Mathematics

In this talk we'll give some necessary condition for a 1-cocycle in $H^1(M_0,\Theta_0)$ (cohomology of the sheaf of holomorphic vector fields on $M_0$) to be the infinitesimal deformation of a complex analytic family. We'll give a notion of completeness for an analytic family and define what the number of moduli (i.e. the dimension of the space of deformation) should be. After this we'll construct explicitly a family of deformations for complex hypersurfaces of degree d \ge 3 in the n-dimensional projective space and see how the theory applies to this cases. The talk will be approximately 1h 15 long.