A Fano manifold is a special type of complex manifold that has positive first Chern class. This property implies that Fano manifolds have ample line bundles, which means they can support a rich structure of algebraic geometry. They are important in the study of algebraic geometry and string theory due to their unique geometric properties. Fano manifolds often exhibit interesting features, such as the existence of many rational curves. These curves can be used to study the manifold's geometry and topology. Examples of Fano manifolds include projective spaces and certain Grassmannians, which serve as fundamental objects in both mathematics and theoretical physics.