Ever notice?

Ever notice that every holomorphic differential on a complex surface is d-closed? For example, let f(z) dz be a holomorphic (1,0)-form. Then df^dz = gdz^dw for some g. Now by Stoke's theorem, the integral of the product of df^dz with its complex conjugate is zero. But this is the integral of |g|^2, so |g| is identically zero, meaning df^dz = 0. Magical.