Picard and Lefschetz numbers of real algebraic surfaces
Denote by \(H^1_{\text{alg}}(X(\mathbb{R}); \mathbb{Z}_2)\) the subgroup of \(H^1(X,\mathbb{R}), \mathbb{Z}_2)\) generated by the elements which are dual to algebraic cycles where \(X\) is a nonsingular real projective surface, and denote by \(H^2_{\text{alg}}(X(\mathbb{C}); G,\mathbb{Z})_-)\) the subgroup of the equivariant cohomology group \(H^2(X(\mathbb{C});G,\mathbb{Z}_-)\) which comes from algebraic cycles, where \(G=G(\mathbb{C}/ \mathbb{R})\) is the Galois group and \(\mathbb{Z}_-\) is the \(G\)-module \(\mathbb{Z}\) on which the generator of \(G\) acts by multiplication by \(-1\). Define the numbers \(\rho(X(\mathbb{R}))\) and \(\rho_0 (X(\mathbb{R}))\) as the dimension and codimension of \(H^1_{\text{alg}} (X(\mathbb{R});\mathbb{Z}_2)\), and the numbers \(\rho(X)\) and \(\rho_0(X)\) as the rank and corank of \(H^2_{\text{alg}} (X(\mathbb{C}); G,\mathbb{Z}_-)\). Assume that \(X(\mathbb{R})\neq \emptyset\). Then the following statements hold. (1) If \(H^*(X(\mathbb{C}), \mathbb{Z})\) is free, then \((\rho(X (\mathbb{R}))\leq \rho(X)+q(X)\), where \(q(X)\) is the irregularity; the above inequality becomes an equality for \(M\)-surfaces (i.e., in the case \(\dim H^*(X(\mathbb{R}); \mathbb{Z}_2)=\dim H^*(X(\mathbb{C}); \mathbb{Z}_2)\). (2) If \(H^1(X(\mathbb{C}), \mathbb{Z})=0\), then for an \((M-k)\)-surface (i.e., for a surface such that \(\dim H^*(X (\mathbb{C}); \mathbb{Z}_2)-\dim H^*(X (\mathbb{R}); \mathbb{Z}_2)=2k)\) one has \(\rho(X)- k\leq\rho (X(\mathbb{R}))\leq \rho(X)\) and \(\rho_0(X)-k\leq\rho_0 (X(\mathbb{R})) \leq\rho_0(X)\).
scientific article; zbMATH DE number 1262481
scientific article