论复数 Solvmanifolds 的典范束及其在超复数几何中的应用

Pub Date : 2024-07-10 DOI:10.1007/s00031-024-09866-z
Adrián Andrada, Alejandro Tolcachier
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引用次数: 0

摘要

我们研究了具有全形琐碎典型束的复(Gamma \backslash G)溶球。我们证明,在 G 的作用下,这个束的微分截面可以是不变的,也可以是非不变的。首先,我们用与\((\mathfrak {g},J)\) 规范关联的科斯祖尔 1-form \(\psi \) 来描述不变琐化部分的存在,其中\(\mathfrak {g}\) 是 G 的李代数。此外,我们还用 \(\psi \)提供了一个代数障碍,使复溶点具有琐碎的(或更一般的全形扭转的)典范束。最后,我们展示了一个紧凑超复数 solvmanifold \((M^{4n},\{J_1,J_2,J_3\})),使得 \((M,J_{\alpha })\的典型束只有在 \(\alpha =1\)时才是琐碎的,因此 M 不是一个 \({\text {SL}}(n,\mathbb {H})\)-manifold。
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On the Canonical Bundle of Complex Solvmanifolds and Applications to Hypercomplex Geometry

We study complex solvmanifolds \(\Gamma \backslash G\) with holomorphically trivial canonical bundle. We show that the trivializing section of this bundle can be either invariant or non-invariant by the action of G. First we characterize the existence of invariant trivializing sections in terms of the Koszul 1-form \(\psi \) canonically associated to \((\mathfrak {g},J)\), where \(\mathfrak {g}\) is the Lie algebra of G, and we use this characterization to produce new examples of complex solvmanifolds with trivial canonical bundle. Moreover, we provide an algebraic obstruction, also in terms of \(\psi \), for a complex solvmanifold to have trivial (or more generally holomorphically torsion) canonical bundle. Finally, we exhibit a compact hypercomplex solvmanifold \((M^{4n},\{J_1,J_2,J_3\})\) such that the canonical bundle of \((M,J_{\alpha })\) is trivial only for \(\alpha =1\), so that M is not an \({\text {SL}}(n,\mathbb {H})\)-manifold.

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