Semiregularity via derived algebraic geometry

Bloch defined a semiregularity map on the obstruction space of any LCI embedding in a
smooth scheme X, and showed that it annihilates curvilinear obstructions. Buchweitz and
Flenner extended this to a map on the obstruction space of any coherent sheaf on X. I will
show how this map is related to an Abel-Jacobi map from the deformation space to
Deligne cohomology, and why this means it must annihilate all obstructions.