On the distribution of mixed Hodge loci
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2026
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866914412119457792 |
|---|---|
| author | Khelifa, Nazim |
| author_facet | Khelifa, Nazim |
| contents | Let $\mathbb{V}$ be an admissible and graded-polarized integral variation of mixed Hodge structures over a smooth and irreducible complex algebraic variety $S$. We show that if the typical Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ of $\mathbb{V}$ is non-empty, the full Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)$ is dense in $S$ for the Zariski topology. In an other direction, we show that if the associated graded variation $\mathrm{Gr}(\mathbb{V})$ for the weight filtration has large monodromy and level at least 3 in the sense of Baldi- Klingler-Ullmo, the typical Hodge locus of $\mathbb{V}$ is empty, and the full Hodge locus of $\mathbb{V}$ is a strict Zariski-closed subset of $S$, at least if one restricts to its factorwise positive dimensional part, improving a classical result of Brosnan-Pearlstein-Schnell in this situation. These results follow from a detailed study of the transverse part $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ of the Hodge locus of $S$ for $\mathbb{V}$, a subset which contains $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ and whose Zariski-density in $S$ is equivalent, under the Zilber-Pink conjecture for $\mathbb{V}$, to the Zariski-density of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$. We show that non-emptiness of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is equivalent to its Zariski-density in $S$, we completely classify variations whose transverse Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is Zariski-dense, and we prove an independent criterion ensuring that $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is empty. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20272 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the distribution of mixed Hodge loci Khelifa, Nazim Algebraic Geometry Let $\mathbb{V}$ be an admissible and graded-polarized integral variation of mixed Hodge structures over a smooth and irreducible complex algebraic variety $S$. We show that if the typical Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ of $\mathbb{V}$ is non-empty, the full Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)$ is dense in $S$ for the Zariski topology. In an other direction, we show that if the associated graded variation $\mathrm{Gr}(\mathbb{V})$ for the weight filtration has large monodromy and level at least 3 in the sense of Baldi- Klingler-Ullmo, the typical Hodge locus of $\mathbb{V}$ is empty, and the full Hodge locus of $\mathbb{V}$ is a strict Zariski-closed subset of $S$, at least if one restricts to its factorwise positive dimensional part, improving a classical result of Brosnan-Pearlstein-Schnell in this situation. These results follow from a detailed study of the transverse part $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ of the Hodge locus of $S$ for $\mathbb{V}$, a subset which contains $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ and whose Zariski-density in $S$ is equivalent, under the Zilber-Pink conjecture for $\mathbb{V}$, to the Zariski-density of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$. We show that non-emptiness of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is equivalent to its Zariski-density in $S$, we completely classify variations whose transverse Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is Zariski-dense, and we prove an independent criterion ensuring that $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is empty. |
| title | On the distribution of mixed Hodge loci |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2603.20272 |