A smooth projective variety often has obstructed deformations.
Nevertheless, important varieties have unobstructed deformations and
it is useful in the study of their moduli spaces.
A smooth Fano manifold over a field of characteristic 0 has
unobstructed deformations thanks to the Kodaira-Akizuki-Nakano
vanishing theorem.
In this talk, I explain how far we can generalise this result to other
varieties,
such as a Q-Fano 3-fold and a weak Fano manifold.