This paper gives a geometric interpretation of the Beilinson regulator $c_{2,1} : K_1(X) \rightarrow H^3(X,{\mathbb {Z}}(2)_D)$ for a complex projective algebraic manifold $X$. This interpretation rests on a theory of holomorphic gerbes and of their differentiable stuctures, in the spirit of a recent book of the author. We give a direct geometric construction of a holomorphic gerbe associated to an element of $K_1(X)$. We also present an $l$-adic analog of the construction, the consistency of which depends on an étale covering of group $\mu ^{\otimes 2}_m$ of the Fermat curve with affine equation $x^m + y^m = 1$, which gives a ramified Galois covering of the projective line.