Hodge Theory and Deformations of Affine Cones of Subcanonical Projective Varieties

TL;DR

This paper explores the connection between Hodge theory and deformation theory of affine cones over subcanonical projective varieties, providing methods to compute Hodge numbers and illustrating with examples including Fano and Calabi-Yau threefolds.

Contribution

It establishes a link between primitive cohomology and deformation modules of affine cones, extending Griffiths' isomorphism to a broader class of varieties.

Findings

Identifies primitive cohomology as a graded component of deformation modules.

Provides a method to compute Hodge numbers of subcanonical varieties.

Includes example computations and SINGULAR code for Fano and Calabi-Yau threefolds.

Abstract

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 $A_X$ over $X$. We start by identifying $H^{n-1,1}_{\mathrm{prim}}(X)$ as a distinguished graded component of the module of first order deformations of $A_X$, 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 example computation, as well a SINGULAR code, for Fano and Calabi-Yau threefolds.