AI 中文总结
研究正规复代数簇相关问题，通过对\(\pi_1(X)\)的幂零商群及摩根 - 海因霍奇滤层的研究，利用高阶阿尔巴内塞流形的\(q\)-凸性等，证明非球面正规簇若基本群几乎幂零则几乎两步幂零，回答相关问题。
AI 中文摘要
设\(X\)为正规复代数簇。设\(\mathcal{G}^s_{\mathbb{Z}}(X)\)为\(\pi_1(X)\)的幂零类至多为\(s\)的最大无挠幂零商群。设\(F^{\bullet}\mathfrak{g}^s\)为\(\pi_1(X)\)的复马尔采夫完备化的第\(s\)个下中心商的李代数上的摩根 - 海因霍奇滤层。我们证明，当\(k > \dim F^1\mathfrak{g}^s\)时，自然映射\(H^k(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z}) \to H^k(X, \mathbb{Z})\)消失。若\(\mathcal{G}^s_{\mathbb{Z}}(X)\)的幂零类大于二，这涵盖了\(H^{\bullet}(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z})\)的最高非零次数。由此推断，若非球面正规簇的基本群是几乎幂零的，那么它几乎是两步幂零的。这在非球面簇的情形下对阿吉拉尔和坎帕纳的一个问题给出了肯定答案。证明的要素是高阶阿尔巴内塞流形的\(q\)-凸性和高阶阿尔巴内塞映射的可定义性。
英文摘要
Let $X$ be a normal complex algebraic variety. Let $\mathcal{G}^s_{\mathbb{Z}}(X)$ be the maximal torsion free nilpotent quotient of $π_1(X)$ of nilpotency class at most $s$. Let $F^{\bullet}\mathfrak{g}^s$ be the Morgan--Hain Hodge filtration on the Lie algebra of the $s$-th lower central quotient of the complex Malcev completion of $π_1(X)$. We show that the natural map $H^k(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z}) \to H^k(X, \mathbb{Z})$ vanishes for $k > \dim F^1\mathfrak{g}^s$. If $\mathcal{G}^s_{\mathbb{Z}}(X)$ is of nilpotency class greater than two, this includes the top nonvanishing degree of $H^{\bullet}(\mathcal{G}^s_{\mathbb{Z}}(X), \mathbb{Z})$. We deduce that if the fundamental group of an aspherical normal variety is virtually nilpotent, it is virtually two-step nilpotent. This gives a positive answer to a question of Aguilar and Campana in this case of aspherical varieties. The ingredients of the proof are the $q$-convexity of higher Albanese manifolds and the definability of higher Albanese maps.
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