GC4M · Generalized complex 4-manifolds

We will study a notion of Lefschetz pencils for generalized complex 4-manifolds, aiming at results analogue to those of Donalson and Gompf in symplectic geometry stating that a 4-manifold has a symplectic structure if and only if it admits a Lefschetz pencil. To achieve the desired results we have to find the right definition of a Lefschetz pencil in the generalized complex setting. Earlier work of the candidate on this area has already shed some light into this question. We will study the question of how to adapt Seiberg—Witten theory in order to produce differential—topological obstructions for a 4-manifold to admit a generalized complex structure. This should mirror Taubes' result relating SW-invariants and symplectic structures. Work of Bauer and Furuta should provide the springboard for this part of the project. On the physical side, we will study mirror symmetry on generalized complex manifolds. The work on Lefschetz pencils will provide us with a number of generalized complex manifolds described in terms of singular torus fibrations, therefore an ideal place where to study mirror symmetry in the setting of SYZ. For these torus fibrations, there is a locus on the base where one of the circles forming the torus fiber collapses. In other models where this occurs, one describes the mirror as a Landau—Ginzburg model. Therefore this raises the questions about what Landau—Ginzburg models are for generalized structures.