Hodge Theory, deformations of affine cones and beyond

Hodge Theory and Deformation Theory are known to be closely
related. Amongst the many avatars of this friendship we have
Griffiths's Residues calculus for hypersurfaces or the Calabi-Yau
case, were the first order deformations of a smooth algebraic variety
are identified with a special piece of its Hodge structure. In this
talk we show how in the more general case of a smooth projective
subcanonical variety X we can reconstruct part of its Hodge Theory by
looking at a distinguished graded component of the first order
deformations module of its affine cone A. In order to get a global
reconstruction theorem we then move to the study of the Derived
deformations of A (à la Kontsevich), showing how to find amongst them
the missing Hodge spaces. We will show then some applications of this
new technique, for example to complete intersection in homogeneous
varieties.