On exact sequences of Hodge theoretic fundamental groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915564663865344 |
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| author | Xu, Simon Shuofeng |
| author_facet | Xu, Simon Shuofeng |
| contents | The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth projective family of algebraic varieties $f\colon X\to B$. In particular, we study when this right exact sequence is exact, relate this question to some prior results in non-abelian Hodge theory, and give an obstruction to splitting in terms of étale fundamental groups. The main examples we consider in this note is the universal curve $f\colon \mathcal{C}_g\to \mathcal{M}_g$ and the moduli space of degree $1$ line bundles on the universal curves $p \colon \text{Pic}^1_{\mathcal{C}_g/\mathcal{M}_g}\to \mathcal{M}_g$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13307 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On exact sequences of Hodge theoretic fundamental groups Xu, Simon Shuofeng Algebraic Geometry 14D07, 14H10 The goal of this paper is to first define a Hodge theoretic fundamental group for smooth connected complex algebraic varieties and then prove and study a right exact sequence of Hodge theoretic fundamental groups associated to a smooth projective family of algebraic varieties $f\colon X\to B$. In particular, we study when this right exact sequence is exact, relate this question to some prior results in non-abelian Hodge theory, and give an obstruction to splitting in terms of étale fundamental groups. The main examples we consider in this note is the universal curve $f\colon \mathcal{C}_g\to \mathcal{M}_g$ and the moduli space of degree $1$ line bundles on the universal curves $p \colon \text{Pic}^1_{\mathcal{C}_g/\mathcal{M}_g}\to \mathcal{M}_g$. |
| title | On exact sequences of Hodge theoretic fundamental groups |
| topic | Algebraic Geometry 14D07, 14H10 |
| url | https://arxiv.org/abs/2503.13307 |