In the study of holomorphic maps, the term "rigidity" refers to certain types of results that give us very specific information about a general class of holomorphic maps owing to the geometry of their domains or target spaces. Under this theme, we begin by studying when, given two compact connected complex manifolds X and Y, a degree-one holomorphic map f : Y → X is a biholomorphism. Given that the real manifolds underlying X and Y are diffeomorphic, we provide a condition under which f is a biholomorphism. Using this result, we deduce a rigidity result for holomorphic self-maps of the total space of a holomorphic fiber space. Lastly, we consider products X = X1 × X2 and Y = Y1 × Y2 of compact connected complex manifolds. When X1 is a Riemann surface of genus ≥ 2, we show that any non-constant holomorphic map F : Y → X is of a special form.
On holomorphic maps with only fold singularities
