I'm just curious about the following. With a compact Kahler manifold, we can associate an Albanese torus. This helps us a lot study the manifold.
My question: Are there other holomorphic objects associated with a compact complex manifold? I'm interested in the objects whose shape is well understood. E.g, an Albanese torus is just a torus, and we know its cohomology, its kahler form etc.
Any help is appreciated. Thanks in advance,