紧相对Kähler纤维化中全纯双截曲率的非严格负性
Non-strict negativity of holomorphic bisectional curvature for compact relative Kähler fibrations
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中文总结 AI 辅助
本文构造反例表明，即使纤维度量严格负且Kodaira-Spencer映射处处单射，紧相对Kähler纤维化的全空间仍可能无严格负全纯双截曲率度量，并研究Monge-Ampère纤维化中广义Weil-Petersson度量的曲率性质。
中文摘要 AI 辅助
全纯双截曲率的严格负性不必从紧全纯纤维化的基和纤维传递到其全空间，即使Kodaira-Spencer映射处处单射。我们构造了一个亏格二曲线上的紧相对Kähler纤维化，其全空间为光滑射影三维簇，Kodaira-Spencer映射处处单射，诱导的纤维度量具有严格负的全纯双截曲率，而全空间却不存在具有该曲率性质的Kähler度量。这给出了To和Yeung提出的一个开放问题的否定答案。我们还从Shimura曲线上的万有四元数阿贝尔曲面构造了紧的、有效参数化的Monge-Ampère纤维化。对于这些族的乘积，广义Weil-Petersson度量具有非正的全纯双截曲率和严格负的全纯截面曲率，但其混合双截曲率消失。因此，尽管在每个点都有有效变化，严格双截负性在存在全空间度量方面以及在参数空间的自然度量方面都可能失败。
英文摘要
Strict negativity of holomorphic bisectional curvature need not pass from the base and fibers of a compact holomorphic fibration to its total space, even when the Kodaira-Spencer map is everywhere injective. We construct a compact relative Kähler fibration over a genus-two curve, with a smooth projective threefold as total space and an everywhere injective Kodaira--Spencer map, whose induced fiber metrics have strictly negative holomorphic bisectional curvature, whereas the total space admits no Kähler metric with this curvature property. This gives a negative answer to an open problem posed by To and Yeung. We also construct compact, effectively parametrized Monge-Ampère fibrations from universal quaternionic abelian surfaces over Shimura curves. For products of these families, the generalized Weil-Petersson metric has nonpositive holomorphic bisectional curvature and strictly negative holomorphic sectional curvature, but its mixed bisectional curvatures vanish. Thus strict bisectional negativity can fail both for the existence of a metric on the total space and for the natural metric on the parameter space, despite effective variation at every point.
发表机构
- Mathematical Science Research Center, Chongqing University of Technology（重庆理工大学数学科学研究中心）
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