论文标题
Kodaira-Thurston歧管上的谐波形式
Harmonic Forms on the Kodaira-Thurston Manifold
论文作者
论文摘要
我们介绍了一种有效的方法来解决Kodaira-Thurston歧管上的$ \ bar \ partial $ harmonic形式，并具有几乎复杂的结构和一个遗传学指标。使用Weil-Brezin变换，我们将椭圆形PDE系统减少为许多线性ode系统。通过解决线性ode系统上的基本问题，查找$ \ bar \ partial $ harmonic形式的问题等同于广义高斯圆圈问题。 我们演示了两个非凡的应用。首先，kodaira-thurston歧管上几乎复杂的$ \ bar \ bar \ partial $ -hodge数字的尺寸可能是任意的。其次，霍奇数字随着遗传学指标的不同选择而变化。这回答了Hirzebruch 1954年的问题清单中Kodaira和Spencer的问题。
We introduce an effective method to solve the $\bar\partial$-harmonic forms on the Kodaira-Thurston manifold endowed with an almost complex structure and an 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 $\bar\partial$-harmonic forms is equivalent to a generalised Gauss circle problem. We demonstrate two remarkable applications. First, the dimension of the almost complex $\bar\partial$-Hodge numbers on the Kodaira-Thurston manifold could be arbitrarily large. Second, Hodge numbers vary with different choices of Hermitian metrics. This answers a question of Kodaira and Spencer in Hirzebruch's 1954 problem list.
