某些阿贝尔四维簇的德拉姆 - 贝蒂群
De Rham-Betti Groups of Type IV Abelian Fourfolds
AI总结：
研究\(\bar{\mathbb{Q}}\)上几类阿贝尔簇的德拉姆 - 贝蒂群，通过采用不同于常规的方法，利用相关结果、分析及约束，确定了简单CM阿贝尔四维簇等的\(\mathrm{G}_{\mathrm{dRB}}(A)=\mathrm{MT}(A)\) 。
AI中文摘要：
我们确定了\(\bar{\mathbb{Q}}\)上几类阿贝尔簇的德拉姆 - 贝蒂（dRB）群。证明了对于简单CM阿贝尔四维簇和具有四次CM自同态域的阿贝尔四维簇，\(\mathrm{G}_{\mathrm{dRB}}(A)=\mathrm{MT}(A)\)。因当前对dRB结构了解有限，采用不同于穆嫩 - 扎尔欣确定芒福德 - 泰特群的方法。利用格罗斯和丘德诺夫斯基的两个结果、伽罗瓦理论分析及极化产生的正性约束排除芒福德 - 泰特群的某些约化子群作为dRB群的候选者。本文基于作者博士论文第二部分。
英文摘要：
We determine the de Rham--Betti (dRB) groups of several classes of abelian varieties over $\overline{\mathbb{Q}}$. We prove that $G_{\mathrm{dRB}}(A)=\mathrm{MT}(A)$ for every simple abelian fourfold of type IV. At present, much of what is known about the dRB structures of abelian varieties derives from Wüstholz's Analytic Subgroup Theorem, whose applications primarily control linear relations among periods of $\mathrm{H}^{1}(A)$ and divisor-class information. In comparison with the theory of Hodge structures, our understanding of dRB structures remains limited. We therefore adopt an approach different from the method of Moonen-Zarhin for determining the Mumford-Tate groups of these abelian varieties. Depending on the endomorphism type of the abelian fourfold, we use Galois-theoretic analysis, van Geemen's half-twist construction, and positivity constraints arising from polarizations, as appropriate, to exclude proper reductive subgroups of the corresponding Mumford-Tate groups as candidates for the dRB groups. We also use results on periods due to Gross and Chudnovsky. This article is an expansion of the second part of the author's PhD thesis https://pure.uva.nl/ws/files/311471255/Thesis.pdf; see also https://arxiv.org/abs/2511.01072 by the author.
