Generic Vanishing, 1-forms, and Topology of Albanese Maps
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866910347002118144 |
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| author | Dutta, Yajnaseni Hao, Feng Liu, Yongqiang |
| author_facet | Dutta, Yajnaseni Hao, Feng Liu, Yongqiang |
| contents | Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety $X$; one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This result is related to Kotschick's conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if $X$ has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients.
In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of $H^0(X,Ω_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_07074 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Generic Vanishing, 1-forms, and Topology of Albanese Maps Dutta, Yajnaseni Hao, Feng Liu, Yongqiang Algebraic Geometry Algebraic Topology primary 32Q55, 32S60, 14K12, 14C30 secondary 14F40 Given a bounded constructible complex of sheaves $\mathcal{F}$ on a complex Abelian variety, we prove an equality relating the cohomology jump loci of $\mathcal{F}$ and its singular support. As an application, we identify two subsets of the set of holomorphic 1-forms with zeros on a complex smooth projective irregular variety $X$; one from Green-Lazarsfeld's cohomology jump loci and one from the Kashiwara's estimates for singular supports. This result is related to Kotschick's conjecture about the equivalence between the existence of nowhere vanishing global holomorphic 1-forms and the existence of a fibre bundle structure over the circle. Our results give a conjecturally equivalent formulation using singular support, which is equivalent to a criterion involving cohomology jump loci proposed by Schreieder. As another application, we reprove a recent result proved by Schreieder and Yang; namely if $X$ has simple Albanese variety and admits a fibre bundle structure over the circle, then the Albanese morphism cohomologically behaves like a smooth morphism with respect to integer coefficients. In a related direction, we address the question whether the set of 1-forms that vanish somewhere is a finite union of linear subspaces of $H^0(X,Ω_X^1)$. We show that this is indeed the case for forms admitting zero locus of codimension 1. |
| title | Generic Vanishing, 1-forms, and Topology of Albanese Maps |
| topic | Algebraic Geometry Algebraic Topology primary 32Q55, 32S60, 14K12, 14C30 secondary 14F40 |
| url | https://arxiv.org/abs/2104.07074 |