On Topology of Compact Hessian Manifolds

We investigate the global topological constraints and structural properties of compact Hessian manifolds. By establishing novel fibration and splitting theorems, we confirm Chern's conjecture on the vanishing of the Euler characteristic for this class of affine manifolds. Applying these techniques to low dimensions, we provide a topological classification of complete Hessian surfaces. Furthermore, utilizing the theory of Hitchin systems and the Cheng-Yau solution to the real Monge-Amp\`ere equation, we establish a geometric classification of closed orientable Hessian $3$-manifolds.