2015/12/31 by Carmelo Di Natale, Enrico Fatighenti, Domenico Fiorenza · 1 citation
Mathematics · #Advanced Algebra and Geometry #Affine transformation #Algebra over a field #Algebraic Geometry and Number Theory #Geometry #Geometry and complex manifolds #Hodge theory #Mathematics #Projective test #Pure mathematics #math.AG
paper · pdf · doi:10.1112/jlms.12073
Final version, to appear in the Journal of the London Mathematical Society. A few minor additions have been made after the final report from the journal was received, so they will not appear in the journal version
arxiv created 2017/08/24 · openalex publication_date 2017/09/09 · arxiv updated 2017/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 prim n − 1 , 1 ( 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 examples of computation, as well as a SINGULAR code, for Fano and Calabi–Yau threefolds.