Hodge theory and deformations of affine cones of subcanonical projective varieties:

We investigate the relation between the Hodge theory of a smooth subcanonical n-dimensional projective variety X and the deformation theory of the affine cone AX over X. We start by identifying H prim n-1,1(X) as a distinguished graded component of the module of first-order deformations of AX, and later on we show how to identify the whole primitive cohomology of X as a distinguished graded component of the Hochschild cohomology module of the punctured affine cone over X. In the particular case of a projective smooth hypersurface X, we recover Griffiths' isomorphism between the primitive cohomology of X and certain distinguished graded components of the Milnor algebra of a polynomial defining X. The main result of the article can be effectively exploited to compute Hodge numbers of smooth subcanonical projective varieties. We provide a few examples of computation, as well as a SINGULAR code, for Fano and Calabi-Yau threefolds.