Complex Geometry/Complex manifolds

complex manifold A complex manifold is a manifold of type the category of open subsets of some with holomorphic maps over [[the site ]], where is a topological space. holomorphic functions on compact connected complex T1 manifolds are constant Let be a compact connected complex manifold and let be holomorphic. Then is constant. [[Since is compact and is continuous, attains its maximum on ]]. Let be the point where it does so. Suppose that is a chart st. . Then is holomorphic and hence constant by the maximum principle. Thus, we have shown that the nonempty set is open. Since is continuous and is , it is also closed. Since is connected, it therefore equals .