Jacobians of Riemann surfaces and the Sobolev space \(H^{1/2}\) on the circle (Q1271367)

In [Osaka J. Math. 32, No. 1, 1-34 (1995; Zbl 0820.30027)] \textit{S. Nag} and \textit{D. Sullivan} constructed a universal period map \(\Pi:{\mathcal T} (\Delta) \to S(H^{1/2})\) embedding the universal Teichmüller space \({\mathcal T} (\Delta)\) holomorphically into the universal Siegel space \(S(H^{1/2}): =Sp (H^{1/2} (S^1))/U (H^{1/2} (S^1))\). Here \(H^{1/2} (S^1)\) stands for the (Sobolev) Hilbert space of all functions on the unit circle admitting a square-integrable half-order derivative. It may be thought of as the ``universal Jacobian''. In this paper the authors relate the theory of Jacobians of closed Riemann surfaces to the universal Jacobian. They obtain for an arbitrary Riemann surface \(X\) an embedding \(\eta_X\) of the Hilbert space of square-integrable harmonic 1-forms on \(X\) into \(H^{1/2} (S^1)\). Deforming \(X\) these embeddings fit together to give a holomorphic family. If \(\pi:X\to Y\) is a finite covering of Riemann surfaces, the embeddings \(\eta_X\) and \(\eta_Y\) are related by a commutative diagram, which allows to proceed to inductive limit constructions.