Abstract: In his book “Basic Algebraic Geometry II”, Shafarevich asked the question whether the universal covering of a complex projective variety is holomorphically convex. This problem, now known as the Shafarevich conjecture, has been extensively studied with
the introduction of non-abelian Hodge theories by Simpson, Eyssidieux, Katzarkov, Pantev, Ramachandran, the speaker, etc. In this lecture, I will present a comprehensive proof of the reductive Shafarevich conjecture, which establishes that the universal covering of a complex
projective normal variety is holomorphically convex if their fundamental groups can be faithfully represented as Zariski-dense subgroups of complex reductive algebraic groups. If times allows, I will explain some extension of the reductive Shafarevich conjecture to the
quasi-projective cases.
The mini-course is structured as follows:
1. Non-Abelian Hodge Theories in the Archimedean Setting by Simpson: Simpson correspondence,
C*
-action Character varieties, Ubiquity theorem, etc.
2. Non-Abelian Hodge Theories in the Non-Archimedean Setting: Harmonic mapping to Euclidean buildings, Spectral coverings, Reduction theorem for representation into non-archimedean local fields, etc.
3. Construction of the Shafarevich Morphism.
4. Proof of the Reductive Shafarevich Conjecture.
References
[BDDM22] D. Brotbek, G. Daskalopoulos, Y. Deng & C. Mese. “Representations of fundamental groups and logarithmic symmetric differential forms”. HAL preprint, (2022). URLhttps://hal.archives-ouvertes.fr/hal-03839053. ↑
[CDY22] B. Cadorel, Y. Deng & K. Yamanoi. “Hyperbolicity and fundamental groups of complex quasi-projective varieties”. arXiv e-prints, (2022):arXiv:2212.12225. 2212.12225,URL http://dx.doi.org/10.48550/arXiv.2212.12225. ↑
[DYK23] Y. Deng, K. Yamanoi & L. Katzarkov. “Reductive Shafarevich Conjecture”. arXiv eprints,(2023):arXiv:2306.03070. 2306.03070, URL http://dx.doi.org/10.48550/arXiv.2306.03070. ↑
[Eys04] P. Eyssidieux. “Sur la convexité holomorphe des revêtements linéaires réductifs d’unevariété projective algébrique complexe”. Invent. Math., 156(2004)(3):503–564. URLhttp://dx.doi.org/10.1007/s00222-003-0345-0. ↑
[Sim92] C. T. Simpson. “Higgs bundles and local systems”. Inst. Hautes Études Sci. Publ. Math.,(1992)(75):5–95. URL http://www.numdam.org/item?id=PMIHES_1992__75__5_0.↑
[Sim93] ———. “Lefschetz theorems for the integral leaves of a holomorphic one-form”. Compos.Math., 87(1993)(1):99–113. ↑
lecture notes:
https://pan.baidu.com/s/1tOv-yh0vD0X5Hy25Otji7g (password: wayj)
