Harmonic Forms on the Kodaira-Thurston Manifold

We introduce an effective method to solve the $\bar{\mathrm{\partial }}$-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 $\bar{\mathrm{\partial }}$-harmonic forms is equivalent to a generalised Gauss circle problem. We demonstrate two remarkable applications. First, the dimension of the almost complex $\bar{\mathrm{\partial }}$-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.