Degeneration of families of projective hypersurfaces and Hodge conjecture
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929529717522432 |
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| author | Bouali, Johann |
| author_facet | Bouali, Johann |
| contents | We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth projective variety $E_0$ which is a component of a normal crossing divisor $E=\cup_{i=0}^rE_i\subset Y$ which is the singular fiber of a pencil $f:Y\to\mathbb A^1$ of smooth projective hypersurfaces. Using the smooth hypersurface case by a previous result of the autor, the nearby cycle functor on mixed Hodge module with rational de Rham factor, and the induction hypothesis, we prove that an Hodge class of $E_0$ is absolute Hodge, more precisely the locus of Hodge classes inside the algebraic vector bundle given the De Rham cohomology the rational deformation of $E_0$ is defined over $\mathbb Q$. By another previous result of the autor, we get the Hodge conjecture for $E_0$. By the induction hypothesis we also have the Hodge conjecture for $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Degeneration of families of projective hypersurfaces and Hodge conjecture Bouali, Johann Algebraic Geometry We prove by induction on dimension the Hodge conjecture for smooth complex projective varieties. Let $X$ be a smooth complex projective variety. Then $X$ is birational to a possibly singular projective hypersurface, hence to a smooth projective variety $E_0$ which is a component of a normal crossing divisor $E=\cup_{i=0}^rE_i\subset Y$ which is the singular fiber of a pencil $f:Y\to\mathbb A^1$ of smooth projective hypersurfaces. Using the smooth hypersurface case by a previous result of the autor, the nearby cycle functor on mixed Hodge module with rational de Rham factor, and the induction hypothesis, we prove that an Hodge class of $E_0$ is absolute Hodge, more precisely the locus of Hodge classes inside the algebraic vector bundle given the De Rham cohomology the rational deformation of $E_0$ is defined over $\mathbb Q$. By another previous result of the autor, we get the Hodge conjecture for $E_0$. By the induction hypothesis we also have the Hodge conjecture for $X$. |
| title | Degeneration of families of projective hypersurfaces and Hodge conjecture |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2401.03465 |