Mathematics

Let X be a Kahler complex manifold and let Z be a complex submanifold. Within the framework of derived deformation theory, the Abel-Jacobi map for the pair (X,Z) has a natural interpretation as a morphism form a homotopy fiber to a double homotopy fiber. Using this fact and the dictionary between formal moduli problems and differential graded Lie algebras up to homotopy, it is easy to describe a linear L-infinity morphism encoding the Abel-Jacobi map, and in terms of this L-infinity morphism it is immediate to reobtain the classical description of the differential of the Abel-Jacobi map as well as the result that Bloch's semiregularity map annihilates the obstructions to deforming Z inside X. This provides a geometrical interpretation of recent results by Iacono-Manetti and Pridham.