Harmonic forms on the Kodaira-Thurston manifold
Harmonic forms on the Kodaira-Thurston manifold
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DOI：
10.1016/j.aim.2022.108277
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发表时间：
2020-01
影响因子：
1.7
通讯作者：
Tom Holt;Weiyi Zhang
中科院分区：
文献类型：
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作者：
Tom Holt;Weiyi Zhang
给出了一种确定Kodaira-Thurston流形上的调和形式的有效方法，该流形具有几乎复结构和几乎Hermitian度量.利用Weil-Brezin变换，我们将椭圆型偏微分方程组化为可数个线性常微分方程组。通过求解线性常微分方程组上的一个基本问题，求λ ′-调和形式的问题等价于广义高斯圆问题。我们展示了两个显著的应用。首先，Kodaira-Thurston流形上的几乎复Hodge数的维数可以任意大。其次，霍奇数随几乎厄米度量的不同选择而变化。这回答了科代拉和斯宾塞在希泽布鲁赫1954年问题清单中提出的一个问题。
We introduce an effective method to determine the∂¯-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an almost Hermitian metric. Using the Weil-Brezin transform, we reduce the elliptic PDE system to countably many linear ODE systems. By solving a fundamental problem on linear ODE systems, the problem of finding∂¯-harmonic forms is equivalent to a generalised Gauss circle problem. We demonstrate two remarkable applications. First, the dimension of the almost complex∂¯-Hodge numbers on the Kodaira-Thurston manifold could be arbitrarily large. Second, Hodge numbers vary with different choices of almost Hermitian metrics. This answers a question of Kodaira and Spencer in Hirzebruch's 1954 problem list.