Dolbeault cohomology is a mathematical tool used in complex geometry to study the properties of complex manifolds. It is based on the idea of differentiating complex-valued functions and forms, allowing mathematicians to analyze the structure of these manifolds through cohomological techniques. This cohomology theory is particularly useful in the context of complex differential forms, where it helps classify the types of forms that can exist on a manifold. By examining the relationships between these forms, Dolbeault cohomology provides insights into the topology and geometry of complex spaces, linking them to other areas of mathematics.