| Description |
Lefschetz initially proved the Lefschetz (1; 1)-theorem in 1924 by taking a Lefschetz pencil of hyperplane sections for a complex projective surface. Lefschetz's method would provide a sort of inductive proof of the Hodge conjecture, but the Abel-Jacobi map is rarely surjective in higher dimensions, which is the main technical obstruction to generalize Lefschetz's method. This dissertation investigates the surjectivity of the topological Abel-Jacobi map for a complex projective surface S. Instead of only taking the hyperplane sections for S and then using the Abel-Jacobi map directly in Lefschetz's method, we continue to take a Lefschetz pencil of hyperplane sections for each hyperplane section of S, then consider the topological Abel-Jacobi map for each hyperplane section of S. We conclude that the induced homomorphism on homology of the topological Abel-Jacobi map is the same as the tube map defined by C. Schnell [14] up to a sign. By restricting this induced homomorphism to the stabilizer of an elementary vanishing cycle, we prove that it is surjective. Our proof is based on the irreducibility of the vanishing homology group under the monodromy action. In fact, we show that the image of the induced homomorphism is nonzero and independent of the choice of the Lefschetz pencil and the elementary vanishing cycle, i.e., the image is a nonzero stable Z-submodule under the monodromy action. In the end, we give a geometric description of our construction by taking a general net of hyperplane sections for each hyperplane section of S, which we call a Lefschetz net in this dissertation. |
