Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914213635555328 |
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| author | Catanese, Fabrizio Zalesskii, appendix by Pavel |
| author_facet | Catanese, Fabrizio Zalesskii, appendix by Pavel |
| contents | We give a very short proof of two Theorems, whose content is outlined in the title, and where $Π_g$ is the fundamental group of a compact complex curve of genus $g$:
(1) Theorem 2.1 of the irrational pencil in the profinite version, saying that for a compact Kähler manifold an irrational pencil, that is, a fibration onto a curve of genus $g \geq 2$, corresponds to a surjection of the profinite completion $\widehatπ_1(X) \twoheadrightarrow \widehat{Π_g}$, which satisfies a maximality property;
(2) Theorem 1.4 on the characterization of varieties isogenous to a product, profinite version, giving in particular a criterion for $X$ a compact Kähler manifold to be isomorphic to a product of curves of genera at least 2: if and only if $\widehatπ_1(X) \cong \prod_1^n \widehat{Π_{g_i}}$, and some volume or cohomological condition is satisfied.
Theorem 1.4 yields a stronger result than the Main Theorem A of a recent article by 5 authors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_18798 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group Catanese, Fabrizio Zalesskii, appendix by Pavel Algebraic Geometry Algebraic Topology Complex Variables 14F45, 14G32, 14M99, 32J27, 32Q15, 32Q55, 55R05 We give a very short proof of two Theorems, whose content is outlined in the title, and where $Π_g$ is the fundamental group of a compact complex curve of genus $g$: (1) Theorem 2.1 of the irrational pencil in the profinite version, saying that for a compact Kähler manifold an irrational pencil, that is, a fibration onto a curve of genus $g \geq 2$, corresponds to a surjection of the profinite completion $\widehatπ_1(X) \twoheadrightarrow \widehat{Π_g}$, which satisfies a maximality property; (2) Theorem 1.4 on the characterization of varieties isogenous to a product, profinite version, giving in particular a criterion for $X$ a compact Kähler manifold to be isomorphic to a product of curves of genera at least 2: if and only if $\widehatπ_1(X) \cong \prod_1^n \widehat{Π_{g_i}}$, and some volume or cohomological condition is satisfied. Theorem 1.4 yields a stronger result than the Main Theorem A of a recent article by 5 authors. |
| title | Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group |
| topic | Algebraic Geometry Algebraic Topology Complex Variables 14F45, 14G32, 14M99, 32J27, 32Q15, 32Q55, 55R05 |
| url | https://arxiv.org/abs/2512.18798 |