Hodge theory and deformations of affine cones of subcanonical projective varieties
| Autor: | Enrico Fatighenti, Domenico Fiorenza, Carmelo Di Natale |
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| Rok vydání: | 2017 |
| Předmět: |
Pure mathematics
General Mathematics Hodge theory 010102 general mathematics Deformation theory Cone (category theory) Fano plane 01 natural sciences Cohomology Mathematics::Algebraic Geometry Hypersurface 0103 physical sciences 010307 mathematical physics Isomorphism 0101 mathematics Projective variety Mathematics |
| Zdroj: | Journal of the London Mathematical Society. 96:524-544 |
| ISSN: | 0024-6107 |
| DOI: | 10.1112/jlms.12073 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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