Linear Chern-Hopf-Thurston conjecture
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909327484256256 |
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| author | Deng, Ya Wang, Botong |
| author_facet | Deng, Ya Wang, Botong |
| contents | If $X$ is a closed $2n$-dimensional aspherical manifold, i.e., the universal cover of $X$ is contractible, then the Chern-Hopf-Thurston conjecture predicts that $(-1)^nχ(X)\geq 0$. We prove this conjecture when $X$ is a complex projective manifold whose fundamental group admits an almost faithful linear representation over any field. In fact, we prove a much stronger statement that if $X$ is a complex projective manifold with large fundamental group and $π_1(X)$ admits an almost faithful linear representation, then $χ(X, \mathcal{P})\geq 0$ for any perverse sheaf $\mathcal{P}$ on $X$.
To prove this, we introduce a vanishing cycle functor of multivalued one-forms and apply techniques from non-abelian Hodge theory, both in archimedean and non-archimedean settings. These techniques allow us to deduce the desired positivity from the geometric properties of pure and mixed period maps. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_12012 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Chern-Hopf-Thurston conjecture Deng, Ya Wang, Botong Algebraic Geometry Differential Geometry Geometric Topology If $X$ is a closed $2n$-dimensional aspherical manifold, i.e., the universal cover of $X$ is contractible, then the Chern-Hopf-Thurston conjecture predicts that $(-1)^nχ(X)\geq 0$. We prove this conjecture when $X$ is a complex projective manifold whose fundamental group admits an almost faithful linear representation over any field. In fact, we prove a much stronger statement that if $X$ is a complex projective manifold with large fundamental group and $π_1(X)$ admits an almost faithful linear representation, then $χ(X, \mathcal{P})\geq 0$ for any perverse sheaf $\mathcal{P}$ on $X$. To prove this, we introduce a vanishing cycle functor of multivalued one-forms and apply techniques from non-abelian Hodge theory, both in archimedean and non-archimedean settings. These techniques allow us to deduce the desired positivity from the geometric properties of pure and mixed period maps. |
| title | Linear Chern-Hopf-Thurston conjecture |
| topic | Algebraic Geometry Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2405.12012 |