\), where \(VG^1_\mathbb{C} (X)\) is the group of \(\mathbb{C}\)-line bundles on \(X\) and \(VB^1_{\mathbb{C}} \text{-alg}(X)\) is the subgroup of those line bundles that admit an algebraic structure. The exponent \(b(X)\) of the group \(G(X)\) serves as a coarser invariant. This is similar to the study of subgroups of algebraic (co)-homology classes in topological (co-)homology groups. Another related question is how the set of regular maps (i.e., maps of an algebraic nature) from \(X\) into the 2-sphere lies in the space of \(C^\infty\)-maps from \(X\) into the 2-sphere. Any complex algebraic variety \(V\) can be viewed as a real algebraic variety (using the identification \(\mathbb{C}= \mathbb{R}^2)\) which is denoted by \(V_\mathbb{R}\). The main results of the paper are that in the cases 1. \(X=A_\mathbb{R}\), \(A\) a complex Abelian variety of dimension at least 3, or 2. \(X=V_{1\mathbb{R}}\times\cdots\times V_{n \mathbb{R}}\), \(V_1,\dots,V_n\) nonsingular complex projective curves, the group \(G(X)\) and the number \(b(X)\) can be computed quite explicitly whenever the group is finite, which is always true in case 2.